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Library of Congress CataloginginPublication Data Catalog record is available from the Library of Congress. This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. Apart from any fair dealing for the purpose of research or private study, or criticism or review, as permitted under the UK Copyright Designs and Patents Act, 1988, this publication may not be reproduced, stored or transmitted, in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage or retrieval system, without the prior permission in writing of the publishers, or in the case of reprographic reproduction only in accordance with the terms of the licenses issued by the Copyright Licensing Agency in the UK, or in accordance with the terms of the license issued by the appropriate Reproduction Rights Organization outside the UK. The consent of CRC Press LLC does not extend to copying for general distribution, for promotion, for creating new works, or for resale. Specific permission must be obtained in writing from CRC Press LLC for such copying. Direct all inquiries to CRC Press LLC, 2000 N.W. Corporate Blvd., Boca Raton, Florida 33431. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation, without intent to infringe. Visit the CRC Press Web site at www.crcpress.com 1997 by Chapman & Hall/CRC First edition 1997 First CRC Press reprint 1999 Originally published by Chapman & Hall No claim to original U.S. Government works International Standard Book Number 0412040611 Printed in the United States of America 3 4 5 6 7 8 9 0 Printed on acidfree paper
Contents Preface 1 Introduction 1.1 Purpose 1.2 Background 1.2.1 The EM algorithm 1.2.2 Markov chain Monte Carlo 1.3 Why analysis by simulation? 1.4 Looking ahead 1.4.1 Scope of the rest of this book 1.4.2 Knowledge assumed on the part of the reader 1.4.3 Software and computational details 1.5 Bibliographic notes 2 Assumptions 2.1 The completedata model 2.2 Ignorability 2.2.1 Missing at random 2.2.2 Distinctness of parameters 2.3 The observeddata likelihood and posterior 2.3.1 Observeddata likelihood 2.3.2 Examples 2.3.3 Observeddata posterior 2.4 Examining the ignorability assumption 2.4.1 Examples where ignorability is known to hold 2.4.2 Examples where ignorability is not known to hold 2.4.3 Ignorability is relative 2.5 General ignorable procedures 2.5.1 A simulated example 2.5.2 Departures from ignorability 2.5.3 Notes on nonignorable alternatives
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2.6 The role of the completedata model 2.6.1 Departures from the data model 2.6.2 Inference treating certain variables as fixed 3 EM and data augmentation 3.1 Introduction 3.2 The EM algorithm 3.2.1 Definition 3.2.2 Examples 3.2.3 EM for posterior modes 3.2.4 Restrictions on the parameter space 3.2.5 The ECM algorithm 3.3 Properties of EM 3.3.1 Stationary values 3.3.2 Rate of convergence 3.3.3 Example 3.3.4 Further comments on convergence 3.4 Markov chain Monte Carlo 3.4.1 Gibbs sampling 3.4.2 Data augmentation 3.4.3 Examples of data augmentation 3.4.4 The MetropolisHastings algorithm 3.4.5 Generalizations and hybrid algorithms 3.5 Properties of Markov chain Monte Carlo 3.5.1 The meaning of convergence 3.5.2 Examples of nonconvergence 3.5.3 Rates of convergence 4 Inference by data augmentation 4.1 Introduction 4.2 Parameter simulation 4.2.1 Dependent samples 4.2.2 Summarizing a dependent sample 4.2.3 RaoBlackwellized estimates 4.3 Multiple imputation 4.3.1 Bayesianly proper multiple imputations 4.3.2 Inference for a scalar quantity 4.3.3 Inference for multidimensional estimands 4.4 Assessing convergence 4.4.1 Monitoring convergence in a single chain
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4.4.2 Monitoring convergence with parallel chains 4.4.3 Choosing scalar functions of the parameter 4.4.4 Convergence of posterior summaries 4.5 Practical guidelines 4.5.1 Choosing a method of inference 4.5.2 Implementing a parametersimulation experiment 4.5.3 Generating multiple imputations 4.5.4 Choosing an imputation model 4.5.5 Further comments on imputation modeling 5 Methods for normal data 5.1 Introduction 5.2 Relevant properties of the completedata model 5.2.1 Basic notation 5.2.2 Bayesian inference under a conjugate prior 5.2.3 Choosing the prior hyperparameters 5.2.4 Alternative parameterizations and sweep 5.3 The EM algorithm 5.3.1 Preliminary manipulations 5.3.2 The Estep 5.3.3 Implementation of the algorithm 5.3.4 EM for posterior modes 5.3.5 Calculating the observeddata loglikelihood 5.3.6 Example: serumcholesterol levels of heart attack patients 5.3.7 Example: changes in heart rate due to marijuana use 5.4 Data augmentation 5.4.1 The Istep 5.4.2 The Pstep 5.4.3 Example: cholesterol levels of heartAttack patients 5.4.4 Example: changes in heart rate due to marijuana use 6 More on the normal model 6.1 Introduction 6.2 Multiple imputation: example 6.2.1 Cholesterol levels of heartattack patients 6.2.2 Generating the imputations 6.2.3 Completedata point and variance estimates 6.2.4 Combining the estimates 6.2.5 Alternative choices for the number of imputations
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6.3 Multiple imputation: example 2 6.3.1 Predicting achievement in foreign language study 6.3.2 Applying the normal model 6.3.3 Exploring the observeddata likelihood and posterior 6.3.4 Overcoming the problem of inestimability 6.3.5 Analysis by multiple imputation 6.4 A simulation study 6.4.1 Simulation procedures 6.4.2 Completedata inferences 6.4.3 Results 6.5 Fast algorithms based on factored likelihoods 6.5.1 Monotone missingness patterns 6.5.2 Computing alternative parameterizations 6.5.3 Noniterative inference for monotone data 6.5.4 Monotone data augmentation 6.5.5 Implementation of the algorithm 6.5.6 Uses and extensions 6.5.7 Example 7 Methods for categorical data 7.1 Introduction 7.2 The multinomial model and Dirichlet prior 7.2.1 The multinomial distribution 7.2.2 Collapsing and partitioning the multinomial 7.2.3 The Dirichlet distribution 7.2.4 Bayesian inference 7.2.5 Choosing the prior hyperparameters 7.2.6 Collapsing and partitioning the Dirichlet 7.3 Basic algorithms for the saturated model 7.3.1 Characterizing an incomplete categorical dataset 7.3.2 The EM algorithm 7.3.3 Data augmentation 7.3.4 Example: victimization status from the National Crime Survey 7.3.5 Example: Protective Services Project for Older Persons 7.4 Fast algorithms for nearmonotone patterns 7.4.1 Factoring the likelihood and prior density 7.4.2 Monotone data augmentation 7.4.3 Example: driver injury and seatbelt use
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8 Loglinear models 8.1 Introduction 8.2 Overview of loglinear models 8.2.1 Definition 8.2.2 Eliminating associations 8.2.3 Sufficient statistics 8.2.4 Model interpretation 8.3 Likelihoodbased inference with complete data 8.3.1 Maximumlikelihood estimation 8.3.2 Iterative proportional fitting 8.3.3 Hypothesis testing and goodness of fit 8.3.4 Example: misclassification of seatbelt use and injury 8.4 Bayesian inference with complete data 8.4.1 Prior distributions for loglinear models 8.4.2 Inference using posterior modes 8.4.3 Inference by Bayesian IPF 8.4.4 Why Bayesian IPF works 8.4.5 Example: misclassification of seatbelt use and injury 8.5 Loglinear modeling with incomplete data 8.5.1 ML estimates and posterior modes 8.5.2 Goodnessoffit statistics 8.5.3 Data augmentation and Bayesian IPF 8.6 Examples 8.6.1 Protective Services Project for Older Persons 8.6.2 Driver injury and seatbelt use 9 Methods for mixed data 9.1 Introduction 9.2 The general location model 9.2.1 Definition 9.2.2 Completedata likelihood 9.2.3 Example 9.2.4 Completedata Bayesian inference 9.3 Restricted models 9.3.1 Reducing the number of parameters 9.3.2 Likelihood inference for restricted models 9.3.3 Bayesian inference 9.4 Algorithms for incomplete mixed data 9.4.1 Predictive distributions
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9.4.2 EM for the unrestricted model 9.4.3 Data augmentation 9.4.4 Algorithms for restricted models 9.5 Data examples 9.5.1 St. Louis Risk Research Project 9.5.2 Foreign Language Attitude Scale 9.5.3 National Health and Nutrition Examination Survey 10 Further topics 10.1 Introduction 10.2 Extensions of the normal model 10.2.1 Restricted covariance structures 10.2.2 Heavytailed distributions 10.2.3 Interactions 10.2.4 Semicontinuous variables 10.3 Randomeffects models 10.4 Models for complex survey data 10.5 Nonignorable methods 10.6 Mixture models and latent variables 10.7 Coarsened data and outlier models 10.8 Diagnostics Appendices A Data examples B Storage of categorical data C Software References
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Preface The last quarter of a century has seen enormous developments in general statistical methods for incomplete data. The EM algorithm and its extensions, multiple imputation and Markov chain Monte Carlo provide a set of flexible and reliable tools for inference in large classes of missingdata problems. Yet, in practical terms, these developments have had surprisingly little impact on the way most data analysts handle missing values on a routine basis. My hope is that this book will help to bridge the gap between theory and practice, making a multipurpose kit of missingdata tools accessible to anyone who may need them. This book is intended for applied statisticians, graduate students and methodologicallyoriented researchers in search of practical tools to handle missing data. The focus is applied rather than theoretical, but technical details have been included where necessary to help readers thoroughly understand the statistical properties of these methods and the behavior of the accompanying algorithms. The methods presented here rely on three fully parametric models for multivariate data: the unrestricted multivariate normal distribution, loglinear models for crossclassified categorical data and the general location model for mixed continuous and categorical variables. In addition, the missing data are assumed to be missing at random, in the sense defined by Rubin (1976). My reviewers have correctly pointed out that many other vitally important topics could (and perhaps should) have been addressed: nonnormal models such as the contaminated normal and multivariatet; repeated measures and restricted covariance structures; censored and coarsened data; models for nonignorable nonresponse; latent variables; and hierarchical or randomeffects models. Imputation for
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complex surveys and censuses, a topic in which I am deeply interested, deserves much more attention than it received. For better or worse, I decided to limit the material to a few important subjects, but to treat these subjects thoroughly and illustrate them with nontrivial data examples. This book would not have been possible without the generous support and encouragement of many friends, colleagues and agencies. Don Rubin, whose countless contributions to the area of missing data provided a springboard for this work, was the first to suggest publishing it as a book. The initial round of software development was sponsored by Frank Sulloway, whose wonderfully incomplete dataset provided the first and most colorful application of these methods. Additional support was provided by the Bureau of the Census, the United States Department of Agriculture and the National Center for Health Statistics. Many helpful comments and suggestions were given by John Barnard, Rose Brunner, Andrew Gelman, Bonnie Ghosh, XiaoLi Meng, Susan Murphy, Maren Olsen, Ritz Scheuren, Stef van Buuren, Recai Yucel and Alan Zaslavsky, and the editorial and production staff at Chapman & Hall. Data on the Foreign Language Attitude Scale were contributed by Mark Raymond. My parents, Chester and Dolores Schafer, created a loving and stable childhood environment, and my wife Sharon did not fail to encourage and inspire. Prayer support was provided by Dr. Samuel C. Lee and members of University Bible Fellowship. Finally, I must acknowledge my debt to the late Clifford C. Clogg, to whom this book is dedicated. Cliff’s steady encouragement and careful review greatly improved the quality of the book, especially the first five chapters. His warmth, love for learning, hard work and faith continue to inspire the many who were close to him. Personally and professionally, it is most gratifying to know that Cliff regarded this book as ’good stuff’. Joseph L. Schafer University Park, Pennsylvania October 1996
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CHAPTER 1
Introduction
1.1 Purpose This book presents methods of statistical inference from multivariate datasets with missing values where missingness may occur on any or all of the variables. Such datasets arise frequently in statistical practice, but the tools for effectively dealing with them are not readily available to data analysts. It is our goal to provide these tools, along with the knowledge of how to use them. When faced with missing values, practitioners frequently resort to ad hoc methods of case deletion or imputation to force the incomplete dataset into a rectangular completedata format. Many statistical software packages, for example, automatically omit from a linear regression analysis any case t hat has a missing value for any variable. Imputation is a generic term for filling in missing data with plausible values. In a multivariate dataset, each missing value may be replaced by the observed mean for that variable, or, in a slightly less naive approach, by some sort of predicted value from a regression model. Almost invariably, after the dataset has been altered by one of these methods no additional provision for missing data is made in the subsequent analysis. The research usually proceeds as if the omitted cases had never really been observed, or as if the imputed values were real data. When the incomplete cases comprise only a small fraction of all cases (say, five percent or less) then case deletion may be a perfectly reasonable solution to the missingdata problem. In multivariate settings where missing values occur on more than one variable, however, the incomplete cases are often a substantial portion of the entire dataset. If so, deleting them
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may be inefficient, causing large amounts of information to be discarded. Moreover, omitting them from the analysis will tend to introduce bias, to the extent that the incompletely observed cases differ systematically from the completely observed ones. The completely observed cases that remain will be unrepresentative of the population for which the inference is usually intended: the population of all cases, rather than the population of cases with no missing data. Ad hoc methods of imputation are no less problematic. Imputing averages on a variablebyvariable basis preserves the observed sample means, but it distorts the covariance structure, biasing estimated variances and covariances toward zero. Imputing predicted values from regression models, on the other hand, tends to inflate observed correlations, biasing them away from zero. When the pattern of missingness is complex, devising an ad hoc imputation scheme that preserves important aspects of the joint distribution of the variables can be a daunting task. Moreover, even if the joint distribution of all variables could be adequately preserved, it may be a serious mistake to treat the imputed data as if they were real. Standard errors, pvalues and other measures of uncertainty calculated by standard completedata methods could be misleading, because they fail to reflect any uncertainty due to missing data. This book presents a unified approach to the analysis of incomplete multivariate data. We will consider datasets for which the variables are continuous, categorical, or both. This approach allows one to analyze the data by virtually any technique that would be appropriate if the data were complete. This is accomplished not by simply modifying the data in an ad hoc fashion to make them appear complete, but by principled methods that account for the missing values, and the uncertainty they introduce, at each step of the analysis in a formal way. These methods tend to be computationally intensive, requiring more computer time than ad hoc alternatives. However, they do not require a heavy investment of analyst time, and can be applied to a wide variety of problems more or less routinely without special efforts to develop new technology unique to each problem. This book is written from an applied perspective, attempting to bring
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together theory, computational methods, data examples and practical advice in a single source.
1.2 Background The methods presented here have their origins in two distinct bodies of statistical literature. The first concerns likelihoodbased inference with incomplete data and, in particular, the EM algorithm. The second concerns techniques of Markov chain Monte Carlo: Gibbs sampling, data augmentation, the MetropolisHastings algorithm, and related methods. 1.2.1 The EM algorithm The EM algorithm is a general technique for finding maximum likelihood estimates for parametric models when the data are not fully observed. Although special cases of EM appear far back in the statistical literature, it was not until Dempster, Laird and Rubin (1977) coined the term EM and established its fundamental properties that the generality and usefulness of this algorithm were realized. EM spawned a revolution in the analysis of incomplete data, making it possible to compute efficient parameter estimates, and thus obviating the need for ad hoc methods like case deletion, in wide classes of statistical problems. The influence of EM has been far reaching, not merely as a computational technique, but as a paradigm for approaching difficult statistical problems. There are many statistical problems which, at first glance, may not appear to involve missing data, but which can be reformulated as missingdata problems: mixture models, hierarchical or random effects models, experiments with unbalanced data and many more. In the last fifteen years, a surprisingly large number of applications for EM have been found in a wide variety of fields. Unfortunately, major producers of statistical software have been rather slow to incorporate generalpurpose EM algorithms for incomplete data into their products. One notable exception is BMDP, which has EM algorithms for the multivariate normal model and for unbalanced repeated
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measures with structured covariance matrices (BMDP Statistical Software, Inc., 1992). 1.2.2 Markov chain Monte Carlo Markov chain Monte Carlo is a body of methods for generating pseudorandom draws from probability distributions via Markov chains. A Markov chain is a sequence of random variables in which the distribution of each element depends on the value of the previous one. As we proceed along the sequence, provided that certain regularity conditions axe met, the distributions of the elements stabilize to a common distribution known as the stationary distribution. In Markov chain Monte Carlo, one constructs a Markov chain whose stationary distribution is a distribution of interest. By repeatedly simulating steps of the chain, one is able eventually to simulate draws from the distribution of interest. The two most popular methods of Markov chain Monte Carlo are Gibbs sampling and the MetropolisHastings algorithm. In Gibbs sampling (Geman and Geman, 1984; Gelfand and Smith, 1990), one draws from the conditional distribution of each component of a multivariate random variable given the other components in a cyclic fashion. In MetropolisHastings (Metropolis et al., 1953; Hastings, 1970), one draws from a probability distribution intended to approximate the distribution actually of interest, and then accepts or rejects the drawn value with a specified probability. Many variations of these are possible, for example, hybrid algorithms that perform steps of MetropolisHastings within iterations of Gibbs. These methods are related to more traditional Monte Carlo methods such as importance sampling (e.g. Kleijnan, 1974) and rejection sampling (e.g. Kennedy and Gentle, 1980). As with EM, specific applications of Markov chain Monte Carlo have been in use for many years, notably in areas of statistical mechanics and image reconstruction. In the past decade, however, many new uses for these methods have been discovered and implemented that are of special interest to statisticians. In particular, Markov chain Monte Carlo has
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spawned a revolution of its own in the area of applied Bayesian inference. In Bayesian inference, information about unknown parameters is expressed in the form of posterior probability distribution. Even with relatively simple probability models, the posterior distribution is often intractable: important summaries such as moments, marginal densities and quantiles are not readily available in closed form. Practitioners have typically resorted to asymptotic approximation, numerical integration and importance sampling to elicit meaningful summaries of intractable posteriors. Through Markov chain Monte Carlo, however, it is now possible in many cases to simulate the entire joint posterior distribution of the unknown quantities, and thereby obtain simulationbased estimates of virtually any features of the posterior that are of interest.
1.3 Why analysis by simulation? Simulation of posterior distributions enjoys many advantages over more traditional methods of parametric inference. Some of these axe listed below. 1. In complex problems it may be easier to implement than other methods, both conceptually and computationally. 2. It may be the only method currently feasible when the unknown parameter is of high dimension. 3. It does not rely on asymptotic approximations. The algorithms converge scholastically to posterior distributions that are exact, regardless of sample size. In an era when computing environments are becoming increasingly powerful and less expensive, simulation promises to be one of the mainstays of applied parametric modeling and data analysis in the years ahead. Simulation is especially attractive at the present time as a general approach to the analysis of incomplete multivariate data. There are at least two major reasons for this. First,
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simulation by Markov chain Monte Carlo is a natural companion and complement to the current tools for handling missing data, and, in particular, the EM algorithm. Markov chain Monte Carlo can be applied to precisely the same types of problems as EM, and, computationally speaking, its implementation is often remarkably similar to that of EM. Whereas EM provides only point estimates of the unknown parameters, however, Markov chain Monte Carlo provides random draws from their joint posterior distribution. A point estimate, even if it is efficient, is not especially useful unless there is also some measure of uncertainty associated with it. With Markov chain Monte Carlo, Bayesian analogues of the standard tools of frequentist inference (standard errors, confidence intervals and pvalues) are now readily simulated, providing these measures of uncertainty. A second reason why simulation is a natural choice for missingdata problems is that it facilitates inference by multiple imputation. Multiple imputation (Rubin, 1987) is a technique in which each missing value is replaced by m > 1 simulated values. The m sets of imputations reflect uncertainty about the true values of the missing data. After the multiple imputations axe created, m, plausible versions of the complete data exist, each of which are analyzed by standard completedata methods. The results of the m analyses are then combined to produce a single inferential statement (e.g. a confidence interval or a pvalue) that includes uncertainty due to missing data. Until now, the task of generating multiple imputations has been problematic except in some simple cases, such as univariate examples and datasets with only one variable subject to nonresponse. No straightforward, generalpurpose algorithms have been available for generating proper multiple imputations in a multivariate setting. Using techniques of Markov chain Monte Carlo, however, it is now possible to do this quite easily. Like other methods of inference, simulation based on Markov chains has certain disadvantages. Carrying out a simulation for a large dataset or a complicated model may require access to a fast computer with substantial memory. Monitoring the convergence of Markov chain Monte Carlo
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algorithms can be difficult. Moreover, the use of the Bayesian paradigm and the introduction of prior distributions for unknown parameters, even if the impact on conclusions is minimal, may be regarded by some as artificial or undesirably subjective. In the chapters ahead, we will try to address these issues carefully and thoughtfully as they arise.
1.4 Looking ahead This book presents iterative algorithms for simulating multiple imputations of missing values in incomplete datasets under some important classes of multivariate models. The same algorithms may also be used to draw values of parameters from their posterior distributions. The algorithms are described in detail, focusing on practical issues of computation. The computational efficiency and low datastorage requirements of the algorithms make them suitable even for datasets that are quite large, and they have been applied routinely to datasets with over 10 000 observations and 30 variables. The use of these algorithms is demonstrated on a variety of real data examples, with accompanying discussion on issues of practical importance to data analysts. Because of their good performance, we believe that these algorithms will find widespread use in a variety of applications. We expect that they will become standard supplements to the current tools of missingdata analysis. Perhaps the most important aspect of this work is that now, for the first time, multiple imputation and parameter simulation are made available to nonspecialists who know the importance of adjusting for missing data in their inference, but who lack the resources or special expertise needed to develop and implement these techniques on their own. 1.4.1 Scope of the rest of this book Chapter 2 presents the key assumptions that will be made throughout this book, the parametric data model and the assumption of ignorable missingness, and discusses their relevance in various applied settings. Chapter 3 presents
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necessary background material on EM and Markov chain Monte Carlo. Chapter 4 discusses in practical terms the various methods of conducting inference by simulation. The remaining chapters describe algorithms for specific multivariate models and illustrate their use in a variety of examples. Chapters 56 discuss methods for the multivariate normal distribution; Chapters 78, models for crossclassified categorical data; and Chapter 9, multivariate models for datasets with both continuous and categorical variables. Chapters 3 and 4 serve as a reference for the subsequent chapters. Readers interested primarily in applications may find it helpful to initially skim through Chapters 3 and 4 and then return to them as necessary while working through Chapters 59. 1.4.2 Knowledge assumed on the part of the reader We assume that the reader is familiar with basic concepts of probability theory, inference based on the likelihood function, and multivariate distributions, especially the multivariate normal and the multinomial. Matrix notation will be used throughout. We assume that the reader is also comfortable with the basic concepts of Bayesian inference, although not necessarily having experience with applying Bayesian techniques in real examples. Some knowledge of standard categoricaldata techniques, especially loglinear models, is also helpful but not absolutely necessary. 1.4.3 Software and computational details The algorithms described in this book have been implemented by the author for general use as functions in the statistical language S (Becker, Chambers, and Wilks, 1988), using subroutines written in Fortran77. The programs are available to anyone free of charge, and information on them is provided in Appendix C. As you read this book, especially the later chapters, you may be surprised at the unusual amount of attention devoted to computational issues. Enough detail has been provided to enable a dedicated reader to reinvent the crucial portions of
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computer programs, if he or she chooses to do so. These details were provided for the following reasons: 1. to encourage others to implement the algorithms in other computer languages or software packages, if they are better served by these environments; 2. to encourage others to improve upon these algorithms, if they discover ways to make them more efficient; and 3. to foster development of similar algorithms for more general classes of models, perhaps using these routines as building blocks for larger and more complex programs.
1.5 Bibliographic notes A general overview of techniques for missing data, with discussion of various ad hoc approaches as well as the EM algorithm, is given by Little and Rubin (1987). The original article on EM by Dempster, Laird, and Rubin (1977) with discussion, now almost twenty years old, still provides a helpful introduction to EM; its simple examples anticipate many of the major types of EM algorithms in use today. For a comprehensive bibliographic review of EM, see Meng and Pedlow (1992). Excellent overviews of Markov chain Monte Carlo methods, including data augmentation, Gibbs, sampling, and the MetropolisHastings algorithm, appear in books by Tanner (1993) and Gilks, Richardson, and Spiegelhalter (1996). Tanner's book also contains an entire chapter on EM. A classic introduction to Bayesian inference is given by Box and Tiao (1992), and Gelman et al. (1995) discuss practical Bayesian data analysis from a modern perspective. Good reference material on crossclassified categorical data and loglinear models is given by Bishop, Fienberg and Holland (1975) and Agresti (1990).
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CHAPTER 2
Assumptions
2.1 The completedata model We will consider rectangular datasets whose rows can be modeled as independent, identically distributed (iid) draws from some multivariate probability distribution. A schematic representation of such a dataset is shown in Figure 2.1. The n rows represent observational units and the p columns represent variables recorded for those units. Missing values, denoted by question marks, may occur in any pattern. Let Y denote the n × p matrix of complete data, which is not fully observed, and let yi denote the ith row of Y, i = 1.
Figure 2.1. Multivariate dataset with missing values.
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By the iid assumption, the probability density or probability function of the complete data may be written n
P( Y θ ) =
∏ f ( yi θ ),
(2.1)
i =1
where f is the density or probability function for a single row, and θ is a vector of unknown parameters. We will consider three classes of distributions f: 1. the multivariate normal distribution; 2. the multinomial model for crossclassified categorical data, including loglinear models; and 3. a class of models for mixed normal and categorical data (Krzanowski, 1980, 1982; Little and Schluchter, 1985). On occasion, the two crucial modeling assumptions above, that the rows are iid, and that the we have correctly specified (up to the unknown θ ) the full joint distribution of all p variables, will not be needed in their entirety and may be partially relaxed. We will sometimes be able to accommodate situations like regression, in which we seek to model the conditional distribution of one or more response variables given some predictor variables without specifying any probability model for the predictors. A discussion of this point will be given in Section 2.6. For now, we turn our attention to the mechanism of missingness.
2.2 Ignorability 2.2.1 Missing at random Denote the observed part of Y by Yobs, and the missing part by Ymis, so that Y = (Yobs,Ymis). Throughout this book, we will assume that the missing data are missing at random (MAR) in the sense defined by Rubin (1976).
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A precise definition for MAR will be given momentarily, but first we describe it in an informal way: the probability that an observation is missing may depend on Yobs, but not on Y mis, Another useful heuristic definition of MAR is the following. Let U and V be any two variables or nonoverlapping groups of variables. Suppose that we restrict attention to units for which U is observed and equal to a specific value, say u. MAR means that among these units, the distribution of V is, apart from ordinary sampling variability, the same among the cases for which V is observed as it is among the cases for which V is missing. Despite its name, then, MAR does not suggest that the missing data values are a simple random sample of all data values. The latter condition is known as missing completely at random (MCAR). MCAR is only a special case of MAR. MAR is less restrictive than MCAR because it requires only that the missing values behave like a random sample of all values within subclasses defined by observed data. In other words, MAR allows the probability that a datum is missing to depend on the datum itself, but only indirectly through quantities that are observed. More formally, Rubin (1976) defines MAR in terms of a probability model for the missingness. Let R be an n × p matrix of indicator variables whose elements are zero or one depending on whether the corresponding elements of Y are missing or observed. We would not in general expect the distribution of R to be unrelated to Y, so we posit a probability model for R, P (RY, ξ), which depends on Y as well as some unknown parameters ξ . The MAR assumption is that this distribution does not depend on Ymis, P(RYobs, Ymis, ξ) = P(RYobs, ξ)
(2.2)
2.2.2 Distinctness of parameters To proceed further, we also need to assume that the parameters θ of the data model and the parameters ξ of the missingness mechanism are distinct. From a frequentist
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perspective, this means that the joint parameter space of (θ,ξ) must be the Cartesian crossproduct of the individual parameter spaces for θ and ξ . From a Bayesian perspective, this means that any joint prior distribution applied to (θ,ξ) must factor into independent marginal priors for θ and ξ . In many situations this is intuitively reasonable, as knowing θ will provide little information about ξ and viceversa. If both MAR and distinctness hold, then the missingdata mechanism is said to be ignorable (Little and Rubin, 1987; Rubin, 1987).
2.3 The observeddata likelihood and posterior 2.3.1 Observeddata likelihood Following arguments given by Rubin (1976) and Little and Rubin (1987), it can be shown that under ignorability, we do not need to consider the model for R nor the nuisance parameters ξ when making likelihoodbased or Bayesian inferences about θ. Because the observed data truly consist not only of Y obs, but also of R, the probability distribution of the observed data is actually given by
P( R, Yobs θ , ξ ) =
=
∫ P( R, Y θ , ξ )dYmis
∫ P( R Y , ξ )P(Y θ )dYmis,
(2.3)
where the integral is understood to mean summation for distributions that are discrete. Under the MAR assumption, (2.3) becomes
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∫
P( R, Yobs θ , ξ ) = P( R Yobs , ξ ) P(Y θ )dYmis
= P( R Yobs ,ξ ) P(Yobs θ ).
(2.4)
The likelihood of the observed data under MAR can thus be factored into two pieces, one pertaining to the parameter of interest θ and the other pertaining to the nuisance parameter ξ. When the two parameters are distinct, then likelihoodbased inferences about θ will be unaffected by ξ or P (RYobs,ξ). Maximumlikelihood estimation of θ , likelihoodratio tests concerning θ, and so on can then be performed without regard for the missingdata mechanism; that is, the missingdata mechanism may be safely ignored. The factor in (2.4) pertaining to θ (or, more precisely, any function proportional to this factor) is referred to by Little and Rubin (1987) as the likelihood ignoring the missingdata mechanism,
L(θ Yobs ) ∝ P(Yobs θ ).
(2.5)
For brevity, we will refer to (2.5) as the observeddata likelihood, although that name should, strictly speaking, be reserved for the complete function (2.4). Because we assume ignorability throughout, however, there is never a need to work with the complete function (2.4), and thus there will be no ambiguity. Notice that at first glance, the factorization (2.4) seems to contain no implicit assumptions about the missingness mechanism. The joint distribution of any two random variables, say Z1 and Z2, can always be written as the marginal distribution of Z1 multiplied by the conditional distribution of Z2 given Z1. A subtle but important difference exists between this basic rule of probability and the factorization (2.4), however, and the distinction lies in the definition of θ . In our framework, θ refers to the parameters of the model for the complete data Y = (Yobs Y mis), not the parameters for the
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distribution of Yobs, alone. We assume that the ultimate goal of the analysis is to draw inferences about the parameters of the completedata model, not the parameters governing the marginal distribution of Y obs. If θ were redefined to pertain only to Y obs then assumptions like MAR would not be necessary. This approach has some major conceptual difficulties, however, and may lead to results that are very hard to interpret, so we will not consider it further. For an interesting discussion related to this point, see the exchange between Efron (1994) and Rubin (1994). 2.3.2 Examples Example 1: Incomplete univariate data. Suppose that a single variable is observed for units 1, 2,...n1 < n and missing for units n1 + 1,..., n. Let Y = (y 1 , y2,yn) denote the complete data and R = (r1, r2,...,rn) the response indicators, where ri = 1 if yi is observed and ri = 0 if yi is missing. If the distribution of R does not depend on Y then the missingness mechanism is MAR. In fact, in this case it is MCAR. One such mechanism is simple Bernoulli selection in which each unit is observed with probability independently of all other units, n
P( R Y ,ξ ) =
∏ ξ r (1 − ξ )1− r . i
i
i =1
Another MAR mechanism arises when the responding units are a simple random sample of all units, otherwise.
n −1 n if r = n1, i =1 i P( R Y , ξ ) = n1 otherwise. 0
∑
The latter regards n1 as fixed whereas the former regards n1 as random. Under either of these mechanisms or any other mechanism that is MAR, it is appropriate to base inferences about parameters of the distribution of Y on the observeddata likelihood. This likelihood may be written
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∫
L(θ Yobs ) = P(Y θ )dYmis n1
n1
P( yi  θ ) ∏ P( yi  θ )dyn +1Ldy . ∫ ∫∏ i=1 i= n +1
= K
1
n
1
The first product in the integrand does not involve Ymis and can be brought out of the integral, and the second product integrates
Figure 2.2. Bivariate data with one variable subject to nonresponse.
to one, yielding
(
n1
) ∏ P( yi θ ),
L θ Yobs =
i =1
which is simply a completedata likelihood based on the reduced sample y1,..., yn1 . Example 2: Bivariate data with one variable subject to nonresponse. Consider a dataset with variables Y1 and Y2 as shown in Figure 2.2, where Y 1 is observed for units 1, 2,..., n but Y2 is observed only for units 1,2,...,n1 < n .The missing data will be MAR if the probability that Y2 is missing does not depend on Y2, although it may possibly depend on Y1. Let y i1,
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and yi2 denote the values of Y1 and Y 2, respectively, for unit i. The observeddata likelihood may be written n1
n
n
i =1
i = n1 +1
i = n1 +1
∫ ∏ P( yi1, yi2 θ ) ∏ P( yi1 θ ) ∏ P( yi2 yi1,θ )dYmis . The first two products in the integrand do not involve Ymis and the last product integrates to one, hence
L(θ  Yobs ) =
n1
∏
n
P( yi1,yi 2  θ )
i =1
=
∏ P( yi1  θ )
(2.6)
i = n1 +1
n
n1
i=1
i=1
∏ P( yi1  θ )∏ P( yi2  yi1,θ ).
(2.7)
For example, suppose that Y1 and Y 2 have a bivariate normal distribution with parameter
θ = ( µ1, σ 11, µ 2 , σ 22 , σ 12 ),
(
)
where µi = E(Yi  θ ) and σ ij = Cov Yi Yj  θ , i, j = 1, 2 . The observeddata likelihood may be written as in (2.6),
L(θ  Yobs ) ∝ Σ
1 − 2
− n1 / 2
( yi − µ )T Σ −1 ( yi − µ )} i =1 n1
∑
n 1 ×σ 11−( n − n1 ) / 2 exp − ( yi1 − µ1 )2 , σ 2 11 i = n +1 1
∑
(2.8)
where y = (yi1, y i2) T , µ = ( µ1, µ 2 ) and Σ is the 2 × 2 matrix T
with elements σij. Alternatively, the parameter of the bivariate normal distribution may be expressed as
φ = ( µ1, σ 11, β 0 , β1, σ 22⋅1 ),
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and β1 = σ 12 / σ 11, β 0 = µ 2 − β1µ1 / σ 11, so that E(Y2  Y1, ϕ ) = β 0 + β1Y1 and V (Y2  Y1, ϕ ) = σ 22⋅1. The transformation ϕ = ϕ (θ ) is onetoone. Following (2.7), the observed data likelihood may be written in terms of ϕ as where
2 σ 22⋅1 = σ 22 − σ 12
1 −n / 2 L(θ  Yobs ) ∝ σ 11 exp − σ 2 11 1 −n / 2 ×σ 22 exp − ⋅1 2σ 22⋅1
( yi1 − µ1 )2 i =1 n1
∑
(2.9)
( yi 2 − β 0 − β1 yi1 )2 , i =1 n
∑
an expression first given by Anderson (1957). Expressions (2.8) and (2.9) are equivalent, but the latter has A, convenient interpretation as the product of two completedata likelihood functions: the univariate normal likelihood for Y1 based on units 1, 2,..., n, and the likelihood for the normal linear regression of Y2 on Y1 based on units 1, 2,..., n1. Because the parameters ϕ1 = ( µ1, σ 11 ) and ϕ 2 = ( β 0 , β1, σ 22⋅1 ) corresponding to these two factors are distinct, inferences about them may proceed independently. For example, maximumlikelihood estimates may be obtained by independently maximizing the likelihoods for ϕ1 and ϕ2, each of which corresponds to a straightforward completedata problem. Expression (2.8) also appears to be the product of two completedata likelihoods, but the parameters in the two factors are not distinct because µ1 and σ11 appear in both. Example 3: Multivariate normal data with arbitrary patterns of missing values. Now consider a pvariate normal data matrix with missing values on any or all variables as in Figure 2.1. It is convenient to group the rows of the matrix according to their missingness patterns. A missingness pattern is a unique combination of response statuses (observed or missing) for Y 1, Y2,..., Yp. With p variables there are 2p possible missingness patterns: It is usually the case, especially when p
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is large, that not all possible patterns are represented in the sample. Index the unique missingness patterns that actually appear in the sample by s, where s = 1, 2,..., S, and let I(s) denote the subset of the rows i = 1, 2,..., n that exhibit pattern s. A generalization of the arguments leading to (2.6) and (2.8) allows us to write the observeddata likelihood as S
∏ ∏ Σ*s s =1 i ∈I ( s )
1
2
{
}
exp − 12 ( yi* − µ s* )T Σ *s −1 ( yi* − µ s* ) ,
(2.10)
where yi* denotes the observed part of row i of the data matrix, and µ s* and Σ *s denote the subvector of the mean vector and the square submatrix of the covariance matrix Σ, respectively, that pertain to the variables that are observed in pattern s. Notice that if any rows of the data matrix are completely missing, then those rows drop out of the observeddata likelihood; under the ignorability assumption, these rows contribute nothing to the inference and may be ignored. Despite the concise appearance of (2.10), this likelihood tends to be a complicated function of the individual means µi and covariances σij , i ,j = 1, 2,...,p. Except in special cases, there is no way to express this likelihood as in (2.7) and (2.9), the product of simple completedata likelihoods whose parameters are distinct (Rubin, 1974). Moreover, the first two derivatives of (2.10) or its logarithm with respect to the individual µi and σij tend to be very complicated as well, making (2.10) awkward to maximize by gradient methods such as NewtonRaphson. A much more convenient method for maximizing this likelihood is provided by the EM algorithm (Beale and Little, 1975; Dempster, Laird, and Rubin, 1977), to be introduced in Chapter 3. The complicated nature of (2.10) is typical of the observeddata likelihood functions one encounters with incomplete multivariate data. Except in special cases, meaningful summaries of these functions (e.g. modes) are not available in closed form, nor are they readily computable from classical
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numerical methods; we typically need to resort to special iterative techniques like EM. 2.3.3 Observeddata posterior Definition In the Bayesian framework all inferences are based on a posterior probability distribution for the unknown parameters that conditions on the quantities that are observed. Returning to the notation of Section 2.3.1, the unknown parameters are (θ,ξ) and the observed quantities are Yobs, and R. By Bayes s Theorem, the posterior distribution may be written as
P(θ , ξ  Yobs , R) = k −1P( R, Yobs  θ , ξ )π (θ , ξ ),
(2.11)
where π (⋅) denotes a prior distribution applied to (θ,ξ) and k is the normalizing constant
k=
∫ ∫ P( R, Yobs  θξ )π (θ , ξ )dθ dξ.
Under the assumption of MAR, we may substitute (2.4) into (2.11) to obtain
P(θ , ξ  Yobs , R) ∝ P( R  Yobs , ξ ) P(Yobs  θ )π (θ , ξ ). Bayesian inferences about θ alone are based on the marginal posterior obtained by integrating this function over the nuisance parameter ξ . When θ and ξ are distinct according to the definition in Section 2.2.2, the prior distribution factors as
π (θ , ξ ) = π θ (θ )π ξ (ξ ). Hence the marginal posterior for θ is, under ignorability,
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P(θ  Yobs , R) =
∫ P(θ , ξ  Yobs , R)dξ
∝ P(Yobs  θ )π θ (θ ) P( R  Yobs , ξ )π ξ (ξ )dξ
∫
∝ L(θ  Yobs )π θ (θ ),
(2.12)
where the proportionality is up to a multiplicative factor that does not involve θ . Note that R does not appear on the right
hand side of (2.12) and therefore P(θ  Yobs , R) = P(θ  Yobs ). We have thus shown that under ignorability all information about θ is summarized in the posterior that ignores the missingdata mechanism,
P(θ  Yobs ) ∝ L(θ  Yobs )π θ (θ ).
(2.13)
We shall refer to (2.13) as the observeddata posterior. In most practical applications one would not be interested in the function (2.13) itself but in meaningful summaries of it: posterior moments, marginal posterior densities and quantiles of univariate components of θ, etc. Note that these summaries are typically integrals of the density (2.13) or functions involving it over the parameter space. In many commonly used probability models with complete data, computation of these integrals can be simplified by choosing a prior distribution for θ within a natural conjugate class (e.g. Box and Tiao, 1992). With incomplete data, however, the usual natural conjugate priors no longer lead to posteriors that are recognizable or easy to summarize. A bivariate normal example Let us return to Example 2 of Section 2.3.2 in which Y1 is observed for all units but Y 2 is missing for some. Assuming that the complete data are bivariate normal with parameter θ = (µ,Σ), the observed data likelihood is given by (2.8). In the
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absence of strong prior beliefs about θ , a prior distribution commonly used with complete data is
π θ (θ ) ∝ Σ −3 / 2 ,
(2.14)
which can be derived by applying the invariance principle of Jeffreys to µ and Σ (e.g. Box and Tiao, 1992). The prior (2.14) is said to be improper because it is not a true probability density function; its integral over the parameter space is not finite. With complete data this prior leads to a posterior distribution for θ that is the product of an invertedWishart distribution for Σ and a normal distribution for Σ given µ. The properties of this normalinverted Wishart distribution are well known, and summaries (marginal densities, moments, etc.) are readily available in closed form. When some values of Y2 are missing, however, the posterior under (2.14) is no longer normalinverted Wishart. One way to characterize this posterior is to express it in terms of the alternative parameterization ϕ = ( µ1, σ 11, β 0 , β1, σ 22⋅1 ) , w h e r e 2 / σ 11. As β1 = σ 12 / σ 11, β 0 = µ 2 − β1µ1 and σ 22⋅1 = σ 22 − σ 12 shown in (2.9), the likelihood for ϕ factors neatly into a completedata likelihood for ϕ 1=(µ,σ1 1 ) based on all the sample units and a completedata likelihood for ϕ 2=(β0, β1,σ22⋅1) based on the units for which Y 2 is observed. Moreover, the prior distribution (2.14) also factors into independent priors for ϕ1 and ϕ2. To see this, note that ϕ = ϕ(θ ) is a onetone transformation, and the density for ϕ implied by (2.14) can thus be written as
(
)
π φ (φ ) = π θ φ −1 (φ ) J
−1
,
where θ = ϕ −1 (ϕ ) denotes the transformation from ϕ back to
θ, and J denotes the absolute value of the determinant of the Jacobian (firstderivative) matrix for the transformation from
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2 θ to ϕ. Notice that Σ = σ 11σ 22 − σ 12 = σ 11σ 22⋅1. Moreover, −1 , and thus it will be shown in Section 5.2.4 that J = σ 11 −1 2 −3 2 π φ (φ ) = σ 11 σ 22⋅1 ⋅
(2.15)
Combining (2.15) with the likelihood (2.9), the observeddata posterior can be written as
P(φ  Yobs ) = P(φ1  Yobs ) P(φ 2  Yobs ),
(2.16)
where n 1 2 − n +1 / 2 P(φ1  Yobs ) ∝ σ 11( ) exp − yi1 − µ1 ) ( 2σ 11 i =1
∑
and n1 1 2 P( yi 2 − β 0 − β1 yi1 ) . exp − ) ( 2σ 22⋅1 i =1 After some manipulation, it can be shown that P(ϕ1  Yobs ) is the product of a normal and a scaledinverted chisquare density, −( n + 3) / 2 ϕ 2  Yobs ∝ σ 22⋅11
∑
(
)
µ1  σ 11, Yobs ~ N y1, n −1σ 11 ,
σ 11  Yobs ~ (n − 1)S11 χ n−−2 2 ,
(2.17)
where y1 and S11 are the usual sample mean and variance of
−2 Y1, based on all n units and χ n− 2 denotes an inverted chisquare variate n (i.e. the reciprocal of a chisquare variate) with n  2 degrees of freedom. Moreover, P(ϕ 2  Yobs ) can be shown to be the product of a bivariate normal and an inverted chisquare,
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(
β  σ 22⋅1, Yobs ~ N βˆ , σ 22⋅1 X T X
)
−1
ˆ n−2−1, σ 22⋅1  Yobs ~ εˆ T εχ 1
,
(2.18)
where β = ( β 0 , β1 ) , T
1 1 X = M 1
(
βˆ = X T X
)
−1
y11 y12 y y21 , y = 22 , M M yn1 ,1 yn1 , 2
X T y and εˆ = y − Xβˆ . Details of the calculations
leading to these posteriors may be found in standard texts on Bayesian analysis and will be reviewed in Chapters 56. In the above example, a particular factorization of the observed data likelihood enabled us to express the posterior in a tractable form. This will not always be the case. One cannot always factor the observeddata likelihood into completedata likelihoods whose parameters are distinct. The techniques of Markov chain Monte Carlo to be introduced in Chapter 3 will free us from many of the constraints of mathematical tractability, allowing us to create random draws from the observeddata posterior whether or not it can be written in a tractable form.
2.4 Examining the ignorability assumption The statistician unaccustomed to missingdata problems might be led to believe that the observeddata likelihood is always the relevant likelihood function for θ whenever data are not fully observed; it is, after all, simply the marginal distribution of Yobs the observed part of Y. But as we have seen, the observed data consist of both Y obs, and R, and one needs the special condition of ignorability to make the observed value of
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R noninformative with respect to θ . Therefore it is crucial for the data analyst to understand the implications of the ignorability assumption, particularly MAR, and assess its appropriateness in any given problem. 2.4.1 Examples where ignorability is known to hold On occasion, the MAR condition is known to hold exactly. Some examples of this are given below. Double sampling. In sample surveys that employ double sampling (e.g. Cochran, 1977), some characteristics, say Y1, Y2,..., Yk, are recorded for all units in the sample, and then additional characteristics Yk+1, . . . ,Yp are recorded for a subsample of the original sample. If this subsample is selected by a mechanism that depends on Y1, Y2,..., Yk alone, even in a systematic or deterministic way, then the missing values of Yk+1, ...,Yp for those units not included in the subsample are MAR. Sampling for nonresponse followup. In censuses and large surveys, initial attempts to collect data may fail for a substantial proportion of units. In a mailbased household survey, for example, some residents will inevitably fail to mail back their forms. In many cases, more intensive datacollection efforts (e.g. personal interviews) would be successful, but attempting to follow up every nonresponding unit may be economically infeasible. If the intensive followup effort is applied to a random sample of nonresponding units, then the missing data for the remaining nonrespondents is MAR. A famous early discussion of this method is given by Hansen and Hurwitz (1946). Randomized experiments with unequal numbers of cases per treatment group. In many designed experiments, the researcher strives to assign an equal number of cases or subjects to each treatment, because data that are balanced in
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this fashion are typically easier to analyze than data that are unbalanced. Moreover, principles of efficiency often support the use of balanced designs. Sometimes balance is not feasible, however, and the data are unbalanced by design. The analysis of unbalanced data can often be simplified by imagining a number of additional cases which, if they were included in the experiment under the appropriate treatment groups, would result in a balanced experiment. Because the hypothetical missing data within each treatment group were missing with probability one, they are MAR. Medical screening with multiple tests, where not all tests are administered to all subjects. In many medical studies, an inexpensive or easily administered test is given to a large number of subjects, and for purposes of calibration a second, more expensive, and more reliable test is administered to a subsample. The calibrating sample may be chosen completely at random, on the basis of subject specific covariates, or even on the basis of the outcome of the first test. As long as all the information used to choose the subsample is recorded and regarded as part of the observed data Yobs, then the missing data will be MAR. Matrix sampling for questionnaire or test items. In matrix sampling (e.g. Thayer, 1983) a test or questionnaire is divided into sections, and groups of sections are administered to subjects in a randomized fashion. The resulting data matrix will have rectangular patches of missing data corresponding to sections of the test or questionnaire that were not administered to subject groups. If all the variables used in the sampling plan are included in Yobs, then the missing data will be MAR. In most of the above examples, the missing data may be said to be missing by design, because it was never the intention of the investigator to record all potential variables for all subjects. When missing data are missing by design, they tend also to be MAR.
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2.4.2 Examples where ignorability is not known to hold In many other missingdata contexts, however, it is not known whether or not the MAR condition is satisfied. Examples include: 1. Sample surveys where some sampled individuals are not at home, unwilling to be interviewed, or do not otherwise provide useful responses to some or all of the questionnaire items. Notice, however, that if followup data can later be obtained for a probability sample of nonrespondents, the missing data can be converted to MAR. 2. Planned experiments where, for reasons unforeseen or unintended by the investigator, one or more outcomes of interest cannot be recorded: culture dishes break, production runs fail, subjects drop out of the study, etc. 3. Observational studies in which data of economic, historic or other scientific interest are collected for analysis, but for reasons beyond the control of the investigator some of the variables desired are simply not available for some cases. Sometimes the fact that a numerical observation is not recorded is more like a response than a missing value. In a laboratory experiment, for example, an animal may die for reasons related to the treatment that was applied to it. If so, then the hypothetical missing data, the measurements that could not be recorded because the animal died, are counterfactual and poorly defined. In such instances, careful thought should be given to whether it is sensible to analyze such data by missingdata methods. When data are missing for reasons beyond the investigator s control, one can never be certain whether MAR holds. The MAR hypothesis in such datasets cannot be formally tested unless the missing values, or at least a sample of them, are available from an external source. When such an external source is unavailable, deciding whether or not MAR is plausible will necessarily involve some guesswork, and will require careful consideration of conditions specific to the problem at hand.
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2.4.3 Ignorability is relative One final point that must be made is that MAR and ignorability are relative, defined with respect to a particular set of observed data Yobs. In many situations, the status of the missing data (whether or not they are MAR) may change if the definition of Yobs is changed. For example, consider a sample survey that involves probability sampling for nonresponse followup. Let Y mis refer to the data for nonrespondents not included in the followup effort. If Y obs is the data for nonrespondents who were included in the followup effort, then the missing data are MAR. If the definition of Yobs is expanded to include the original respondents, however, and no information (e.g. dummy indicator) is retained to distinguish them from the nonrespondents who were subsequently followed up, then the missing data are no longer MAR. In other situations, a nonresponse mechanism may not be exactly known to the analyst, but covariates are available that could plausibly explain or predict the missingness to a great extent. The plausibility of MAR would then depend on whether these covariates are included in the analysis. Further discussion of this point is given by Graham et al. (1994) with regard to attrition in a longitudinal study of adolescent drug use.
2.5 General ignorable procedures Virtually all of the missingdata procedures used in statistical practice, both ad hoc approaches and principled ones, rely at least implicitly on an assumption of ignorability. Often the assumptions made are even stronger. For example, the casedeletion method used by many statistical packages (omitting all incomplete cases from the analysis) may introduce bias into inferences about θ unless the missing data are MCAR. Even if MCAR holds, case deletion may still be grossly inefficient. We shall call a missingdata procedure a general ignorable procedure if it is based upon either an observeddata likelihood or an observeddata posterior. The EM algorithm, for example, will be seen to be a general ignorable procedure
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because it maximizes the observeddata likelihood. Omitting all incomplete cases from an analysis, however, is not a general ignorable procedure because it leads to a different likelihood or posterior, one based only on the complete cases. All of the methods developed in this book are general ignorable procedures. A common feature of ad hoc missingdata treatments like case deletion is that they tend to discard information from certain units and/or variables in order to make the estimation problem more tractable. General ignorable procedures by nature, however, do not discard such information because the observeddata likelihood or posterior conditions fully on Yobs. From standpoints of efficiency and bias, full conditioning on Yobs is advantageous because it leads to inferences that are proper under any missingdata mechanism that is ignorable, whereas procedures that are not fully conditional may perform well in some but not all ignorable scenarios. This point can be illustrated with a simple hypothetical example. 2.5.1 A simulated example Consider again the bivariate data described in Example 2 of Section 2.3.2, in which Y 1 is always observed but Y 2 is sometimes missing. Under a bivariate normal data model with ignorable missingness, Adata we may easily find the value φˆ that maximizes the observe likelihood function L(ϕ  Yobs ) in
( )
(2.9) and then apply the inverse transformation θˆ = ϕ −1 ϕˆ to find the maximumlikelihood (ML) estimate for θ . Straightforward calculation shows that the ML estimate of µ 2 is µˆ 2 = βˆ 0 + βˆ1µˆ 1, where βˆ 0 and βˆ1 are leastsquares estimates from the regression of Y 2 on Y1 based on units 1,2, ...,n1 and µ1, is the average value of Y1 among units 1,2,...,n (Anderson, 1957). Alternatively, one may compute µˆ 2 by first imputing regressionbased predictions for the missing values of Y , i.e. letting y = βˆ + βˆ y for i = n + 2
i2
0
1 i1
1
l,...,n, and then computing the average of Y2 among units 1,2,...,n in the imputed dataset.
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Estimating µ2 by µˆ 2 is a general ignorable procedure because µˆ 2 is obtained by maximizing the observeddata likelihood L(ϕ  Yobs ). Another plausible estimate is the
n completecase (CC) estimate µ˜ 2 = n1−1Σ i =1 1 yi 2 , the average of Y2 among the completely observed cases. Estimating µ2 by µ˜ 2 may certainly be regarded as an ignorable procedure because it is consistent with the belief that the missing data are MCAR, a special case of MAR. It is not, however, a general ignorable procedure because it does not condition fully on Yobs; in particular, it discards the observed values of Y1 for units n 1 + l,..., n. Consequently, the ML estimate tends to be more efficient than the CC estimate, and it exhibits better performance over a variety of missingness mechanisms. We can easily compare the performance of the ML and CC estimates by simulation. As shown in Figure 2.2, let us define
Table 2.1. Simulated means (standard deviations) of the CC and ML estimates under three MAR mechanisms
R = (r1,...,rn) to be a vector of response indicators where ri = 1 if Y2 is observed and ri = 0 if Y 2 is missing for unit i, i = 1, 2,...,n. In particular, consider the class of ignorable mechanisms in which P(ri = 1Y) = g(yil) independently for units i = 1, 2,..., n, where g is some function that maps the real line into the unit interval [0, 1]. Three possible choices for g are constant g1 ( yi1 ) = a1, 0 ≤ a1 ≤ 1, probit selection interval selection
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g2 ( yi1 ) = Φ( a2 = b2 yi1 ). g3 ( yi1 ) = 1 if a3 ≤ yi1 ≤ b3 , else 0,
where Φ (⋅) denotes the standard normal cumulative distribution function. The constant function g 1 is MCAR, whereas g2 and g3 are MAR but not MCAR. A simulation was conducted in which samples of size n = 100 were drawn from bivariate normal populations with µ1 = µ 2 = 0, σ11 = σ 22 = 1 and σ12 = ρ = 0, 0.5 and 0.9, respectively. Random patterns of missingness were imposed on each sample according to g1 1 with a1 = 0.5, g2 with a2 = 0, b2 = 1 and g3 with a3 = Φ− (0.35) = 0.385, b3 = Φ − (0.85) = 1.036. These constants were chosen to yield an expected response rate n1/n of 50% under each mechanism, a level higher than is found in most typical applications. The means and standard deviations of the CC and ML estimates over 5000 repetitions are shown in Table 2.1. Under the nonMCAR mechanisms, the CC estimate is biased whenever ρ ≠ 0. Except under the unrealistic condition that n1 / n → 1, this bias does not vanish as n → ∞, causing µ˜ 2 to be inconsistent. The ML estimate, however, is unbiased and consistent under the three mechanisms used here, as well as under essentially all other ignorable mechanisms. From considerations of bias and consistency, the ML estimate has a clear advantage over the CC estimate. In fairness, one should note that it is possible to construct a missingness mechanism for which the ML estimate would be less biased than the CC estimate. Such a mechanism would be neither MAR nor MCAR, but a peculiar nonignorable mechanism in which Y1 and Y2 would have correlations of opposite sign among the respondent and nonrespondent groups. Such mechanisms, although mathematically possible, are somewhat atypical and should not be expected to occur often in practice with real data. Further discussion of this point in an applied setting is given by Schafer (1992). Under the more restrictive condition of MCAR, both the ML and CC estimates are unbiased, but ML still has an advantage over CC for ρ = 0.5 and ρ = 0.9 because its variance is lower. This reduction in variance occurs because Y1 becomes an increasingly valuable predictor of the missing 1
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values of Y2 as ρ increases. The only situations in Table 2.1 for which CC appears to dominate ML are when Y1 and Y2 are unrelated (ρ = 0), in which case µˆ 2 has more variability than µ˜ 2 . Here CC enjoys an advantage because it correctly assumes that the correlation between Y1 and Y2 is zero, whereas ML uses an estimated regression line whose slope βˆ1 randomly varies about zero. This advantage of CC over ML would be much less dramatic if the average missingness rate were lower. Moreover, the benefit of CC s lower variance in this special situation tends to be outweighed by the protection against bias afforded by ML when ρ ≠ 0 and the mechanism is not MCAR. 2.5.2 Departures from ignorability The above example illustrates the advantages of general ignorable procedures when missing data are MAR. Even when the missing data are not precisely MAR, however, general ignorable procedures still tend to be better than ad hoc procedures such as case deletion for the following reason: general ignorable procedures remove all of the nonresponse bias explainable by Yobs whereas ad hoc procedures may not. To d emo n str ate th is po in t, let u s mo d ify th e ex amp le o f Sectio n 2.5 .1 to in clud e a mech an ism th at is no t MA R. Su pp o se th at p ro p en sity to r esp o nd f o r Y2 is n o lon g er a f u nctio n o f Y1 b ut is no w d irectly r elated to Y2 . We w ill assu me that P( r i = 1 Y) = g ( yi 2 ) in d ep end en tly f o r i = 1 , 2 ,..., n . A simu latio n w as con d u cted u sing p r ob it selection , g ( yi 2 ) = Φ ( a 2 + b 2 y i 2 ) wi t h a 2 = 0 , b 2 = 1 a n d a l l
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Table 2.2. Simulated means (standard deviations) of d2 and A2 under a nonMAR mechanism
other parameters as before. Results shown in Table 2.2 show that the CC estimate µ˜ 2 and the general ignorable ML estimate µˆ 2 are equally biased when ρ = 0 but that ML becomes substantially less biased for larger values of ρ . By making use of Y1 to predict missing values of Y2, the ML procedure removes the nonresponse bias in the observed values of Y2 attributable to Y1. In this example nonresponse is related to Y2 rather than Y 1, but as ρ increases Y1 becomes an increasingly useful proxy for Y2. Limited practical experience with real data also suggests that general ignorable procedures may tend to perform well even when the ignorability assumption is suspect, especially in multivariate settings. In surveys containing questions about income, for example, nonresponse rates on incomerelated questions tend to be high, and both experience and common sense suggest that the probability of response is likely to be related to level of income. In a study of missingdata methods for an income question in the Current Population Survey, Greenlees et al. (1982) established definite relationships between response and income itself. Upon further investigation, David et al. (1986) found little evidence of bias in ignorable procedures that imputed missing values of income on, the basis of other demographic and questionnaire items that were observed. This evidence came from knowledge of the missing values obtained from an external source, the actual wages and salary reported to the Internal Revenue Service on the individuals tax returns. David et al. (1986) also concluded that further improvements in the missingdata procedures
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would probably come from better modeling of the multivariate structure of the data, not from nonignorable modeling. The crucial assumption made by ignorable methods is not that the propensity to respond is completely unrelated to the missing data, but that this relationship can be explained by data that are observed. Whether ignorability is plausible in a particular setting is therefore closely related to the richness of the observed data Y obs and the complexity of the data model P(Y θ ). If Yobs contains a lot of information relevant for predicting Ymis, and if the data model is sufficiently complex to make use of this information, then we should expect the residual dependence of R upon Ymis after conditioning on Yobs to be relatively minor. Thus in multivariate datasets where both the observed data and the completedata model are rich, general ignorable procedures may tend to perform well in practice. Even if they do not perform well, they still may provide an important and useful baseline for assessing and comparing any available alternatives. 2.5.3 Notes on nonignorable alternatives Various approaches to incomplete data that do not assume ignorability have also appeared in the literature. A detailed review of nonignorable methods is beyond the scope of this text, but we note that these nonignorable methods tend to have a common approach. They generally involve joint probability modeling of both the data P (Y θ ) and the missingness mechanism P(RY, ξ) and joint estimation of θ and ξ from Yobs and R. These joint models for Y and R typically involve more parameters than can be estimated from Yobs and R alone. In order to make them identifiable, one must either (a) impose a priori restrictions on the joint parameter space for θ and ξ or (b) impose an informative Bayesian prior distribution on (θ,ξ). For continuous data, one group of nonignorable methods is based on models known in the econometric literature as stochastic censoring or selection models (Heckman, 1976; Amemiya, 1984). For categorical data, nonignorable contingencytable approaches are described by Fay (1986);
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Baker and Laird (1988); Rubin, Schafer and Schenker (1988); Winship and Mare (1989); and Park and Brown (1994). A review of some nonignorable methods is given by Little and Rubin (1987, chap. 11). Glynn, Laird and Rubin (1993) describe nonignorable modeling based on followup data. Approaches to nonignorable modeling for longitudinal data are discussed by Conaway (1992, 1994); Diggle and Kenward (1994); and Baker (1994). Little (1993) discusses in general terms a class of nonignorable models called patternmixture models, in which the joint distribution of Y and R is specified in terms of the marginal distribution of R and the conditional distribution of Y given R.
2.6 The role of the completedata model Having discussed at length the assumption of ignorability, we now return to the role of the model for the complete data presented in Section 2.1. A good model should be plausible and have sufficient flexibility to preserve the subtle features of the data at hand, but in realistic settings one must also, consider issues of convenience and mathematical tractability. The classes of models considered in this book, the multivariate normal for continuous data, loglinear models for crossclassified categorical data and the class of models for mixed continuous and categorical data, are generalpurpose multivariate models that are both mathematically tractable and appropriate in many but not all situations. With categorical and mixed data, the analyst has considerable freedom to tailor a model to the particular dataset; twoway, threeway or higher associations between variables may be included if they are thought to be important. The multivariate normal model for continuous data is less flexible, however, because it fits only pairwise associations. Sometimes the normality assumption may be made more plausible by applying suitable transformations to one or more variables (Box and Cox, 1964; Emerson, 1991).
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2.6.1 Departures from the data model When making inferences about a parameter θ , the assumed parametric form of the model and the iid assumption often play a crucial role. If these assumptions are seriously violated, then even under ignorability the likelihood (2.5) may be a poor summary of the data s evidence about θ . Indeed, if the data model does not hold, the interpretation of θ itself may be ambiguous. Selection of a data model should proceed with care, and diagnostics for assessing goodness of fit should be used whenever possible. Of course, in all analyses of real (not simulated) data a probability model is only an approximation to reality, and some departures from modeling assumptions are inevitable. In practice one must judge whether these departures are of a magnitude and nature that seriously impairs the quality of the inference about θ , or whether they are of only minor importance and may be safely ignored. Complex sample surveys The assumption (2.1) that the rows of the data matrix are independent and identically distributed (iid) can be problematic, especially when the rows do not correspond to observational units that are exchangeable (i.e. like a simple random sample). This assumption is commonly violated in largescale surveys, which typically employ complex probability sampling methods. Sample surveys often have special features (unequal probabilities of selection, stratification, clustering, etc.) that cause the units to be nonexchangeable. In some cases, it may be possible to apply the models of this book to survey data in a way that preserves the integrity of the complex design. Stratification and the general issue of design variables are discussed in Section 2.6.2 below. Data with clusters, naturally occurring groups of observations that are potentially intercorrelated, should not in general be handled by iid models. If the clusters are few and large, it may be feasible to fit separate models for each cluster that treat the units within clusters as exchangeable. Another possibility is to
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add a cluster identifier to the model as an additional categorical variable, and posit some simple associations between this variable and the other variables in the dataset; an example of this approach is described in Section 9.5.3. When the number of clusters is large, and there are relatively few observations per cluster, the data are more appropriately described by hierarchical probability models with an explicit multilevel error structure. These models, sometimes called randomeffects or mixed models, have been extensively applied in the univariate setting with continuous responses (Laird and Ware, 1982; Searle et al., 1992). Recent advances in computational methods have allowed extensions to discrete (Zeger and Karim, 1991) and multivariate continuous (Everson and Morris, 1996) response models. Application to problems of missing data in surveys, however, will typically require hierarchical models for multivariate categorical responses. A proper treatment of these models is well beyond the scope of this book. For a detailed example of such a model in the context of the U.S. Decennial Census, see Schafer (1995). More discussion of missingdata problems in complex surveys will appear in Sections 4.3 and 4.5. The role of imputation It may be possible to mitigate some of the effects of model failure through multiple imputation (Section 4.3). Inference by multiple imputation proceeds in two stages. First, m simulated versions of the missing data are created under a data model. Second, the m versions of the complete data are then analyzed by completedata statistical techniques, and the results are combined to produce one overall inference. Sometimes the completedata statistical analyses of the second stage will involve different models than the one used to produce the multiple imputations in the first stage. When analyzing data from a sample survey, for example, one may impute the missing data on the basis of an elaborate multivariate model, but then proceed to analyze the data using classical nonparametric survey methods in which inferences are based entirely on the randomization used to draw the sample (e.g. Cochran, 1977). Even if the model used for imputation is
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somewhat restrictive or unrealistic, it will effectively be applied not to the entire dataset but only to its missing part. Multiple imputation thus has a natural advantage over some other methods of inference in that it may tend to be more robust to departures from the completedata model, especially when the amounts of missing information are not large. Hence, even though the classes of models examined in this book may not realistically describe many of the multivariate datasets one encounters in the real world, we suspect that they will still prove useful in a wide variety of data analyses if applied within the framework of multiple imputation. The role of modeling assumptions in multiple imputation will be revisited in Section 4.5.4. 2.6.2 Inference treating certain variables as fixed Sometimes the only relevant analysis of a multivariate dataset involves modeling the conditional distribution of certain variables given others. In regression analysis, for example, the goal is to model one or more response variables given one or more predictors. When analyzing data from surveys, it is common practice to estimate means, proportions, etc. not for the population as a whole but within subdomains (strata or poststrata) that are considerably smaller. The individual subdomain estimates are then combined, using population proportions derived from an external source (e.g. a census), to yield estimates for larger domains. The relative sizes of the subdomains in the population are assumed to be known and are not estimated from the sample. Similarly, with data from planned experiments, the relevant analysis is usually a comparison of mean responses across two or more treatment groups; the manner in which experimental units are allocated to treatments is determined by the experimenter and does not need to be modeled. In discussing situations like these, we will refer to variables in a generic sense as either response variables or design variables, with the latter being those that the statistical analysis ultimately regards as fixed. Predictors in regression analyses, variables defining strata or poststrata in sample surveys and indicators of treatment groups in planned
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experiments are all examples of design variables. When a dataset contains one or more design variables, the iid assumption (2.1) is typically violated, as design variables are often under direct control of the investigator and are thus not random in the same sense as the response variables are random. It is usually not desirable to impose any probability model at all on the design variables, but to model only the conditional distribution of the response variables given the design variables. Datasets with design variables can be accommodated in our framework with just a few additional assumptions. Suppose that each row yi of the completedata matrix can be partitioned into two parts, a vector ui of design variables and a vector vi of response variables. Furthermore, suppose that the density of yi can be factored as
f ( yi  θ ) = f u ( ui  α ) f v  u ( vi  ui , β ),
(2.19)
where α and β are distinct parameters. Finally, suppose that the design variables are completely observed for all units. Under these assumptions, we can write the completedata matrix as
Y = (Yobs , Ymis ) = (U , Vobs , Vmis ), where U denotes the design variables, Vobs the observed response variables, and Vmis the missing response variables, so that Yobs = (U, Vobs) and Ymis =Vmis. The probability distribution of the observed quantities may then be written as
∫ P(R  U ,Vobs ,Vmis , ξ )P(U  α )P(Vobs ,Vmis  U , β )dVmis , (2.20) where ξ is a set of parameters governing the response mechanism. Under ignorability, this response mechanism does not depend on Vmis,
P( R  U , Vobs , Vmis , ξ ) = P( R  U , Vobs , ξ ).
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Thus the first two factors in the integrand of (2.20) do not involve Vmis, so (2.20) becomes
P( R, Yobs  θ , ξ ) = P( R  U , Vobs , ξ ) P(U  α ) P(Vobs  U , β ). (2.21) The factorization in (2.21) implies that inferences about β, the parameters of the conditional distribution of V given U, may be based on the conditional observeddata likelihood function
L( β  U , Vmis ) ∝ P(Vobs  U , β ), or on the observeddata posterior
P( β  U , Vmis ) ∝ L( β  U , Vmis )π β ( β ), where πβ is a prior distribution applied to β independently of any prior on α or ξ. Notice that (2.22) and (2.23) are the same for all a and even for all P (Uα). In other words, we will obtain a correct inference about β even if the marginal model for the design variables is misspecified. To summarize, when design variables are present we can often apply a joint probability model such as (2.1) to the complete data Y, even a model in which the distribution of the design variables is misspecified. We will obtain correct inferences about the parameters of interest provided that (a) the design variables U are observed for all units in the sample, and (b) the joint density of (U, V) factors into densities for U and for VU, with the parameters of the two densities being distinct. Assumption (b) will be a characteristic of some, but not all, of the multivariate models used in this book. Assumption (a) often does hold in practice; it would be somewhat unusual, for example, for stratification variables in a sample survey or treatment indicators in a planned experiment to be missing. In regression analysis with incomplete predictors, the factorization in (2.21) will not precisely hold, but it may still be approximately true provided that the amount of information missing on the predictors is not large.
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Example: a comparison of two sample means The experimental data in Table 2.3 reported by Snedecor and Cochran (1989, Table 6.9.1) show the weight gains of two groups of female rats, one fed a lowprotein diet and the other fed a highprotein diet. The lowprotein group has 7 rats and the highprotein group has 12. Snedecor and Cochran perform a classical analysis assuming that the observations are independent and normally distributed and the withingroup variances are equal. A pooled estimate of the common variance is
6( 425) + 11( 457) = 445.7 17 Table 2.3. Weight gains in grams of two groups of female rats (2884 days old) under two diets
on 17 degrees of freedom, and a 95% confidence interval for the difference in mean weight gain between the two groups is
(120 − 101) ± t17, .975 445.7 + , 7 12 1
1
(2.24)
where tv,p denotes the pth quantile of the t distribution with v degrees of freedom. The interval (2.24) extends from 2.2 to 40.2, barely covering zero, so the difference in means is almost significant at the 0.05 level. Let us now regard Table 2.3 as incomplete data from a balanced experiment; that is, we now suppose that the lowprotein group had 12 potential observations, 5 of which are missing. This supposition is for illustrative purposes only,
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because the analysis of variance for an experiment with a single factor is no more difficult when the data are unbalanced than when they are balanced. The complete data could then be regarded as a 24 × 2 matrix in which the first variable Y1 is a treatment indicator (0=low protein, 1=high protein) and the second variable Y2 is weight gain. Suppose that we modeled the joint distribution of Y1 and Y2 as bivariate normal. The implied marginal normal distribution of the design variable Y1 would be clearly erroneous. But note the conditional distribution of Y2 given Y1 implied by this model, a normal linear regression with constant variance, is precisely the same model that underlies the classical analysis and the confidence interval (2.24). Because Y1 is coded as 0 or 1, the slope of this regression is identical to the difference in mean weight gain between the two groups. Likelihoodbased or Bayesian inferences about the regression slope would yield essentially the same result as the classical interval (2.24), perhaps with minor differences depending on how the observeddata likelihood is summarized or on what prior distribution is chosen. One possible advantage of using the bivariate normal model here is that a general ignorable procedure devised for incomplete multivariate normal data could be applied to this dataset and the resulting inference about the regression slope would still be valid, provided, of course, that the conditional normal model for Y 2 given Y1 was correct. When design variables are present it will often be convenient to apply modelfitting and simulation algorithms devised for iid probability models like the multivariate normal, even though parts of the model pertaining to the design variables may be incorrect, because developing a more specialized algorithm then becomes unnecessary. This example also raises an unrelated but important issue regarding unbalanced experimental data. Classical methods of analysis such as the tinterval in (2.24), and other methods for unbalanced data arising from more complicated designs (e.g. Dodge, 1985), almost invariably contain an implicit assumption that the mechanism causing the imbalance is ignorable. If the data are unbalanced not by design but by
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accident, e.g. if responses for one or more units could not be recorded because of mishaps or other unforeseen occurrences, then these methods should not be applied without first considering the plausibility of MAR.
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CHAPTER 3
EM and data augmentation
3.1 Introduction Assuming that the completedata model and ignorability assumptions are correct, all relevant statistical information about the parameters is contained in the observeddata likelihood L(θYobs ) or observeddata posterior P ( θYobs). Except in special cases, however, these tend to be complicated functions of θ, and extracting meaningful summaries such as parameter estimates and standard errors requires special computational tools. EM and data augmentation provide those tools. The key ideas behind EM and data augmentation are the same: to solve a difficult incompletedata problem by repeatedly solving tractable completedata problems. As a result, the two methods share many features in common, and their implementation in specific examples is often remarkably similar. In this chapter, EM and data augmentation are introduced together to highlight the similarities between them.
3.2 The EM algorithm 3.2.1 Definition EM capitalizes on the interdependence between missing data Ymis and parameters θ . The fact that Ymis contains information relevant to estimating θ , and θ in turn helps us to find likely values of Ymis suggests the following scheme for estimating θ in the presence of Yobs alone: Fill in the missing data Ymis
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based on an initial estimate of θ , reestimate θ based on Y obs and the filledin Ymis, and iterate until the estimates converge. This idea is so intuitively appealing that specific applications of it have appeared in the statistical literature as far back as 1926 (Little and Rubin, 1987; Meng and Pedlow, 1992). Dempster, Laird and Rubin (1977) formalized the meaning of filling in the missing data at each step and presented the algorithm in its full generality, naming it ExpectationMaximization or EM. In any incompletedata problem, the distribution of the complete data Y can be factored as
P(Y  θ ) = P(Yobs  θ ) P(Ymis  Yobs , θ ).
(3.1)
Viewing each term in (3.1) as a function of θ, it follows that
l(θ  Y ) = l(θ  Yobs ) + log P(Ymis  Yobs , θ ) + c
(3.2)
where l(θ  Y ) = logP (Yθ) denotes the completedata loglikelihood, l(θYo b s ) = log P ( Yθ) the observeddata loglikelihood, and c an arbitrary constant. The term P(YmisYobs , θ), which we shall call the predictive distribution of the missing data given θ, plays a central role in EM because it captures the interdependence between Ymis , and θ . When viewed as a probability distribution it summarizes knowledge about Ymis, for any assumed value of θ, and when viewed as a function of θ it conveys the evidence about θ contained in Ymis beyond that already provided by Yobs. Because Ymis is unknown we cannot calculate the second term on the righthand side of (3.2), so instead we take the (t) average of (3.2) over the predictive distribution P(YmisYobs,θ , where θ is a preliminary estimate of the unknown parameter. This averaging yields (t)
(
)
(
)
Q θ  θ (t ) = l(θ  Yobs ) + H θ  θ (t ) + c,
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(3.3)
where
(
) ∫
(
)
Q θ  θ (t ) = l(θ  Y )P Ymis  Yobs , θ (t ) dYmis and
(
) ∫
(
)
H θ  θ (t ) = log P(Ymis  Yobs , θ )P Ymis  Yobs , θ (t ) dYmis. A central result of Dempster, Laird, and Rubin (1977) is that if (t+1) (t) be the value of θ that maximizes Q(θθ ), then we let θ
θ
(t+1)
is a better estimate than θ in the sense that its observed(t)
data loglikelihood is at least as high as that of θ , (t)
(
) (
)
l θ (t +1)  Yobs ≥ l θ (t )  Yobs .
(3.4)
This can be seen by writing
(
) (
) (
) (
l θ (t +1)  Yobs ≥ l θ (t )  Yobs = Q θ (t +1)  θ (t ) − Q θ (t )  θ (t )
(
) (
)
)
+ H θ (t )  θ (t ) − H θ (t +1)  θ (t ) .
(
) (
t +1 t t t The quantity Q θ ( )  θ ( ) − Q θ ( )  θ ( )
t +1 because θ ( ) has been chosen to satisfy
) ( ) The remainder H (θ ( )  θ ( ) ) − H (θ (
) is nonnegative
(
Q θ (t +1)  θ (t ) ≥ Q θ  θ (t ) for all θ. t
t
t +1)
(3.5)
)
 θ (t ) , which can be
written
∫
(
) P(Y )
P Y  Y , θ (t ) mis obs log (t +1) P Ymis  Yobs , θ
(
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mis
)
 Yobs , θ (t ) dYmis,
is easily shown to be nonnegative by Jensen s inequality and the convexity of the function x log x. It is convenient to think of one iteration of EM, defined by (3.5), as consisting of two distinct steps: 1. the Expectation or Estep, in which the function Q(θθ ) is calculated by averaging the completedata loglikelihood (t) l(θY) over P(YmisYobs, θ ); and (t)
2. the Maximization or Mstep, in which θ
(t+1)
is found by
maximizing Q(θθ ). (t)
Alternately performing the E and Msteps beginning with a (0) (t) starting value θ defines a sequence of iterates {θ :t = 0,1,2,...}. Dempster, Laird, and Rubin (1977) and Wu (1983) provide conditions under which this sequence converges reliably to a stationary point of the observeddata loglikelihood. In wellbehaved problems this stationary point is a global maximum and EM yields the unique maximumlikelihood estimate (MLE) of θ , the maximizer of l(θY obs). Not all problems are wellbehaved, however, and sometimes EM does not converge to a unique global maximum; these situations are taken up in Section 3.3.1 below. EM for regular exponential families The Estep of EM clarifies the intuitive idea of filling in the missing data under an assumed value of θ. In some problems (e.g. incomplete data that are purely categorical), we shall see that the Estep actually does correspond to filling in the missing data in the sense that it replaces Ymis with its average (t) or expected value E(YmisYobs, θ) under the assumption θ=θ . In other problems, however, it does not. In particular, when the completedata probability model falls in a regular exponential family, the completedata loglikelihood based on n iid (possibly multivariate) observations Y = (y 1 , y2,..., yn) may be written
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l(θ  Y ) = η(θ )T T (Y ) + ng(θ ) + c,
(3.6)
where
η(θ ) = (η1 (θ ), η2 (θ ),..., ηs (θ ))
T
is the canonical form of the parameter θ,
T (Y ) = (T1 (Y ), T2 (Y ),..., Ts (Y ))
T
is an sdimensional vector of completedata sufficient statistics and c is a constant term that does not involve θ. Moreover, each of the sufficient statistics has an additive form,
T j (Y ) =
n
∑ h j ( yi ) i =1
for some function hj. Because l(θY) is a linear function of the sufficient statistics, the Estep replaces Tj(Y) by E(Tj(Y)Yobs, (t) θ ) for j =1,2, . . . ,s; in other words, the Estep fills in not the missing elements of Y per se, but rather the missing portions of the completedata sufficient statistics. In our regular (t) exponentialfamily models, the expectations E(Tj(Y)Yobs,θ ) will be available in closed form and thus the Estep will be computationally straightforward. In many cases the Mstep will also be straightforward; (t) Q(θθ ) will have the same functional form as a completedata will be computationally no loglikelihood, so finding θ different from finding the MLE in the completedata case. For regular exponential families, the completedata MLE can be found as the solution to the moment equations (t+1)
E ( T (Y )  θ ) = t ,
(3.7)
where t is the realized value of the vector T(Y) and the expectation is taken with respect to P(Y θ ) (e.g. Cox and
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Hinkley, 1974). If these equations can be solved for an arbitrary t, then they can just as easily be solved when t is replaced by the output of an Estep. In many of the models appearing in this book, the moment equations can be solved for θ algebraically, and thus the Mstep will be available in closed form. When an algebraic solution is not available, one can still maximize the loglikelihood numerically for any given t using standard completedata iterative techniques such as NewtonRaphson. In the latter situation, implementation of EM would require undesirable nested iterations because each Mstep would itself be iterative. When this arises, however, we are often able to streamline the computation by applying a generalization of EM known as ECM, to be discussed in Section 3.2.5. 3.2.2 Examples Example 1: Incomplete univariate normal data. Suppose that Y = (y1, y2,..., y n ) represents n iid observations from a univariate normal distribution with mean µ and variance ψ, so that θ = ( µ,ψ) is the unknown parameter. The reader may easily verify that the completedata loglikelihood l(θY) can be written in exponentialfamily form (3.6) with sufficient statistics
(
T (Y ) = (T1, T2 ) = Σ in=1 yi , Σ in=1 yi2 T
)
T
.
Letting t1 and t2 denote the realized values of T 1 and T2 respectively, the moment equations
E(T1 ) = nµ = t1, E(T2 ) = nψ + nµ 2 = t2 lead immediately to the wellknown MLEs µˆ = y = n −1Σ in=1 yi 2 and ψˆ = n −1Σ in=1 yi2 − ( y ) .
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Now suppose that only the first n1 components of the data vector Y are observed, and the remaining n 0 = n  n1 components are missing at random (which, in this simple example, is equivalent to MCAR). It follows from Example 1, Section 2.3.2 that the observeddata likelihood L(θYobs) is just a complete data likelihood based only on Yobs = (y1, y2,..., yn1) and the observeddata MLEs for µ and ψ are thus
yobs = n1−1Σ i =1 1 yi and n1−1Σ i =1 1 yi2 − ( yobs ) , respectively. In this trivial example the observeddata MLEs exist in closed form, but one could also compute them using the EM algorithm. Because of the iid structure the predictive distribution of the missing data given θ does not depend on the observed data, and n
2
n
P(Ymis  Yobs , θ ) = P(Ymis  θ ) =
n
∏ P( yi  θ ).
(3.8)
i = n1 +1
The Estep replaces T1 and T2 by their expected values under P(YmisYobs,θ),
[
E(T1  Yobs , θ ) = E Σ i =1 1 yi + Σ in= n1 +1 yi n
]
n
= Σ i =1 1 yi + n0 µ,
[
E(T2  Yobs , θ ) = E Σ i =1 1 yi2 + Σ in= n1 +1 yi2 n
(
n
)
]
= Σ i =1 1 yi2 + n0 ψ + µ 2 . Inserting these expected sufficient statistics into the expressions for the completedata MLEs yields a single iteration of EM,
µ ( t +1) = n −1
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[∑
n1 i=1 yi
]
+ n0 µ ( t ) ,
(3.9)
ψ ( t +1) = n −1 −n
−2
∑ in=1 yi2 + n0ψ (t) + n0 (µ (t) )
[∑
1
n1 i=1 yi
]
2
(3.10)
2 + n0 µ ( t ) .
In this simple example, the fixedpoint equations µ
(t+1)
= µ
(t)
and ψ = ψ can be solved explicitly to show that the iterations converge to the correct observeddata MLEs (Little and Rubin, 1987). The behavior of this EM algorithm in a small numerical example is displayed in Table 3.1. A sample of n 1 = 10 observations is shown with mean µˆ = 48.1 and sample (maximumlikelihood) variance ψˆ = 59.4. Arbitrarily choosing the number of missing observations to be n0 = 3 and (0) the starting values to be µ , the algorithm converges to within four decimal places of the MLEs by the 11th iteration. It is apparent from (3.9)(3.10) that convergence can be accelerated by taking no to be small, and with n0 = 0 the MLE is achieved after just one iteration regardless of the starting values. We shall see in Section 3.3.3 that the rate of convergence in this example is determined by n0(n0+n1), the proportion of observations that are missing. More generally, the convergence rate of EM is governed by the fractions of information about components of θ missing due to nonresponse. (t+1)
(t)
Example 2: Two binary variables with missing data on both. Suppose that Y1 and Y 2 are two potentially related dichotomous variables, each taking values 1 or 2. If the n units in a sample are iid, the complete data may, without loss of information, be reduced to an array of counts x=(x 11, x12, x21, x22) having a multinomial distribution, where xij is the number of sample units having Y1 = i and Y2 = j. Let θ = (θ11, θ12,, θ21,
θ22), where θij is the probability that a unit has Y1 = i and Y2 = j. We will use the notation x ~ M(n, θ) to indicate that x has a
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multinomial distribution with index n and parameter θ. Because the completedata loglikelihood is linear in the cell counts xij,
l(θ  x ) = x11 log θ11 + x12 log θ12 + x21 log θ 21 + x22 log θ 22 , the counts are the sufficient statistics and the MLEs are found by equating each xij with its expectation nθ ij ; hence the complete data MLEs are simply the sample proportions θˆij = xij / n for i, j=1,2. Table 3.1. Example of EM for incomplete univariate normal data with n1 = 10 values observed and n0 = 3 values
If missing values occur on both Y1 and Y2, we can partition the sample into three parts denoted by A , B and C, respectively, where A includes units having both variables observed, B includes those having only Y1 observed and C includes those having only Y2 observed. (Any units that have neither Y1 nor Y2 observed contribute nothing to the observeddata likelihood and may be excluded from the analysis under
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ignorability.) Each completedata count x ij can then be expressed as the sum of contributions from each of the three sample parts, xij = xijA + xijB + xijC . Although xijA is observed,
xijB and xijC are not; for sample parts B and C, we observe only the marginal totals xiB+ = respectively. The A B C Yobs = xij , xi + , x + j : i, j = 1, 2
{
}
xiB1 + xiB2 and x +Cj = x1Cj + x2Cj , observed data can be displayed as in Table
3.2, with a 2 × 2 table crossclassifying the units in A by Y1 and Y2, a 2 × 1 table classifying the units in B by Y1 alone, and a 1 × 2 table classifying the units in C by Y2 alone. A convenient feature of the multinomial distribution is that if we regard the sum of any set of components of x as fixed, the conditional distribution of those components becomes another multinomial and is independent of the remaining components (e.g. Agresti, 1990). For example, the c o n d i t i o n a l d i s t r i b u t i o n o f ( x11, x 12) g i v e n Table 3.2. Classification of sample units by two incompletely observed binary variables
x1+ =x 11 +x 12 is multinomial with parameter (θ 11 /θ 1 +,θ12/θ1+) where θ 1+ = θ 11+ θ 12; furthermore, (x 11 , x12) is conditionally independent of (x21, x22). Applying this property within parts B and C of the sample, the predictive distribution of the missing data given θ and the observed data becomes a set of independent multinomials or a product multinomial,
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( xiB1 , xiB2 )Yobs ,θ ~ M ( xiB+ , (θ i1 / θ i+ ,θ i2 / θ i+ )), i = 1, 2. ( xiC1 , xiC2 )Yobs ,θ ~ M ( x+C j , (θ1 j / θ + j ,θ 2 j / θ + j )), j = 1, 2. The Estep of EM replaces the unknown counts xijB and xijC in xij by their conditional expectations under an assumed value for θ,
) (
(
E xij  Yobs , θ = E xijA + xijB + xijC  Yobs , θ
)
= xijA + xiB+θ ij / θ i+ + x+C jθ ij / θ + j . The Mstep then estimates θ ij by E(xijYobs, θ). Combining the two steps yields a single iteration of EM, (t +1)
θ ij
=
n −1 xijA
+
θ (t ) ij B xi + t θ( ) i+
+
θ (t ) ij C x + j t , θ( ) + j
i , j=1,2, an expression first given by Chen and Fienberg (1974). The data in Table 3.3, previously analyzed by Kadane (1985), were obtained through the National Crime Survey conducted by the U.S. Bureau of the Census. Housing unit occupants were interviewed to determine whether they had been victimized by crime in the preceding sixmonth period. Six months later the units were visited again to determine whether the occupants had been victimized in the intervening months. Discarding the 115 households that
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Table 3.3. EM algorithm applied to victimization status of households on two occasions
did not respond to the survey at either visit, we are left with a sample of n = 641 for which responses are available at one or both occasions. The EM algorithm for this example converges quite rapidly. Starting from a table of uniform probabilities (all θ ij = 0.25), the estimated cell probabilities converge to four decimal places by the fifth iteration as shown in Table 3.3 (b). One way to summarize the association between two binary variables is by the crossproduct or odds ratio θ θ ω = 11 22 , θ12θ 21 with ω = 1 under independence. The ML estimate of the odds ratio is θˆ θˆ / θˆ θˆ =3.57. Households that were
( 11 22 ) ( 12 21 )
victimized during the first period appear to be more than 3.5 times as likely, on the odds scale, to have been victimized in the second period than households that were crimefree in the
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first period. The question of whether this result is statistically significant will be addressed shortly. 3.2.3 EM for posterior modes The EM algorithm is typically presented as a technique for finding MLEs. As pointed out by Dempster, Laird, and Rubin (1977), however, EM may also be used to compute posterior modes, values of θ for which the observeddata posterior density rather than the observeddata likelihood is highest. Because the completedata posterior density under the prior π(θ) is P(θY) ∝ P(θY) π (θ), it follows from (3.2) that
log P(θ  Y ) = l(θ  Yobs ) + log P(Ymis  Yobs , θ ) + log π (θ ) + c. Averaging this equation over the predictive distribution of Ymis (t) given θ = θ gives
(
)
(
)
Q * θ  θ (t ) = log P(θ  Yobs ) + H θ  θ (t ) + log π (θ ) + c, where
(
) (
)
( )
Q * θ  θ (t ) = Q θ  θ (t ) + log π θ (t ) , and the functions Q(θθ ) and H(θθ ) are defined as before. (t)
(t)
If we choose the next iterate θ to satisfy
(t+1)
(
)
(
to maximize Q*(θθ ) i.e. (t)
)
Q * θ (t +1)  θ (t ) ≥ Q * θ  θ (t ) for all θ, then each iteration will increase P( θY obs ) and in a wellbehaved problem the sequence of parameter estimates will converge to the mode of P(θY obs). It is evident that when the prior π(θ ) is chosen to be a constant function over the parameter space, this algorithm reduces to the maximumlikelihood version of EM. If the prior
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is not constant the Mstep will change, requiring maximization (t) (t) of Q*(θθ ) rather than Q(θθ ). The Estep procedure will be the same as in the maximumlikelihood version, however, because the Estep is dependent upon a fixed value of θ and therefore does not involve the prior. 3.2.4 Restrictions on the parameter space Thus far little has been said about the parameter space or domain of θ. In many problems it will be the natural parameter space, the set of all values of θ for which P(Yθ ) is a valid probability density or probability function. In Example 2 of Section 3.2.2, for instance, we assumed nothing about the multinomial parameter π except the minimal requirements
π ij ≥ 0 for i,j = 1,2 and π++ = π11 + π12 + π22 = 1. In other situations, however, it is desirable to restrict θ to lie within some smaller set Θ 0, a subset of the natural parameter space which is typically of lower dimension. In the 2 × 2 contingency table, for example, we could require the cell probabilities to satisfy the condition of rowcolumn independence, πij = πi+π+j for all i and j. Appropriate choices for Θ 0 generate useful classes of models in a variety of continuous and categoricaldata contexts. It often happens that we want to test a null hypothesis θ ∈ Θ0 versus an alternative hypothesis θ ∈ v1, where Θ 0 is a lowerdimensional subset of Θ1. If θˆ is the maximizer of Θ1 1
over Θ 0 and θˆ1 the maximizer over Θ1, then the well known largesample approximation
(
) (
)
2l θˆ1  Yobs − 2l θˆ0  Yobs ~ χ d2
(3.12)
under the null hypothesis, where d = dimΘ 1 − dimΘ 0, forms the basis for a likelihoodratio test (e.g. Cox and Hinkley, 1974). If the drop in 2l(θYobs) as we move from θˆ1 to θˆ0 is
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unusually large when compared to the χ d2 distribution, then the evidence against the null hypothesis in favor of the alternative is strong. In performing this test one needs to maximize the likelihood twice, once over Θ0 and once over Θ1. Using the EM algorithm to maximize a likelihood or posterior over a restricted parameter space is conceptually no different from applying EM without such restrictions. The form of the Estep will not change, because taking the expectation of a quantity with respect to P(YmisYobs , θ ) is computationally the same whether or not θ ∈ Θ 0. The Mstep, however, will become a constrained maximization of the expected completedata likelihood or posterior over Θ 0 and hence may require special, often iterative, optimization techniques. To avoid implementing an iterative Mstep which would make the EM algorithm doubly iterative, it is often helpful to apply a recent extension of EM known as ECM, described below in Section 3.2.5. Example: testing hypotheses for an incomplete 2 × 2 table Returning to the National Crime Survey data in Table 3.3, we can apply the EM algorithm under the restriction that victimization status on the first occasion is independent of victimization status on the second occasion. The Estep is the same as in the unrestricted case, but the Mstep is different. It is well known that with complete data, the ML estimates under independence are
(
xi + x+ j ( xi1 + xi2 ) x1 j + x2 j θ˜ ij = = n2 n2
)
(3.13)
for i,j=1,2. The Mstep of EM uses (3.13) with each count xij replaced by E(xijθ,Yobs ), the output of the Estep. Starting from a uniform table, EM converges after four iterations to the restricted ML estimate
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(
θ˜ = θ˜11, θ˜12 , θ 21, θ˜22
)
= (0.6631, 0.1329, 0.1699, 0.0341). Notice that the estimated crossproduct ratio θ˜11θ˜22 / θ˜12θ˜21 = 1.00 satisfies the independence condition as required. To test whether rowcolumn independence is plausible, we can perform a likelihoodratio test. For the households that responded to the survey on both occasions, the loglikelihood contribution has the form of a completedata multinomial loglikelihood,
(
)(
)
A A A A l A (θ  Yobs ) = x11 log θ11 + x12 log θ12 + x21 log θ 21 + x22 log θ 22 . The households that responded to the survey only on the first occasion provide information only about the marginal probabilities θ1+= θ 11+ θ 12 and θ2+=θ21+θ22; their loglikelihood contribution has the form of a binomial,
lB (θ  Yobs ) = x +B1 log(θ11 + θ12 ) + x2B+ log(θ 21 + θ 22 ). Similarly, the households that responded only on the second occasion contribute
lC (θ  Yobs ) = x+C1 log(θ11 + θ 21 ) + x+C2 log(θ12 + θ 22 ). The observeddata loglikelihood is the sum of the loglikelihood contributions from each missingness pattern,
l(θ  Yobs ) = l A (θ  Yobs ) + lB (θ  Yobs ) + lC (θ  Yobs ). Plugging in the observed data and the restricted ML estimate θ˜ yields l θ˜  Yobs = 575.19, the highest loglikelihood
(
)
achievable under independence. Plugging in the unrestricted ML estimate θˆ from Table 3.3 (b) yields l θˆ  Yobs = 562.50.
(
)
The likelihoodratio test statistic is thus 2(562.50 + 575.19) =
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25.38, which is well beyond the plausible range of a χ12 random variate. We therefore conclude, not surprisingly, that victimization status on the two occasions is related. Perhaps a more interesting and appropriate question for these data is not whether victimization during the two periods seems independent, but whether the victimization rate seems to have changed over time. That is, it may be of interest to test the hypothesis of marginal homogeneity, θ k+ = θ + k, k = 1,2, which for a 2 × 2 table is equivalent to the hypothesis of offdiagonal symmetry, θ 1 2 = θ 2 1 . With complete data, a commonly used procedure for assessing marginal homogeneity/symmetry in a 2 × 2 table is McNemar s test , in which the statistic
M=
x12 − x21 x12 + x21
is compared to the standard normal distribution (McNemar, 1947; Agresti, 1990). With incomplete data, we can perform a likelihoodratio test by maximizing the likelihood subject to θ 1 2 = θ 21 . Under, this restriction, we may collapse the off diagonal cells into a single cell and express the completedata loglikelihood as that of a trinomial,
l(θ  Y ) = x11 log θ11 + ( x12 + x21 ) log(2θ12 ) + x22 log θ 22, with sufficient statistics x11, x 22 and (x 12 + x21); the moment ( equations (3.7) lead immediately to the ML estimates θ11 = ( ( x11/n, and θ12 = θ 21 = (x12+x21)/(2n). Revising the Mstep to include the marginal homogeneity/symmetry restriction, EM quickly converges to ( θ = (0.6970, 0.1173, 0.1173.0, 0685).
( The loglikelihood at this estimate is l θ  Yobs = 564.25, so
(
)
the statistic for testing the null hypothesis of marginal homogeneity/symmetry is 2(562.50 + 564.25) = 3.50 with a
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(
)
P χ12 ≥ 3.50 = 0.06. The evidence against marginal homogeneity/symmetry is thus fairly strong. Extensions of this procedure for testing marginal homogeneity in r × r tables for r > 2 are more complicated, because ML estimates do not exist in closed form; the Mstep may be carried out using techniques of nonlinear programming, as discussed by Shih (1987).
pvalue of
3.2.5 The ECM algorithm The ECM or ExpectationConditional Maximization algorithm is a useful extension of EM for situations where the Mstep cannot be carried out without iteration (Meng and Rubin, 1993). ECM replaces a complicated Mstep with a sequence of simpler conditional or constrained maximizations known as a CMstep. ECM retains the reliable convergence properties of EM while simplifying, and often reducing, the required computations. The CM step of ECM is comprised of S conditional maximizations in which the Q function is maximized not over the entire parameter space as in (3.5), but over a smaller set in which a vectorvalued function g s(θ ) is fixed at its previous value for s = 1,2, , S. The set of functions G = {g s(θ ) : s = 1, , S } must be preselected and must satisfy precise conditions defined by Meng and Rubin (1993). Once a is specified, one iteration of ECM proceeds as follows. Given the (t) current value of the parameter θ , first perform an Estep to obtain Q(θθ ) as in the EM algorithm. Then find θ (t)
(t+1)
by
= θ
(t+1)
maximizing Q(θθ ) subject to the constraint (t)
gs (θ ) = gs θ (
t + ( s −1) / S )
for s=1,2,...,S. The resulting parameter value θ
(t+S/S)
becomes the input to the next Estep. Clearly Q(θ
(t+1)
θ ) must (t)
be at least as large as Q(θ θ ), so subsequent iterations will never decrease the observeddata loglikelihood. Moreover, it (t)
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(t)
can be shown that a stationary point of ECM, i.e. a value θ
(t)
such that θ = θ , is also a stationary point and typically a maximum of the observeddata loglikelihood. The basic condition required of the set of constraints G in ECM is that repeated application of the CMstep in the absence of missing data would result in the likelihood being maximized over the whole parameter space. Many, but not all, iterative algorithms used for maximizing likelihoods with complete data can be interpreted as repeated application of a CMstep. One example is the method of cyclic ascent or iterated conditional modes (Besag, 1986), in which θ i s partitioned into a set of S subvectors and the likelihood function is successively maximized with respect to each subvector holding the others constant. Another example is iterative proportional fitting of loglinear models for contingency tables (Bishop, Fienberg, and Holland, 1975), in which the estimated cell probabilities are proportionately adjusted at each step to match the observed cell proportions on sets of margins determined by the model. These algorithms have often been regarded as less desirable than gradient methods such as NewtonRaphson, because they tend to converge more slowly in completedata problems. When paired with an Estep and applied to incomplete data, however, they tend to produce computationally stable and reliable ECM algorithms that are guaranteed to increase the likelihood at each step. Our uses of the ECM algorithm will be confined to loglinear models for categorical data (Chapter 8) and models for mixed continuous and categorical data that employ loglinear constraints (Chapter 9). Other examples of ECM and further references are given by Meng and Rubin (1993). (t+1)
(t)
3.3 Properties of EM 3.3.1 Stationary values For incompletedata problems, the most attractive features of EM relative to other optimization techniques are its simplicity
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and its stability. Rather than maximizing the potentially complicated function l(θYo b s ) directly, we repeatedly maximize the Q function, which is typically much easier and often equivalent to finding MLEs with complete data. Moreover, successive iterations of EM are guaranteed never to decrease l(θYobs ) which is not generally true of gradient methods like NewtonRaphson. In practice, evaluating l(θYobs ) at each step to ensure that it is increasing is often helpful for assessing the progress of EM and for detecting and diagnosing programming errors. As with any optimization technique, however, EM is not guaranteed to always converge to a unique global maximum. In wellbehaved problems the function l(θYobs ) is unimodal and concave over the entire parameter space, in which case EM converges to the unique MLE θˆ from any starting value. Exceptions occur both in theory and in practice, however, and one needs to be alert to detect these abnormalities when they arise. Multiple modes Table 3.4 shows a hypothetical bivariate dataset reported by Murray (1977) with twelve sample units and four missing values for each variable. Under the assumption that the complete data are iid observations of a bivariate normal vector (Y1, Y2), one may calculate the observeddata likelihood using (2.10). It is apparent that the likelihood function is symmetric with respect to Y1 and Y2, and that the marginal means (µ1, µ2) and variances (σ11, σ22) are relatively better estimated than the correlation coefficient ρ = σ 12 / σ 11σ 22 . Using analytical methods or the EM algorithm for multivariate normal data to be presented in Section 5.3, one may Table 3.4. Bivariate dataset with missing values
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verify that l(θYobs) has two modes, one at θ = (µ1,µ2,σ22,ρ) = (0, 0, 2.67, 2.67, 0.5) and the other at θ = (0, 0, 2.67, 2.67, 0.5). EM will converge to the first mode if started with ρ > 0 (0)
and to the second mode if ρ < 0. The bimodality in this example is due to the symmetry of the data and the unusual pattern of missingness in which the observations with high leverage, i.e. those that would contribute the most information about ρ, have one of the two components missing. In real multivariate datasets where the data are sparse and/or the missingness pattern is unusually pernicious, multiple modes do sometimes occur. Unlike this symmetric example in which the likelihood values at the two modes are equal, there will typically be one major and one or more minor modes, and the mode to which EM converges may (0) change depending on the choice of θ . To detect multiple modes, it is helpful to run EM from a variety of starting values to see whether it always converges to the same answer. (0)
Saddlepoints A saddlepoint is a value of θ at which the directional derivatives of l(θYobs ) are zero but which is neither a local maximum nor a local minimum. EM could possibly converge to a saddlepoint, but in practice this is quite rare. Convergence to a saddlepoint requires not only that the saddlepoint exist, but that the successive iterates of θ approach it only from certain directions. The loglikelihood for the data in Table 3.4 has a saddlepoint at θ = (0, 0, 2.5, 2.5, 0), and EM converges to it if started from ρ
(0)
= 0. However, even a very slight
perturbation from ρ = 0 will cause EM to leave the saddle and go to one of the two modes. In real data examples convergence to a saddlepoint is rarely encountered, and thus saddlepoints are not a cause for concern.
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Likelihood ridges It may happen that the maximum value of the likelihood function is achieved not at a single value of θ but at a whole continuum of values. This phenomenon occurs when one or more components or functions of θ are inestimable in the sense that they do not appear in the likelihood, and thus l(θYobs) is the same for any value of those components; that is, l(θYobs ) is flat in certain directions. Consider the bivariate dataset shown below. 2
1
0
1
2
?
?
?
?
?
?
?
?
?
?
2
1
0
1
2
Under the bivariate normal model l(θYmis) is the sum of two completedata loglikelihoods, one pertaining to (µ1, σ11), and the other pertaining to (µ 2, σ22 ), and the correlation ρ is inestimable. The maximum value of l(θYmis) is achieved for µˆ 1 = µˆ 2 = 0 , σˆ 11 = σˆ 22 = 2 and any value of ρ, so the likelihood is said to have a onedimensional ridge. If EM were applied to this dataset from various starting values, it would converge to different points on the ridge. When the likelihood has a ridge, any value along the ridge is a stationary value of EM. The algorithm does not wander aimlessly on the ridge but stops once the ridge is reached. When two normal variables are not observed together, the correlation between them will be inestimable and the likelihood will have a ridge. Similar results apply to incomplete datasets with three or more variables. Consider the trivariate case where Y 1 and Y 2 are sometimes observed together, Y1 and Y3 are sometimes observed together, but Y 2 and Y 3 are never jointly observed; in this case it can be shown that the partial correlation of Y2 and Y 3 given Y1 is
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inestimable. Estimability of parameters in multivariate normal datasets where not all variables are observed together is discussed by Rubin (1974) and Rubin and Thayer (1978). It should be noted that merely having joint observations of all variables is not sufficient to guarantee uniqueness of the MLE, as in the example below. 2
1
1
2
0
0
?
?
?
?
?
?
?
?
0
0
2
1
1
2
Here the two joint observations of Y1. and Y2 are identical and thus provide no information about ρ. Boundary estimates Yet another abnormality is an ML estimate on the boundary of the parameter space. Consider the dataset below for which the ML estimates are µˆ 1 = µˆ 2 = , σˆ 11 = σˆ 22 = 1.80 and = ρˆ = 1. 2
0
0
2
1
1
?
?
?
?
?
?
?
?
1
1
2
0
0
2
If convergence is assessed by relative changes in the components of θ = (µ 1,µ2,σ11,σ22,ρ) then EM will converge reliably in this example. If ρ is examined on some openended scale, however (for example, the familiar Fisher's ztransformation
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1+ ρ z = 12 log , 1− ρ
(3.14)
which takes values on the whole real line) then the iterations of EM will appear to diverge as z → −∞ . General comments on the method of maximum likelihood It is important to note that the abnormalities described above, multiple modes, saddlepoints, ridges and boundary solutions, are not shortcomings of the EM algorithm but inherent features of l(θYobs ) that would impact any optimization method. Indeed, when such features exist, EM is often remarkably wellbehaved in comparison with other computational methods. With incomplete multivariate data, these abnormalities Are typically associated with small samples, high rates of missingness and models that are clearly overparameterized (i.e. having too many parameters) relative to the amount of information in Yobs. In the data of Table 3.4, for example, an unusually large portion of the information in Y about ρ is concentrated in Ymis and inferences about ρ will be highly sensitive to untestable assumptions about missing data and the missingdata mechanism. From a theoretical point of view, the desirability of ML estimates stems primarily from their largesample properties. Under suitable regularity conditions, MLEs are asymptotically unbiased, normal, and efficient with variance determined by the curvature of the loglikelihood near the mode (e.g. Cox and Hinkley, 1974). In large samples the loglikelihood function tends to be unimodal and approximately quadratic. In such cases an MLE and an estimate of its variance provide an excellent summary of the data s information about θ , and largesample procedures such as asymptotic confidence intervals, likelihoodratio tests, etc. will tend to be reliable. When this is not the case, however, when the loglikelihood is oddlyshaped with multiple modes, suprema on the boundary, etc., the behavior of largesample procedures may be seriously impaired and attractiveness of the ML method is greatly diminished.
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When abnormalities are found in l(θYobs ), the analyst is faced with several options. One option is to reduce the size of the model by eliminating parameters or imposing restrictions on the parameter space. Another possibility is to introduce additional information about θ through a prior distribution π and base inference on the observeddata posterior P( θYobs) rather than the observeddata likelihood. A posterior mode will tend to be a better estimate of θ than the MLE when substantial prior knowledge about θ is available. Even when prior knowledge is scarce, however, we will find that adding small amounts of information through π may be a useful technique for ensuring that EM converges to a unique value of θ in the interior of the parameter space. When data are sparse or missing values occur in such a way that one or more components of θ are poorly estimated, a judiciously chosen prior may greatly improve the numerical stability of computations and perhaps even strengthen the estimate of θ from a statistical point of view. 3.3.2 Rate of convergence Like any algorithm for successive approximation, EM implicitly defines a function that maps the parameter space to T itself. Let θ = (θ 1,θ2, . . . , θk ) be the kdimensional parameter. Denote a single iteration of EM by
( )
( ) ( )
)
θ (t +1) = M θ (t ) = ( M1 θ (t ) , M2 θ (t ) ,... Mk θ (t )
T
so that both the E and Msteps are incorporated into the vector (t) function M. Expanding M(θ ) in a Taylor series about θˆ gives a firstorder approximation
( ) ()
( )(
M θ (t ) − M θˆ ≈ M ′ θˆ θ (t ) − θˆ
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)
(3.15)
in the neighborhood of θˆ , where M ′(θ ) is the k × k firstderivative or Jacobian matrix for M(θ ) with typical element ∂Mi(θ )/∂θ j. If θˆ is a stationary value of EM, then M( θˆ ) = θˆ and (3.15) becomes
(θ ( or ε
(t+1)
t +1)
)
( )(
− θˆ ≈ M ′ θˆ θ (t ) − θˆ
)
(3.16)
(t) t t = Dε , where ε ( ) ≈ θ ( ) − θˆ
is the error in approximation at step t and D is shorthand for M ′ θˆ . EM s
()
convergence is thus said to be linear, because ε
(t+1)
is
approximately a linear transformation of ε near the mode. NewtonRaphson and other superlinear methods have the property that D = 0 so the Taylor series approximation in (3.15) is dominated by the smaller secondorder term. The speed at which EM converges in any particular application is determined by the rate matrix D. In the case of a t +1 t scalar parameter we have ε ( ) ≈ D ε ( ) where D is a single (t)
number between 0 and 1. The convergence will be rapid when D is near zero and slow when D is near one. The situation for k ≥ 2 is more complicated, however, and depends on the eigenstructure of D. Any vector v such that Dv = λv for some constant λ is said to be an eigenvector of D, and λ is its associated eigenvalue. The eigenvalues must also satisfy the equation D − λI = 0 , and because the determinant of a k × k matrix is a polynomial of order k this equation has at most k distinct roots. When the roots λ1,λ2,...,λk are distinct the corresponding eigenvectors v1, v2,...,vk are linearly independent, and any kdimensional vector can be written as a linear combination of the eigenvectors. In particular, we can write the error vector ε (t ) = θ (t ) − θˆ as
ε (t ) − c1v1 + c2 v2 + L ck vk .
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Then the error at the next iteration becomes
ε ( t +1) ≈ D( c1v1 + c2 v2 + L ck vk ) ≈ c1λ1v1 + c2 λ 2 v2 + L + ck λ k vk , and after r iterations,
ε (t + r ) ≈ c1λ1r v1 + c2 λr2 v2 + L + ck λrk vk . In ordinary problems all the eigenvalues will satisfy 0 ≤ λj < 1 and successive iterations of EM beginning from any θ in a neighborhood of θˆ will shrink the error toward zero. If θˆ is a saddlepoint then one or more eigenvalues could exceed one. (t) If ε happens to be precisely orthogonal to the eigenvectors corresponding to those eigenvalues, then EM will converge to (t) the saddlepoint. For a randomly chosen θ in the neighborhood of θˆ , however, this will happen with negligible probability, so in most cases the iterates will diverge from a saddlepoint. One or more eigenvalues equal to one indicates that the likelihood is flat in certain directions and is maximized along a ridge (Dempster, Laird and Rubin, 1977). (t)
The missing information principle It is well known that in regular problems the largesample precision of the MLE is determined by the curvature of the loglikelihood function. With complete data, the Fisher information is defined to be
∂2 I * (θ  Y ) = − 2 l(θ  Y )P(θ  Y )dY , ∂θ
∫
(3.17)
where ∂ l(θY)/∂θ is the k × k matrix with typical element 2 ∂ l(θY)/∂θi∂θj. One estimate of the covariance matrix of θˆ in 2
2
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large samples is
[I * (θˆ  Y )]
−1
the inverse of the Fisher
information matrix evaluated at the completedata MLE. Another, asymptotically equivalent, estimate is the inverse of
( )
∂2 I θˆ  Y = − 2 l(θ  Y ) ˆ , θ =θ ∂θ
(3.18)
which fixes Y In the loglikelihood function at its realized value rather than averaging over its distribution. With incomplete data, differentiating (3.2) twice yields
−
∂2 ∂2 ∂2 l θ  Y ) = − 2 l(θ  Yobs ) − 2 log P(Ymis  Yobs , θ ). 2 ( ∂θ ∂θ ∂θ
Taking the expectation of this over P(YmisYobs,θ), we obtain a fundamental relationship: the complete information is equal to the observed information plus the missing information. This relationship, called the missing information principle by Orchard and Woodbury (1972), was also investigated by Dempster, Laird and Rubin (1977), Louis (1982) and Meng and Rubin (1991a). Assuming sufficient regularity to interchange the order of differentiation and integration, we can write the complete information as
Ic (θ ) = −
∂2 Q(θ  θ ), ∂θ 2
the observed information as
Io (θ ) = −
∂2 l(θ  Yobs ), ∂θ 2
and the missing information as
Im (θ ) = − so that
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∂2 H (θ  θ ), ∂θ 2
Ic (θ ) = Io (θ ) + Im (θ ). Note that each quantity in (3.19) is a function of Y obs although this fact has been suppressed in the notation. Also note that the concept of information used in (3.19) is more consistent with (3.18) than with (3.17), because we have fixed Yo b s at its realized value and averaged only over the distribution of the unknown Y mis . A natural largesample estimate of the covariance matrix with incomplete data is
I 01
−1
∂2 , θˆ = − 2 l(θ  Yobs ) θ =θˆ ∂θ
()
(3.20)
where θˆ is now the observeddata MLE, the maximizer of l(θYobs). Missing information and convergence Dempster, Laird and Rubin (1977) established an important connection between these information quantities and D = M ′ θˆ , the asymptotic rate matrix of EM. In regular problems
()
is obtained as a solution to ∂ Q(θθ )/∂θ=0, they where θ showed that (t+1)
(t)
() ()
D = Ic−1 θˆ Im θˆ .
(3.21)
In the extreme case where Y mis provides no additional information about θ not already contained in Yobs, then I θˆ m
()
= 0 and (3.21) implies that EM essentially converges in a single iteration. More generally, for a scalar parameter (3.21) implies that D is the ratio of the missing information to the complete information. For brevity we will call this ratio the fraction of missing information, although a more precise term would be the fraction of information missing due to nonresponse. If we denote the fraction of missing information
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in the scalar case by D = λ, then each iteration approximately multiplies the error by λ,
(θ (
t +1)
) (
)
− θˆ ≈ λ θ (t ) − θˆ ,
(3.22)
which demonstrates one of the fundamental properties of the EM algorithm: the rate of convergence of EM is determined by the fraction of missing information. When θ is a vector of length k > 1 the fraction of missing information is no longer a number but a matrix; yet a result similar to (3.22) holds for multiparameter problems as well. Suppose we order the eigenvalues of D so that λ1≥λ2≥⋅⋅⋅≥λk, and let v1,v2,...,vk be eigenvectors of D corresponding to these ordered eigenvalues. As before, we can write the error vector as
ε (t ) = θ (t ) − θˆ = c1v1 + c2 v2 + ck vk for some c1,c2, . . . , ck, so that after r iterations
ε (t + r ) ≈ c1λ1r v1 + c2 λr2 v2 + L + ck λrk vk . By analogy with (3.22) we may regard λ1,λ2,...,λk as fractions of missing information corresponding to the particular directions v1, v2,...,vk . Moreover, if λ2 is strictly less than λ1, we have
ε (t + r ) ≈ λ1r (c1v1 + R),
(3.23)
where the remainder term r
r
λ λ R = c2 2 v2 + L + ck k vk λ1 λ1 approaches zero at a rate determined by λ 2/λ1. Thus in the vicinity of the mode
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(θ (
t +1)
) (
− θˆ ≈ λ1 θ (t ) − θˆ
)
(3.24)
where λ1, is the largest eigenvalue of D . If the s largest eigenvalues happen to be equal, then (3.23) becomes
ε (t + r ) ≈ λ1r (c1v1 + L + cs vs + R) where R approaches zero at a rate determined by λs+1/λ1, and (3.24) still holds. In the multiparameter case, we can thus say that EM s rate of convergence is governed by the largest fraction of missing information. Exceptions to this rule are (t) possible; for example, if ε happens to be precisely orthogonal to the eigenvector(s) corresponding to the largest eigenvalue, then convergence will be dominated by the next largest eigenvalue. For most realdata problems, however, this basic result does hold. Further results and discussion on the convergence of EM are given by Meng (1990). 3.3.3 Example In Example 1 of Section 3.2.2, we derived the EM algorithm for incomplete univariate normal data and applied it to the n1 = 10 observations displayed in Table 3.1 (a) by assuming that an additional n0 = 3 observations were missing. By varying the choice of no we can make the rate of convergence in this example arbitrarily large or small. Figure 3.1 displays the iterations of EM in this twoparameter problem from a variety of starting values under n0 = 10 and n 0 = 90, corresponding to missingness rates of 50% and 90%, respectively. For visual clarity the variance ψ is shown on the log scale. With n0 = 10 the convergence is quite rapid, whereas for n0 = 90 it is much slower. Unlike gradient methods EM does not necessarily follow the path of steepest ascent, but often climbs the loglikelihood surface by a more circuitous route.
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Figure 3.1. Iterations of EM from various starting values for univariate normal data with no = 10 and no = 90.
In simple examples like this one, it is feasible to investigate the convergence properties analytically. The elements of the t matrix M ′ θ ( ) , obtained by differentiating (3.9)(3.10), are
( )
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∂µ (t +1)
n = 0, n
∂µ (t )
∂µ (t +1)
= 0,
∂ψ (t )
∂ψ (t +1) ∂µ (t )
∂ψ (t +1) ∂ψ (t )
where yobs = n1−1
(
)
n n = 2 0 2 1 µ (t ) − yobs , n n = 0, n
∑ in=1 yi 1
is the observeddata MLE for µ .
t Evaluating these derivatives at the mode (i.e. taking θ ( ) = θˆ ) gives
()
0 n0 / n D = M ′ θˆ = n0 / n 0
.
The eigenvalues of this matrix are λ 1 = λ2 = λ = n0/n. Whenever eigenvalues are repeated, the corresponding eigenvectors are not uniquely defined. In fact, because D is proportional to the identity matrix in this example, any error (t) vector ε is an eigenvector. Yet the overall convergence rate is still governed by A, and a single iteration of EM can be t +1 t expressed as ε ( ) ≈ λε ( ) . In this and other univariate examples, the fraction of missing information λ is also the rate of missing observations. In multivariate applications, however, the fractions of missing information corresponding to various components of θ will typically differ from the rates of missing observations, both overall and on a variablebyvariable basis, depending on the pattern of missingness and the observed interrelationships among variables.
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3.3.4 Further comments on convergence Monitoring and detecting convergence Two basic methods for monitoring the convergence of EM (t) involve (a) successive parameter values θ , and (b) successive values of the observeddata loglikelihood l(θ Yo b s ). In practice both are quite useful. Because each iteration of EM is guaranteed never to decrease the likelihood, evaluating l(θYobs ) at each step is helpful for detecting programming errors as well as for monitoring the progress of EM in specific examples. Convergence is typically judged by examining changes in individual components of θ = (θ 1,θ2,..., θk) from one iteration to the next. If the changes are all relatively small, for example, if (t)
θ (j ) − θ (j t
t −1)
t ≤∈θ (j )
for j = 1, 2,..., k and a suitably small ∈ (say, 0.0001), then we may say that EM has converged by iteration t. If the elements of θ continue to change for many iterations with very little increase in l(θYobs), then it should be taken as a sign that the loglikelihood is nearly flat in certain directions and that one or more functions of θ are very poorly estimated. Often in these problems there is little to be gained from additional iterations, because the value of θ to which EM would ultimately converge has loglikelihood only slightly higher than the current value; the observed data do little to distinguish between these values of θ . Slow convergence is also a sign that the fractions of missing information corresponding to certain aspects of θ are close to one, and that most of the information about them is being contributed by P(YmisYobs,θ ), the model for the missing data, rather than by Yobs. Because the correctness of P(YmisYobs,θ) rests on both the completedata model and the ignorability assumption, slow
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convergence warns us that inferences about certain aspects of θ are highly modeldependent. This does not automatically mean that all inferences about θ are suspect, because in some problems (e.g. when the missing data are missing by design and therefore known to be MAR) the model may be quite trustworthy. In other problems, however, particularly when it is not known whether the data are MAR, slow convergence is a useful warning that the estimate of θ may rest largely on our assumptions about the unknown Ymis, rather than on the known Yobs . If so, then a simplification of the model, perhaps by imposing restrictions on the parameter or by eliminating variables with high rates of missingness, may be a sensible strategy. Asymptotic covariance matrices from EM One drawback of EM relative to other optimization techniques is that it does not automatically provide standard errors associated with the parameter estimates. The asymptotic covariance matrix I0−1 θˆ defined in (3.20) is not readily
()
available because implementation of EM does not require calculation of the derivatives of l(θY obs ), which are often complicated and tedious to derive. In a comment on the original EM paper, Smith (1977) noted that for a scalar parameter the iterations of EM provide a good estimate of λ. From (3.22) it is apparent that after a large number of iterations λ will be well approximated by ε /ε , t t +1 t t t −1 or equivalently, by λˆ( ) = θ ( ) − θ ( ) / θ ( ) − θ ( ) , (t+1)
(
)(
because
θ (t +1) − θ (t ) ε (t +1) − ε (t ) λε (t ) − ε (t ) = λ. ≈ = λˆ(t ) = t θ ( ) − θ (t −1) ε (t ) − ε (t −1) ε (t ) − λ−1ε (t )
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(t)
)
This estimate of λ may be used to obtain a largesample standard error for θˆ , because (3.19) and (3.21) imply that the asymptotic variance of θˆ is
()
()
I0−1 θˆ = Ic−1 θˆ / (1 − λ ) .
()
For exponential families, Ic−1 θˆ is the completedata asymptotic variance calculated from the expected sufficient statistics obtained at the last Estep. In other words, a standard error for θˆ can be obtained simply by inflating the completedata standard error by 1 − λ , where λ is estimated from the steps of EM. In most realdata applications, of course, θ is multidimensional, and obtaining a covariance matrix for θˆ is less straightforward. A general numerical procedure for approximating given by Meng and Rubin (1991a), is called Supplemented EM or SEM. From (3.19) and (3.21) it can be shown that
()
()
()
I0−1 θˆ = Ic−1 θˆ + Ic−1 θˆ ( I − D) −1 D.
(3.25)
()
Meng and Rubin show how the elements of D = M ′ θˆ can be estimated by repeated runs of a forced EM in which all but one of the individual elements of θ are fixed at their MLEs. The procedure corresponds to numerical differentiation with step sizes determined by EM. For a k dimensional parameter one needs to perform k runs of forced EM, one to estimate each row of D. Using (3.25), the numerical estimate of D is then combined with the completedata asymptotic covariance matrix Ic−1 θˆ to produce I0−1 θˆ . Implementation of SEM
()
()
thus requires only the code for computing an asymptotic covariance matrix from complete data and the code for the EM algorithm itself. An asymptotic covariance matrix for θˆ is typically used in conjunction with the largesample normal approximation
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(θˆ − θ ) ~ N(0, I0−1(θˆ)),
(3.26)
which can be justified from either a frequentist or a Bayesian perspective (e.g. Cox and Hinkley, 1974). This approximation is expected to work well when the sample size is sufficiently large that l( θYobs ) resembles a quadratic function in the vicinity of d. In practice the validity of (3.26) often depends on the scale of the parameter, and transformations may need to be applied to one or more components of θ to make the approximation more accurate. Transformations to improve normality will alter the form of the completedata covariance matrix Ic−1 θˆ . When l(θYobs) has unusual features such as
()
multiple modes, ridges or suprema on a boundary, then the value of an asymptotic covariance matrix is dubious at best, and (3.26) should not be used for making inferential statements about θ. For many of the multivariate models and data examples in this book, the potentially large number of parameters makes the implementation of SEM computationally prohibitive. In some cases the validity of the normal approximation (3.26) will be suspect as well, even on a carefully chosen scale for θ. For this reason, we will adopt simulation rather than asymptotic approximation as the primary method of inference. In multiparameter problems, simulation is often feasible even when the dimension of θ is very large. Moreover, simulationbased inferences can be made about any transformation or function of θ with no special analytic work involved. Elementwise rates of convergence Apart from obtaining asymptotic standard errors, it may still be useful to examine rates of convergence corresponding to the individual elements of θ = (θ 1,θ2, , θk). These rates may be estimated from the iterations of EM by
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θ (j ) − θ (j ) t λˆ(j ) = t t −1 θ (j ) − θ (j ) t +1
t
(3.27)
for j1,2,...,k at suitably large values of t. Unlike the eigenvalues of D, which are the fractions of missing information corresponding to the eigenvectors, these elementwise rates pertain to u 1,u2,...,uk, where uj is a unit vector with a one in position j and zeroes elsewhere. As noted by Meng and Rubin (1991a), in most cases (3.27) will estimate the largest eigenvalue of D, because uj will have a nonzero component corresponding to the first eigenvector. If uj happens to be precisely orthogonal to the first s eigenvectors, then (3.27) will converge to the (s+1) st largest eigenvalue of D. Consequently, the elementwise rates of convergence typically provide the first and perhaps a few additional eigenvalues of D, which can be a useful diagnostic for assessing how much information about θ is contained in P(YmisYobs,θ) relative to Yobs. t t For the EM example in Table 3.1, estimates λˆ(1 ) and λˆ(2 ) of the elementwise rates of convergence corresponding to the mean µ and the variance ψ , respectively, are displayed in Table 3.2. As previously shown, the eigenvalues of D are both equal to n 0/n=3/13=0.2308, and the elementwise rates converge to this number quite rapidly. Very close to the mode, successive values of θ are nearly identical and computation of (3.27) becomes numerically unstable. It is generally wise to compute (3.27) using double precision arithmetic and to estimate the rates from the last few iterations before numerical instability becomes evident. Note that in a multiparameter problem these elementwise rates alone are not sufficient
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Table 3.5. Iterations of EM for incomplete univariate normal data with estimated elementwise rates of convergence
to obtain standard errors for the individual elements of θˆ . As seen from (3.25), the variance of a single element of θˆ generally depends on the entire D matrix, whereas the elementwise rates provide at most only a few eigenvalues of D. In some problems one or more components of θ may have no missing information at all. In the bivariate normal data depicted in Figure 2.2, for example, there are no missing observations of Y1 and hence µ1 and σ 11 all have no missing information. An EM algorithm for these data would converge to the ML estimates for µ1 and σ11 all in a single step from any starting value. When one or more components of θ converge immediately, the elementwise rates of convergence from the remaining components still estimate the largest fractions of missing information.
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Accelerating convergence The linear behavior of EM near the mode suggests some potentially useful methods for accelerating convergence. Rearranging (3.16), we can obtain an estimate of 0 in terms of (t) (t+1) two successive iterates θ and θ ,
(
)
θ˜ (t +1) = θ (t ) + ( I − D) −1 θ (t −1) − θ (t ) ,
(3.28)
which is typically closer to the mode than θ . This technique, commonly known as Aitken acceleration, can make a linearly convergent algorithm like EM almost superlinear. When the individual components of θ appear to be converging at the same elementwise rate, (3.24) suggests that (t+1)
(
−1 θ˜ (t +1) = θ (t ) + (1 − λ1 ) θ (t −1) − θ (t )
)
(3.29)
may also work well, where λ1 is the largest eigenvalue of D. These acceleration techniques require an estimate of D or at least its largest eigenvalue, which can be obtained by analytic methods or from the iterations of EM. The use of Aitkentype acceleration methods for EM have been investigated by Louis (1982); Laird, Lange and Stram (1987); and Lansky and Casella (1990). Another technique, proposed by Belin and Diffendal (1991), is to estimate the jth component of θ by
(
t +1 t θ˜ (j ) = θ (j ) + 1 − λˆ j
) (θ ( −1
t −1) j
t − θ (j )
)
(3.30)
for j = 1,2,...,k, where λˆ j is the estimated elementwise rate of convergence for θj given by (3.27). This third method, which may be regarded as intermediate between (3.28) and (3.29), is easier to compute than (3.28) and more appropriate than (3.29) in situations where the elements of θ appear to be converging at different rates. Care should be taken in the use of these accelerated versions of EM, as they are not guaranteed to increase the
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loglikelihood at each step. Acceleration should not be (t) employed until θ is close enough to the mode for (a) the steps of EM to be approximately linear, and (b) the estimated fractions of missing information to be stable. As previously mentioned, slow convergence of EM in an incompletedata problem should be taken as a warning that certain aspects of θ are being estimated primarily from P(YmisYobs,θ ) rather than Yobs. In such problems it is sometimes more reasonable to bail out of the current analysis and fit a simpler model, rather than to continue iterating with a model whose parameters are poorly estimated. Convergence and prior information When EM is being used to find a posterior mode rather than an ML estimate, the missing information principle described in Section 3.3.2 applies but in a slightly modified form. The decomposition of (3.19) becomes
Ic (θ ) = Io (θ ) + Im (θ ) + Iπ (θ ), where I c(θ ), Io(θ ) and Im(θ ) are defined as above, and the additional term Iπ(θ) is the information contained in the prior distribution,
Iπ (θ ) = −
∂2 π (θ ). ∂θ 2
This term will be small when π is relatively flat and large when π is sharply peaked. The basic relationship (3.21) between these information quantities and the rate matrix D = M ′ θˆ still applies,
()
() ()
D = Ic−1 θˆ Im θˆ ,
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()
but the complete information matrix Ic θˆ now includes prior information. The introduction of a prior may thus be expected to reduce the magnitude of D and accelerate the convergence of EM in most cases. In particular, this will be true when the prior introduces substantial information about those aspects of θ that are most poorly estimated, those that influence the largest eigenvalue of D. Convergence properties of ECM The ECM algorithm introduced in Section 3.2.5 shares many of the convergence properties of EM. Like EM, it increases the loglikelihood at each step and converges reliably to a local maximum or (rarely) a saddlepoint of the loglikelihood. Like EM, it also exhibits linear convergence in the vicinity of the mode. ECM can be thought of as a combination of two linearly convergent algorithms: an EM algorithm, which pertains to the incompletedata aspects of the problem, and a CM or conditional maximization algorithm, which pertains to the maximization of the likelihood in the completedata case. As pointed out by Meng and Rubin (1992a), there seems to be little advantage to replacing the linearly convergent CM step with one or more steps of a superlinear technique such as NewtonRaphson, because the overall convergence of the combined algorithm will still be linear. Moreover, unless the superlinear algorithm is run to full convergence at each Mstep, the loglikelihood would not be guaranteed to increase at each iteration. With ECM, the global and elementwise rates of convergence cannot immediately be interpreted as fractions of missing information, because the simple identity D = Ic−1 θˆ Im θˆ does not generally hold. Basic results on
() ()
ECM s rate of convergence, including relationships between the D matrix and information quantities, have been established by Meng (1994). Some of these results are counterintuitive; for instance, examples can be constructed where ECM converges more quickly than EM. A numerical method for obtaining largesample covariance matrices from ECM, called
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Supplemented ECM or SECM, is described by Meng and Rubin (1992a).
3.4 Markov chain Monte Carlo Markov chain Monte Carlo is a collection of techniques for creating pseudorandom draws from probability distributions. In recent years it has been a subject of intense interest among statisticians, spawning a wide range of applications as well as a great deal of innovative theoretical work. In a broad sense, the goal of Markov chain Monte Carlo is to generate one or more values of a random variable Z, which is typically multidimensional. Let P(Z)=f(Z) denote the density of Z, which we call the target distribution. Rather than attempting to (1) draw from f directly, we generate a sequence {Z , (2) (t) Z ,...,Z ,...} where each variate in the sequence depends in some fashion on the preceding ones, and where the stationary (t) distribution (i.e. the limiting marginal distribution of Z as t → ∞ ) is the target f. For a t sufficiently large, Z(t) is approximately a random draw from f. Markov chain Monte Carlo is attractive when f is difficult to draw from directly, but drawing each variate in the sequence is straightforward. Markov chain Monte Carlo methods have often been classified under Bayesian computation or Bayesian posterior simulation because many of the best known current applications have a strong Bayesian flavor; when viewed strictly as simulation methods, however, there is nothing inherently Bayesian about them. Also, despite the popularity of the term Markov chain Monte Carlo, depending on how the methods are viewed some of them are not strictly Markovian. In this new and rapidly evolving field, the lack of well defined and broadly accepted terminology has sometimes been a source of confusion. The reader should understand that names given to the methods below, and the definitions of these methods, are not universally accepted and may differ somewhat from what other authors have written. This list of Markov chain Monte Carlo methods is not meant to be exhaustive, but concentrates on some that have proven most useful in the analysis of incomplete multivariate
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data. Presentations in a more general setting and additional references are given by Gelfand and Smith (1990); the articles by Gelman and Rubin (1992a), Geyer (1992) and Smith and Roberts (1993) with accompanying discussions; and Tierney (1994). Applications of Markov chain Monte Carlo are discussed by Gelfand et al. (1990); Casella and George (1992); Smith and Roberts (1993); and Gilks et al. (1993), among others. A comprehensive overview including theory and applications appears in the books by Tanner (1993) and Gilks, Richardson, and Spiegelhalter (1996). 3.4.1 Gibbs sampling Gibbs sampling is the most popular and well known form of Markov chain Monte Carlo. Suppose that a random vector Z is partitioned into J subvectors,
(
)
t t t Z (t ) = Z1( ) , Z2( ) ,..., Z J( ) ,
Let P(Z) denote the joint distribution of Z, which is also the target distribution to be simulated. In Gibbs sampling, we iteratively draw from the conditional distribution of each subvector given all the others. Given the value of Z at step t, say t t t Z (t ) = Z1( ) , Z2( ) ,..., Z J( ) ,
the value of Z at step t+1,
(
Z (t +1) = Z1(
t +1)
)
t +1 t +1 , Z2( ) ,..., Z J( ) ,
is obtained by successively drawing from the distributions
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(
P Z J  Z1( Z1(
t +1)
t +1)
( t +1)
Z2 Z J(
t +1)
t +1 t +1 , Z2( ) ,..., Z J( −1 )
( ( P( Z
) )
t t t ~ P Z1  Z2( ) , Z3( ) ,..., Z J( ) t +1 t t ~ P Z  Z ( ) , Z ( ) ,..., Z ( )
M ~
2
J
1
3
J
)
t +1 t +1 t +1  Z1( ) , Z2( ) ,..., Z J( −1 )
(3.31)
)
in a slight abuse of notation. In other words, we draw from the conditional distributions of Z1, Z2, up to ZJ, conditioning each time on the most recently drawn values of all other subvectors. After the full set of subvectors has been drawn, we repeat (t+2) (t+3) the whole process to obtain Z , Z and so on. The (t) sequence {Z :t=0,1,2,...} forms a Markov chain which, under mild regularity conditions, has a stationary distribution equal t to P(Z); that is, Z ( ) → Z in distribution as t → ∞ . The name Gibbs is not at all descriptive of this method, but actually refers to a class of probability distributions on lattice systems that have been used in problems of spatial analysis and statistical image reconstruction (Besag, 1974). The first use of Gibbs sampling in this context was made by Geman and Geman (1984), who provided a proof of convergence for a discrete Z with finite state space. The method was independently derived for J=2 by Li (1988), who presented an argument for convergence in the continuous case. Other convergence proofs under various conditions are given by Schervish and Carlin (1992); Liu, Wong, and Kong (1994, 1995); and Tierney (1994). The regularity conditions necessary to establish convergence of the Gibbs sampler in a general setting are somewhat technical, but they do tend to be satisfied in most problems of practical interest. Informally, one can say that sufficient conditions Are (a) that the target distribution P(Z) must be a genuine probability distribution, and the sequence (3.31) must be the actual conditional distributions corresponding to this target; and (b) that the sample space of Z must be connected in the sense that it must be possible to
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reach any point in the sample space from any other point by repeated sampling from the conditionals in the manner of (3.31); periodicity and absorbing states are not allowed. For more discussion, see Roberts (1996) and Tierney (1996). For some examples of nonconvergence, see Casella and George (1992); Arnold (1993); and Section 3.5.2 below. As pointed out by Liu, Wong and Kong (1995), the conditional distributions (3.31) in the Gibbs sampler need not be drawn from in any particular order in each iteration, nor do they need to be drawn from equally often. As long as each conditional distribution is visited infinitely often, the stationary distribution will be P(Z). As a practical matter, of course, different visitation schemes will have different properties when only a finite number of iterations are performed. The distributions (3.31) are sometimes called the full conditionals because each one is the distribution of a subvector given all the other subvectors. Other sets of conditional distributions may also be grouped together to form sampling schemes that will converge to P(Z), as described by Gelfand and Smith (1990). 3.4.2 Data augmentation Closely related to Gibbs sampling is the data augmentation algorithm of Tanner and Wong (1987). Suppose that a random vector z is partitioned into two subvectors, z = (u, v), where the joint distribution P(z) is not easily simulated but the conditional distributions P(uv)=g(uv) and P(vu)=h(vu) are. At iteration t, let
(
t t (t ) Z (t ) = z1( ) , z2( ) ,..., zm
(
)(
)
) (
)
t t t t (t ) , v (t ) = u1( ) , v1( ) , u2( ) , v2( ) ,..., um m
be a sample of size m from a distribution that approximates the target distribution P(z). This sample is updated in two steps.
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First,
(
U (t +1) = u1(
t +1)
t +1 (t +1) , u2( ) ,..., um
)
is created by drawing
ui(
t +1)
(
t ~ g u  vi( )
)
independently for i = 1, 2,...,m. Next,
(
V (t +1) = v1(
t +1)
t +1 (t +1) , v2( ) ,..., vm
)
is drawn as an iid sample from the equally weighted mixture t +1 of the conditionals h v  u ( ) ,
(
(
)
i
)
w 1 t +1 h v  Ui( ) = m
∑ h(v  ui(t +1) ), m
(3.32)
i =1
which completes the new sample
(
) (
)
t +1 t +1 ( t +1) , v( t +1) Z ( t +1) = u1( ) , v1( ) ,..., um m
Using functional analysis, Tanner and Wong (1987) show that the distribution of Z(t) converges to P(z) as t → ∞ . This result does not require a large value of m; in particular, with m = 1 data augmentation reduces to a special case of the Gibbs sampler (3.31) with the random quantities z = (u, v) partitioned into two subvectors, u and v. More generally, if we modify the second step of each iteration by sampling
vi(
t +1)
(
t +1 ~ h v  ui( )
)
independently for i = 1, 2,...,m rather than drawing them from the mixture (3.32), then the algorithm becomes m independent, parallel runs of a Gibbs sampler. The mixing of the conditionals h(vu) at each iteration may not provide much practical benefit in speeding the convergence of the Z(t) but
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when m is large (3.32) provides a good analytic approximation to the marginal density P ( v)=eP(u,v) d u i f s u c h a n approximation is desired. Application to missingdata problems The name data augmentation arose from applications of this algorithm to Bayesian inference with missing data. In many incompletedata problems, the observeddata posterior P(θYobs ) is intractable and cannot easily be summarized or simulated; when Yobs is augmented by an assumed value of the Y mis , however, the resulting completedata posterior P(θYobs,Ymis ) becomes much easier to handle. Consider the following iterative sampling scheme: given a current guess θ of the parameter, first draw a value of the missing data from the conditional predictive distribution of Ymis, (t)
(
)
(t +1) Ymis ~ P Ymis  Yobs , θ (t ) .
(3.33)
(t +1) , draw a new value of θ from its Then, conditioning on Ymis completedata posterior,
(
)
(t +1) θ (t +1) ~ P θ  Yobs , Ymis .
(3.34)
Repeating (3.33)(3.34) from a starting value θ stochastic sequence
{(θ
)
}
(0)
yields a
(t ) , Y (t ) : t = 1, 2,... whose stationary mis
distribution is P(θYmisYobs), and the subsequences {θ : t = (t ) 1,2,...} and Ymis : t= 1, 2,... have P(θYobs) and P(YmisYobs)
{
(t)
}
as their misrespective stationary distributions. Following the terminology of Tanner and Wong (1987), we will refer to (3.33) as the Imputation or Istep and (3.34) as the Posterior or Pstep, because (333) corresponds to imputing a value of the missing data Ymis and (3.34) corresponds to drawing a value of θ from a completedata posterior. For a value of t that is
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suitably large, we can regard θ as an approximate draw from t P(θY ); alternatively, we can regard Y ( ) as an approximate (t)
obs
mis
draw from P(YmisYobs). Many particular examples of the algorithm (3.33)(334) will appear throughout the remainder of this book. The first use of this algorithm seems to have been made by Li (1988) who presented an argument for convergence and used it to create imputations of Ymis . in incompletedata problems. The algorithm can be regarded either as a special case of data augmentation with m=1 or as a special case of Gibbs sampling with (Ymis,θ ) partitioned into Ymis and θ . Because the former name is more descriptive for incompletedata problems, we will refer to it as data augmentation rather than Gibbs sampling. For the most part, however, we will use only the special case of data augmentation with m=1. On occasion we will perform m > 1 parallel runs of data augmentation, but we will keep the runs independent; that is, we will not employ mixing (3.32) at each iteration. Data augmentation bears a strong resemblance to the EM algorithm. The Estep of EM calculates the expected completedata sufficient statistics, whereas the Istep of data augmentation simulates a random draw of the completedata sufficient statistics. The implementation of an Istep is typically very similar to that of an Estep, usually requiring only minor modifications of the computer code. The Mstep of EM is a maximization of a completedata likelihood, while the Pstep of data augmentation is a random draw from a completedata posterior. The computational requirements of EM and data augmentation are therefore quite similar, as both involve repeated application of completedata methods to solve an incompletedata problem. 3.4.3 Examples of data augmentation Example 1: Incomplete univariate normal data. Suppose that Y=(y1, y2,...,yn) is an iid sample from a normal distribution with mean µ and variance ψ which, for the moment, is
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assumed to be known. If we apply a normal prior distribution to µ with mean µ0 and variance τ, it follows that the posterior distribution of µ given Y is also normal with mean
nψ −1 τ −1 E( µ  Y ) = y + µ −1 −1 −1 −1 0 nψ + τ nψ + τ
(3.35)
and variance V(µY) = (n ψ +τ ) , where y is the sample 1
1 1
mean of y 1 ,y2,...,yn. Letting τ → ∞, the posterior becomes normal with mean y and variance n ψ , which may also be obtained by applying Bayes' s formula with the improper diffuse prior π (µ) ∝ c where c is a constant (e.g. Box and Tiao, 1992). Now suppose that only the first n1 elements of Y are observed and the remaining n 0=nn1 are missing. Under ignorability and the diffuse prior π(µ) ∝ c the observeddata 1
posterior
yobs =
n1−1
P(µYo b s )
∑
n1 i =1 yi
becomes
normal
and variance
n1−1ψ .
with
mean
In this trivial
example, values of µ from P(µYobs) can be simulated directly using standard routines for generating normal random variates. We can also simulate them iteratively, however, using the data augmentation routine of (3.33)(3.34). Given a current t +1 (t) parameter value µ , the Istep simulates Y ( ) by drawing mis
yi(
t +1)
(
 µ (t ) , Yobs ~ N µ (t ) , ψ
)
(3.36)
independently for i = n1+1,...,n. The Pstep then proceeds to draw t +1 (t+1) µ from the completedata posterior P µ  Y , Y ( ) ,
(
anormal distribution with mean
y ( t +1) n −1
[∑
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n1 i=1 yi
+
∑ ni= n +1 yi(t +1) ] 1
obs
mis
)
and variance n ψ. In this simple example of data augmentation, one may (t) analytically verify that the distribution of µ approaches the 1
(
)
correct observeddata posterior N yobs , n1−1ψ as t → ∞. This is possible because of the following well known property of the normal distribution: If UV ~ N(V,a) and V ~ N(b,c) then U ~ N ( b , a + c ). Applying this property, the conditional (t) (t1) distribution of µ given Yobs and the previous iterate µ is easily seen to be
(
)
(
)
µ ( t )  µ ( t −1) ~ N yobs + λ µ ( t −1) − yobs , n1−1ψ 1 − λ2 , where λ = n0/n and conditioning on Yobs has been suppressed in the notation. Similarly, the conditional distribution of µ given Yobs and µ
(t2)
is also normal, with mean
( t ) ( t −1) ( t − 2 ) ( t) ( t − 2) E µ  µ = E E µ  µ µ ( t −1) = E yobs + λ µ − yobs  µ ( t − 2 ) ( t − 2) = yobs + λ2 µ − yobs and variance
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(t)
(
)
( ) +V E( µ ( )  u ) µ
V µ ( t )  µ ( t − 2 ) = EV µ ( t )  µ ( t −1)  µ ( t − 2 ) ( t −1)
t
(
= n1−1ψ 1 − λ2
)
(
( t − 2)
)
+V yobs + λ µ ( t −1) − yobs  µ ( t − 2 )
(
)
= n1−1ψ i − λ4 . Repeating this argument t times gives the marginal (t) (0) distribution of µ in terms of the starting value µ ,
µ
( t)  µ (0) N y
−1 t (0) 2t obs + λ µ − yobs , n1 ψ 1 − λ ,
(3.37)
(0) which approaches N yobs , n1−1ψ as t → ∞ for any fixed µ (0) as long as λ < 1. If the starting value µ is not fixed but drawn from a probability distribution, then we can also (t) investigate the unconditional distribution of µ . In particular,
is drawn from the correct posterior N yobs , n1−1ψ then (t) (3.37) implies that µ will be normal with mean if µ
(0)
( t) (0) ( t) E µ = E E µ  µ (0) = E yobs + λt µ − yobs = yobs and variance
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(3.38)
( t) (0) ( t) (0) ( t) V µ = EV µ  µ + V E µ µ (0) = n1−1ψ 1 − λ2 t + V yobs + λt µ − yobs = n1−1ψ , and stationarity is achieved immediately. We can also perform data augmentation in this example when the variance ψ is unknown. Under the diffuse prior
π(µ,ψ) ∝ ψ the completedata posterior is 1
µ  ψ , Y ~ N y, n −1ψ
(
)
ψ  Y ~ n − 1 S 2 χ n−−21
(3.40)
2
where S is the sample variance of y 1 , y2,...,yn, and the observeddata posterior is
µ  ψ , Yobs ~ N yobs , n −1ψ
(
)
2 ψ  Yobs ~ n1 − 1 S obs χ n−2−1
(3.41)
1
2 is the sample variance of y1, y2,...,yn1. The Istep where S obs
( t +1) by drawing
of data augmentation simulates Ymis
( t +1)  µ ( t) ,ψ ( t) , Y
yi
obs
( t) ( t) ~ N µ ,ψ
independently for i = n1+1,...,n, and the Pstep simulates µ
(t+1)
( t +1) and ψ from (3.40) with Ymis substituted for Ymis. In this twoparameter problem, writing down the marginal (t) (t) (t) distribution of θ = (µ ,ψ ) at any step t is no longer a simple matter. We can, however, demonstrate empirically that (t+1)
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the marginal distribution of θ approaches the observeddata posterior (3.41) in numerical examples. The algorithm was applied to the univariate sample of size n1 = 10 in Table 3.1 (0) (a), arbitrarily taking n 0 = 3 and starting values ψ = 70. (t)
Simulated marginal densities of µ and ψ for t = 1,2,3 are displayed in Figure 3.2. Based on the marginals, it appears that convergence to the observeddata posterior is quite rapid. The densities in Figure 3.2 were simulated by m = 500 parallel (t) (t) chains of data augmentation, each starting from µ and ψ . The chains were run independently; no mixing as in (3.32) was used. For plotting purposes, however, the marginal densities were estimated by the RaoBlackwell method, averaging (t)
(t)
(t)
(t)
Figure 3.2. Simulated marginal densities of µ and ψ for t=1,2,3, with dotted lines showing the exact observeddata posteriors.
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formulas for the completedata marginal posteriors over the 500 iterates of Y mis. This and other techniques for extracting meaningful summaries from Markov chain Monte Carlo runs will be discussed in Chapter 4. Example 2: Incomplete binary data. Let Y =(y1,y2,...,yn) represent the outcomes of n independent Bernoulli trials where yi = 1 with probability θ and yi = 0 with probability 1  θ for i = 1, 2,..., n. The beta prior distribution
()
(
π θ ∝ θ α −1 1 − θ
)β −1
for α > 0, β > 0, denoted by θ ~ Beta (α, β), leads to the completedata posterior
θ  Y ~ Beta α +
∑ in=1 yi , β + n − ∑ in=1 yi .
For simplicity let us use the limiting form of this prior as α → 0 and β → 0, so that the posterior becomes θ Y ~ Beta (a,b) where a =
∑ ni=1 yi and b = n−a; this posterior is proper as
long as Y contains at least one success and one failure. Now suppose that the first n1 elements of Y are observed and the remaining n0 = n−n1 elements are missing. The observeddata
∑n
1 posterior becomes θYobs ~ Beta (a1, b1) where a1 = i=1 yi and b1 = n1 − a1, which is proper provided that 1 ≤ a1 ≤ n1 − 1.
Applying data augmentation to this example, the Istep fills
( t +1)
in the missing trial outcomes by letting yi =1 with (t) probability θ and 0 otherwise for i = n 1+1,...,n. The Pstep then samples
θ
( t +1)  Y
( ) ~ Beta a( t +1) , b( t +1) ,
t +1 obs , Ymis
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where
a
( t +1) = a
1
( t +1) = b b
( t +1) = a
+ a0
1
( t +1) = b +b
+
∑ in= n +1 yi(
)
t +1
1
( t +1) . −a
+ n0 0 If data augmentation works properly, then the marginal (t) distribution of θ as t → ∞ should approach Beta (a1, b1) 1
1
0
which has mean a1/n1. Also, for large t, the distribution of any imputed trial should be Bernoulli with marginal probability of success
( t) E yi( t +1) = E θ = a1 / n1 , where conditioning on Y o b s is assumed and has been suppressed in the notation. We can algebraically verify that ( t) E yi does in fact approach a1/n1 as t → ∞, because
( t +1) ( t ) ( t) ( t) E yi  Ymis = E θ  Ymis ( t) = a1 + a0 / n ( t) = a1 / n1 − λ a0 / n0 − a1 / n1 where λ = n0/n. Similarly,
(
)
(
1 t +1 (1) E yi( )  Ymis = a1 / n1 − λt a0( ) / n0 − a1 / n1
)
which implies that
( t +1) ( 0 ) (0) E yi  θ = a1 / n1 − λt µ − a1 / n1 .
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Clearly, (3.42) approaches a1/n1 from any starting value µ long as λ < 1. If µ
(o)
(o)
as
happens to be equal to a1/n1, or is drawn
from a probability distribution with mean a 1/n1, then convergence is immediate. This example was first used by Li (1988). 3.4.4 The MetropolisHastings algorithm An older method of Markov chain Monte Carlo is the algorithm of Metropolis et al. (1953) and its generalization by Hastings (1970). The MetropolisHastings algorithm will not be needed in the remainder of this book; we briefly mention it, however, because of its usefulness in extending the basic algorithms of Chapters 59 to more complicated modeling situations. In the Hastings version, a Markov chain {Z (t):t=0,1,2,...} with stationary distribution P(Z)=f(Z) is constructed as follows. Given Z(t), a candidate value Z˜ is drawn from a transition distribution g(ZZ(t)). Then the ratio
R
( t +1)
=
( ) f (Z˜ ) . g( Z  Z ( ) ) f ( Z ( ) ) g Z ( t ) Z˜
t
t
(t+1)
is calculated. It R is greaterthan 1, we accept the value of (t+1) ˜ (t+1) = Z . If R < 1, we the candidate variable and set Z (t+1) with randomly accept the value of Z˜ as our next iterate Z (t+1) probability R , and otherwise keep the current value, (t+1) (t) Z =Z . If the transition distribution g allows the process to (t) eventually reach any state in the support of Z, then Z → Z in distribution as t → ∞. MetropolisHastings is useful when a transition distribution g can be found that (a) is easy to simulate, and (b) leads to acceptance ratios (3.43) that are easy to calculate. Because the target density f enters into the algorithm only through the acceptance ratio, we need only to be able to evaluate f up to a constant of proportionality. This makes MetropolisHastings attractive for simulation of Bayesian posterior distributions,
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for which the densities are typically known only up to a normalizing constant. From a standpoint of efficiency, it is advantageous for the acceptance ratios to be close to one over the region where f(Z) (t) is appreciable, which occurs when g(ZZ ) is a good (t) approximation to f(Z). When g (ZZ )=f(Z), then the acceptance ratio is always one and convergence is immediate. In practical applications, it is wise to choose a g that is somewhat more diffuse (i.e. having heavier tails) than f; otherwise, there may be little opportunity for a candidate value Z˜ to fall in some regions of the sample space where f(Z) is appreciable, and convergence to the target distribution may be too slow for practical use. If g is too diffuse, however, than many of the candidate values will fall outside the range where f(Z) is appreciable, in which case the rejection rate will be very high and convergence will again be slow. 3.4.5 Generalizations and hybrid algorithms As noted by several authors (Gelman, 1992; Smith and Roberts, 1993; Middleton, 1993; Tierney, 1994), both the Gibbs sampler and the MetropolisHastings algorithm may be generalized in a variety of ways. Consider the Gibbs sampler for a random vector Z =(Z1,Z2,...,ZJ). In one iteration of ordinary Gibbs, we sample from the full conditionals
Z (j
t +1)
(
t +1 t +1 t ~ P Z j  Z1( ) ,..., Z (j −1 ) ,..., Z J( )
)
(3.44)
for j=1,2,...,J. In practice, however, it is not necessary to t +1 generate Z (j ) directly from (3.44), but only from the transition distribution of a Markov chain that has (3.44) as its stationary distribution. In other words, if a Markov chain Monte Carlo scheme can be found that would eventually converge to the local distribution (3.44), we need only to perform one or more cycles of this local algorithm instead of (3.44), and the stationary distribution of the Gibbs sampler will be preserved. In specific examples, it sometimes happens that one (or more) of the full conditional distributions is
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difficult to simulate directly, but another Gibbs sampler or MetropolisHastings algorithm can be found that converges to the desired conditional. By replacing the difficult conditional with one or more iterations of this sampling scheme, we obtain a hybrid algorithm that still converges to the proper target. Another potential use of these generalized algorithms is in data augmentation with an inconvenient prior. Suppose that a prior distribution π*( θ ) exists that leads to a tractable completedata posterior P*(θYobs,Ymis ), but the prior that we would like to use for inference is π (θ ) which leads to a posterior P(θYobs,Ymis) that is intractable. In this situation, we can replace the Pstep under π(θ ) with one or more steps of a MetropolisHastings algorithm that draws a candidate value θ˜ from P*(θYobs,Ymis). The acceptance ratio becomes
(
)
(
)
P * θ ( t )  Yobs , Ymis P * θ˜  Yobs , Ymis R( t +1) = P * θ˜  Yobs , Ymis P * θ ( t )  Yobs , Ymis =
( ) π (θ˜ ) / π * (θ˜ ) ,
(
)
(3.45)
( ) ( )
π θ ( t) / π * θ ( t)
which does not depend on Y obs , or Ymis. When MetropolisHastings is used within data augmentation in this manner, the result is a hybrid algorithm with stationary distribution equal to the correct observeddata posterior under the desired prior. Note that (3.45) requires the evaluation of π and π* only up to a constant of proportionality, so the acceptance ratios are typically easy to calculate.
3.5 Properties of Markov chain Monte Carlo 3.5.1 The meaning of convergence Below we discuss some basic properties of Markov chain Monte Carlo, with special emphasis on the data augmentation
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scheme of (3.33)(3.34). Unlike optimization methods like EM, which are deterministic and converge to a point in the parameter space, Markov chain Monte Carlo algorithms are stochastic and converge to probability distributions. Yet certain important similarities exist between the convergence behavior of EM and data augmentation. Assuming the conditions needed for convergence are satisfied, the output of data augmentation is a sequence t θ ( t ) , Y ( ) : t = 0, 1, 2,... with stationary distribution P( θ,
{(
mis
}
)
Yobs,Ymis) For the sequence to have converged, it is sufficient (t) for the distribution of θ to have converged to P ( θYobs), t+s (t) because θ ~P(θY ) implies that θ (t + s ) , Y ( ) ~ P(θ, obs
(
mis
)
YobsYmis ) for all s > 0. Equivalently, it is sufficient for the t distribution of Y ( ) to have converged to P(Y Y ). Also, mis
mis
obs
( s ) are convergence by t iterations means that θ and Ymis (s+t) ( s + t ) . In applications it is typically and Ymis independent of θ more convenient to monitor convergence through the behavior of successive values of θ than successive values of Ymis because the latter is usually of higher dimension. Except in trivial examples like those in Section 3.4.3 for which data augmentation is not needed, summaries of P(θYobs ) are not available in closed form, making it difficult to know precisely when convergence has occurred. Techniques for assessing convergence are described in Chapter 4. For now we will discuss only in broad terms some issues surrounding convergence. (s)
3.5.2 Examples of nonconvergence Nonexistence of a stationary distribution As mentioned above, convergence of a Gibbs sampler requires that the full conditionals (3.31) are the conditionals of a genuine joint probability distribution. It is possible to
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construct simple examples in which a set of proper conditional distributions does not define a proper joint distribution (Casella and George, 1992). For data augmentation, Bayes s Theorem guarantees a proper limiting distribution as long as the prior π (θ ) is proper. In many real data applications of Bayesian analysis, however, it is convenient to use socalled noninformative priors that are actually improper but lead to proper posteriors when Bayes s formula is applied. Even when an improper π is known to yield a proper posterior in the case of complete data, it may not necessarily do so when some data are missing. For a very simple example, let Y = (y 1 ,y2) represent two independent observations from N(µ, ψ ) with µ and ψ both unknown. Under the standard noninformative prior π(µ,ψ ) ∝
ψ , the posterior distribution is given by (3.40) with 1
2
2
n = 2, y = ( y1 + y2 ) / 2 and S 2 = ( y1 − y ) + ( y2 − y ) . Now suppose that only y1=Yobs is observed and y2=Ymis is missing. Applying Bayes ' s formula, the observeddata posterior becomes P( µ, ψ  Yobs ) ∝ L( µ, ψ  Yobs )π ( µ, ψ ) ( y − µ )2 ∝ ψ −3 2 exp − 1 2ψ
,
which is not a proper probability distribution because the integral
( y − µ )2 ψ −3 2 exp − 1 dµdψ = −∞ 2ψ
∞ ∞
∫0 ∫
∞
∫0
(2π )
−1 2
ψ −2 dψ
does not exist. Yet, one could naively apply data augmentation to this example under the improper prior. The Istep would be
y2(
t +1)
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(
)
~ N µ ( t) ,ψ ( t) ,
(3.46)
and the Pstep would be
( )(
ψ ( t +1)  y1, y2( t +1) ~ S 2
t +1)
χ1−2
(3.47)
(
)
t +1 µ ( t +1)  ψ ( t +1) , y1 y2( ) ~ N y ( t +1) , ψ ( t +1) / 2 ,
where
(
)
t y ( t ) = y1 + y2( ) / 2
(S 2 )( ) = ( y1 − y(t) ) + ( y2(t) − y(t) ) . t
2
2
Notice that even though the joint posterior does not exist, both the Istep and the Pstep are defined at every iteration. One could naively alternate between (3.46) and (3.47) indefinitely without any clue that the algorithm is not converging. The fundamental reason why data augmentation fails here is that the mean and variance cannot be jointly estimated on the basis of a single observation y1. The observed data provide no information on one aspect of θ = (µ ,ψ ), namely ψ , so that unless a proper prior distribution is applied to ψ there is no basis for inference. Although there is no compelling reason to use data augmentation in this trivial example, it is not difficult to construct more realistic multivariate problems where both the I and Psteps of data augmentation are defined but the stationary distribution does not exist. These would be similar to the examples of Section 3.3.1 where the ML estimate of θ is not unique because the likelihood is maximized over a ridge. Whenever the likelihood has a ridge, certain aspects of the parameter are inestimable, and unless a proper prior is applied to those aspects of θ the posterior will not be proper. Sparse datasets with few observations or high rates of missingness may be prone to these problems. If we suspect that data augmentation may not be converging to a proper posterior, we
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can switch to a proper prior, thereby guaranteeing that the posterior will be proper as well. Boundary values and absorbing states Another basic requirement for the convergence of a Markov chain Monte Carlo algorithm is that the support of the target distribution must be connected in the sense that it must be possible to eventually reach any state from any other state. There must be no periodic states and no absorbing states, i.e. regions where the algorithm could become trapped with zero probability of escape. In the stochastic processes literature, this property is known as ergodicity. Consider again the normal sample Y = (y1,y2) where y 1 is observed and y2 is missing, but now let us suppose that the population mean µ is known, and without loss of generality take µ = 0. Suppose that we apply an improper prior distribution to the variance, π(ψ ) ∝ ψ
(v+2)/2
where v is a fixed
constant. Given Y = (y1,y2), the completedata posterior is
P(ψ  Y ) ∝ ψ
(
−1
(
y2 + y2 1 2 exp − 2ψ
) ψ −(v +2) / 2
)
or ψ Y ~ y12 + y22 χ v−+2 2 , which is proper provided that v > −2. Given only Yobs = y1, the observeddata posterior is
y2 P(ψ  Yobs ) ∝ ψ −1 2 exp − 1 ψ −( v + 2) / 2 2ψ
(3.48)
or ψ Y obs ~ y12 χ v−+21 , which is proper for any v >− 1 If data augmentation were applied to this example, the Istep would be
(
)
y2( t +1)  ψ ( t ) , y1 ~ N 0, ψ ( t ) , and the Pstep would be
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(
)
2 t +1 t +1 ψ ( t +1)  y1 y2( ) ~ y12 + y2( ) χ v−+2 2 .
The algorithm would proceed normally except in the unlikely event that y1 = 0 happened to become zero at some iteration. If t (t) that were to occur, we would obtain y( ) = 0 and ψ for every 2
iteration thereafter. In other words, ψ = 0 would be an absorbing state. Even if y1 happened to be zero, absorption would be unlikely because unless we start on the boundary (ψ (0) = 0) the event ψ =0 occurs in theory with probability zero. Depending on the computer and random variate generator used, there could be a small chance of falling within machine (0) precision of the boundary, especially if the starting value ψ is very close to zero. The presence of an absorbing state is not the only difficulty in this example, because the observeddata posterior (3.48) is not proper for y1 = 0. In Chapter 8, however, we will see that in some real categoricaldata problems absorption onto a boundary becomes a distinct possibility even when the posterior is technically proper, and we will need to handle such situations with care. (t)
3.5.3 Rates of convergence Assuming that a Markov chain Monte Carlo algorithm does converge to a proper stationary distribution, it is important to consider how quickly this convergence occurs. Convergence rates are typically defined in terms of a distance measure between the marginal distribution of the iterates at any given time and the target distribution. Some interesting theoretical results on convergence rates and further references are given by Schervish and Carlin (1992); Smith and Roberts (1993); Tierney (1994); and Liu, Wong and Kong (1995). This work, although reassuring, does not easily translate into practical guidelines for knowing when convergence has occurred in specific examples. Ongoing research regarding convergence behavior will undoubtedly lead to greater understanding in the
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future; for now, however, we can informally state a few general principles that apply to incompletedata problems. Convergence and missing information Consider simple data augmentation in which we alternately perform an Istep
(
)
(
)
( t +1) Ymis ~ P Ymis  Yobs , θ ( t ) and a Pstep
( t +1) θ ( t +1) ~ P θ  Yobs , Ymis .
Intuitively, the rate of convergence should depend on how much information about the parameter is contained in missing data relative to the observed data and the prior. The completedata posterior may be written
P(θ  Yobs , Ymis ) ∝ P(θ  Yobs )P(Ymis  Yobs , θ ).
(3.49)
In the extreme case where Ymis provides no information about θ beyond that already contained in Yobs , Ymis and θ would be conditionally independent given Yobs the last term in (3.49) would then be constant with respect to θ, and convergence to the target distribution would be immediate. More generally, if P(YmisYobs , θ ) as a function of θ is relatively flat over the region of high posterior density (which is typically equivalent to the missing information, as defined in Section 3.3.2, being near zero), then each Pstep will be nearly a draw from P(θYobs) and the algorithm will converge rapidly. On the other hand, if the missing information is a large portion of the total information, then θ will depend heavily on Ymis at each Pstep, which will in turn depend on the value of θ used in the previous Istep; successive iterates of θ will tend to be highly correlated and convergence will be slow. Just as with EM, the
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rate of convergence of data augmentation and the fractions of missing information are fundamentally related. This relationship between missing information and rate of convergence is difficult to formalize in a general way, but it can be easily demonstrated in simple examples. Consider again the univariate normal data Y = (y 1, y2,...,yn) with known variance ψ and unknown mean µ. In Example 1 of Section 3 . 4 . 3 we investigated data augmentation under the noninformative prior π(µ) ∝ c (a constant), where the first n1 elements of Y are observed and the remaining n0 = n − n 1 are missing. We found that the stationary distribution of µ is
(
)
(t) µYobs ~ N yobs , n1−1ψ and the marginal distribution of µ is
(
)
(
)
µ ( t )  µ (0) , Yobs ~ N yobs + λt µ (0) − yobs , n1−1ψ 1 − λ2t (3.50)
where λ = n 0/n is the fraction of missing information. Clearly, the algorithm will approach stationarity rapidly for λ near zero and slowly for λ near one. This example can easily be generalized to an informative prior µ ~ N(µ0,τ), in which case we would obtain an expression like (3.50) but with the following changes: yobs and n1−1ψ would be replaced by the new observeddata posterior mean and variance, respectively, and λ would be replaced by the new fraction of missing information
λ* =
n0ψ −1 , nψ −1 + τ −1
where the prior information τ now appears in the denominator as a part of the total information. When the mean and variance are both unknown, the joint (t) (t) distribution of µ and ψ is intractable, but we can still demonstrate empirically that the rate of convergence depends on n 0/n. Using the n 1 = 10 observations in Table 3.1 (a) and 1
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−1
the noninformative prior π(µ, ψ ) ∝ ψ , we performed data augmentation under various choices of n0. Independent sample (t) (t) paths for (µ ,ψ ) beginning from four different starting positions are displayed in Figure 3.3, first for n 0 = 10 and again for n0=90, with ψ shown on a log scale. The starting values were all chosen to be in the tails of the observeddata posterior, so that the iterates would exhibit an initial trend as they wander into the region of high posterior density. For n0 = 10 the sample paths become heavily intertwined by t = 8, suggesting that for most practical purposes the algorithm has probably converged by eight or ten iterations. For n 0 = 90, however, the sample paths have still not crossed one another by t = 25; the algorithm takes smaller steps and the successive iterates aremore highly correlated. Starting values and starting distributions In the univariate normal example with known mean, (3.50) reveals that convergence behavior depends not only on the fraction of missing information but also on the choice of a (0) starting value. If we happened to take µ = yobs , then the distribution of µ would have the correct mean (i.e. the same mean as the stationary distribution) for every t. Even though (t) the variance of µ is always less than the stationary variance, choosing a starting value near the center of the observeddata posterior makes the first moment more nearly correct. (t)
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Figure 3.3. Iterations of data augmentation form various starting values for univariate normal data with n0 =10 and n0=90. (t) It is also evident from (3.50) that the variance of µ does not depend on the starting value µ
(0)
as long as µ
(0)
is fixed. If
but draw it from a probability we do not use a fixed µ distribution, however, then we can alter the second moment of (t) (0) µ as well as the first moment. Suppose that µ is drawn at (0)
random from a probability distribution with variance κ. Then (3.39) implies that the unconditional variance of µ is (t)
( )
(
)
V µ ( t ) = n1−1ψ 1 − λ2t + λ2tκ ,
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which is equal to the stationary variance n1−1ψ ; if κ = n1−1ψ ; t if κ > n1−1ψ then V µ ( ) > n1−1ψ as well. For inferential
( )
purposes, it is often wise to draw starting values of parameters at random from a probability distribution that is overdispersed relative to (i.e. having variance at least as great as) the target distribution, so that the variance after a finite number of iterations is at least as large as the stationary variance and resulting inferences are conservative (Gelman and Rubin, 1992a). It is also wise to use a starting distribution that is centered at or near the mean of the target distribution, so that the first moment at any iteration will be approximately correct. In Chapter 4 we discuss how one might obtain starting distributions in realistic problems where moments of the stationary distribution are unknown. If a single, fixed starting value is desired, then a point near the center of P(θYobs), e.g. an ML estimate or posterior mode obtained from EM, may be a wise choice. Difficulties with slow convergence When Markov chain Monte Carlo converges very slowly in an incompletedata problem, it is typically for the same reason that EM converges slowly: the fractions of missing information for one or more components of θ are very high. Previous comments about slow convergence of EM apply here as well; it should be taken as a warning that inferences about certain aspects of θ depend heavily on the missingdata model P(YmisYobs , θ ). Slow convergence of Markov chain Monte Carlo algorithms can be notoriously difficult to detect (e.g. Gelman and Rubin, 1992b), but when EM is slow it is usually painfully obvious. Consequently, it is good practice to apply EM in addition to Markov chain Monte Carlo, even if merely as a device for diagnosing slow convergence. Slow convergence of Markov chain Monte Carlo may also be the result of an observeddata posterior that is oddly shaped. If the posterior is poorly connected, e.g. if it has multiple modes that are widely separated by regions of low density, then simulation routines may get stuck in certain
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regions of the parameter space. Using EM in conjunction with simulation often helps to reveal unusual features of the likelihood or posterior that may be less apparent if only one or the other is applied.
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CHAPTER 4
Inference By Data Augmentation
4.1 Introduction In a narrow sense, one may define the problem of inference in relation to θ, the unknown parameter of a probability model. The statistician may desire a point estimate for one or more components or functions of θ , summarizing the uncertainty with a confidence interval or confidence region. One may want to test whether θ is equal to some null value or lies in some subset of the natural parameter space, summarizing the evidence with a pvalue. With incomplete data, such quantities can be obtained from the observeddata likelihood or posterior, although in practice special computational techniques may be needed. In a broader sense, however, the problem of inference often goes beyond making statements about a single parameter θ. A data analyst will typically want to apply a variety of exploratory and modeling techniques to a dataset, such as graphical displays, linear regression, factor analysis and so on, to investigate various interesting features of the data. When the data are incomplete, the analyst’s task often becomes considerably more difficult. Carrying out procedures that are ordinarily straightforward, such as fitting a satisfactory regression model, may not be straightforward when some data are missing. Analysts need sensible routine methods for analyzing incomplete data, while recognizing and assessing the role of missingdata uncertainty at each step of the analysis.
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This chapter addresses inference both in the narrow and broad sense, attacking both through techniques of simulation. Simulation may be used either to make simple inferences about θ or to perform multipurpose data analyses. To accomplish the former, we simulate random values of θ from its observeddata posterior distribution. To accomplish the latter, we generate plausible versions of the unknown Ymis. These complementary techniques will be called, respectively, parameter simulation and multiple imputation. In parameter simulation, one creates random but not necessarily independent draws of θ from P (θ/Yobs) The sample moments of these draws provide estimates of the posterior moments, and the empirical distribution, perhaps smoothed in some fashion, provides estimates of the marginal distributions of individual components or functions of θ . Depending on which features of the posterior distribution are of interest, one may need to generate a large sample to obtain accurate inferences; hundreds or perhaps even thousands of draws of θ may be necessary. In multiple imputation (Rubin, 1987), one creates m plausible sets of missing values by drawing repeatedly from P(YmisYobs) This results in m simulated complete datasets which are analyzed by completedata methods. The results of the m, completedata analyses are then formally combined to produce a single overall inference. Exploratory data analyses (e.g. graphical displays) may also be performed on each of the m, completed datasets, providing an informal but valuable assessment of how interesting features of the data are affected by missingdata uncertainty. When the fractions of missing information are moderate, as is often the case, only a few imputations (e.g. m = 3 or m = 5) are usually adequate to provide inferences that are nearly efficient and practically valid. Data augmentation and related Markov chain Monte Carlo algorithms enable us to perform either parameter simulation, multiple imputation or both; the same algorithm may be used to draw θ from P (θ/Yobs) and to draw Ymis from P (YmisYobs). Parameter simulation and multiple imputation, to be described
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in Sections 4.2 and 4.3 respectively, can be viewed merely as two different ways of extracting information from the same Markov chain. Methods for monitoring the convergence of Markov chain Monte Carlo algorithms are discussed in Section 4.4, and Section 4.5 contains practical advice on applying these methods to real data problems.
4.2 Parameter simulation
4.2.1 Dependent samples A natural way to answer inferential questions concerning particular components or functions of θ is to directly examine and summarize simulated values of θ . Suppose that we run a single series of data augmentation or a related algorithm long enough to achieve approximate stationarity; that is, we choose (t) a t large enough so that θ is essentially a draw from P (θ/Yobs).This initial phase, sometimes called the burnin period, is helpful to rid the series of dependence on the starting value or starting distribution. Suppose that we discard the values of θ from the burnin period and continue for another m iterations, calling the resulting values (1) (2) θ ,θ ,..., θ (m) .These can be regarded as a genuine sample from the observeddata posterior, because stationarity implies (t) that θ is marginally distributed according to P (θ/Yobs) for every t. However, the members of this sample will, in most cases, be dependent upon one another; values of θ that are close to one other in the sequence will tend to be more alike than values that are far apart. Successive values of θ may be highly positively correlated, particularly when convergence is slow. For many readers, the notion of a dependent sample will be somewhat unfamiliar. Suppose for a moment that we are interested in a particular scalar component or function of the
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parameter, denoted by ξ=ξ(θ ). If the sample values are independent, then the sample average
ξ= (t)
1 m
(t)
m
∑ ξ(t),
(4.1)
t =1
where ξ = ξ(θ ,t = 1,2, ,m , is the obvious Monte Carlo estimate of the posterior means
(
) ∫ (
)
E ξ  Yobs = ξP ξ  Yobs dξ ; ⁄ the sample variance
1 m =1
m
∑ (ξ(t ) − ξ )2
(4.2)
t =1
is the obvious estimate of the posterior variance
(
) ∫
(
)
(
)
V ξ  Yobs = ξ 2Ρ ξ  Yobs dξ − E 2 ξ  Yobs ; and so on. When the sample values are dependent, however, it may not be immediately obvious whether the same types of summaries (averages, etc.) are appropriate. In one important sense, they are. A law of large numbers for Markov chain Monte Carlo (Tierney, 1994) states that under quite general (1) (2) (m) conditions, if Z , Z ,..., Z is a realization of a Markov chain Monte Carlo run with target distribution f, then
1 m
∑ g(Ζ (t) ) → E f [ g(Z )] m
t =1
(4.3)
(almost surely) for any realvalued function g(Z) as m → ∞, provided that Ef [g(Z)], the expectation of g (Z ) under the target distribution, exists. By (4.3) it follows that the sample moments of a dependent sample are consistent estimates of the population moments. A histogram of the sample values will come to resemble the population density for large m. Virtually any summary that is appropriate for an independent random sample is appropriate for a dependent one. Although the consistency of most empirical summaries is maintained under dependence, however, other familiar properties may be lost. For example, the variance of the
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1
sample average (4.1) is not m times the variance of a single (t) ξ , but also involves the covariances among (1) (2)
ξ ,ξ ,..., ξ (m) . If successive iterates are highly positively correlated, then the covariance terms become large and ξ becomes substantially less precise than an average from an independent sample of the same size. Moreover, under dependence (4.2) is not in general an unbiased estimate of V(ξYobs); if successive iterates are positively correlated then (4.2) has a downward bias for any finite m, and the usual justification for using (m − 1) in the denominator rather than m no longer applies. Perhaps the most serious drawback of dependence is that assessing the error of a Monte Carlo estimator is no longer a simple matter. Estimating the error variance internally from a single dependent sample can be difficult. An alternative strategy is to employ replication: rather than performing the simulation once, perform it independently k times, and examine the variation among the k replicate values of the estimator. Multiple runs also provide a method for diagnosing lack of convergence of the Markov chain itself (see Section 4.4). In most cases, practitioners would not be interested in Monte Carlo error if they could be assured that it was small enough that their important numerical estimates and conclusions were not in jeopardy. Obtaining accurate measures of Monte Carlo error, therefore, may be a matter of secondary importance when the error is known to be small, and even crude estimates may suffice. On the other hand, if the error is more substantial, then it needs to be assessed more carefully and perhaps even be formally incorporated into pvalues and interval estimates. Subsampling a chain One way to finesse the issue of dependence in Markov chain Monte Carlo is to subsample the chain: rather than summarizing) a posterior by (1) (2) (m) (k) (2k) (mk) θ ,θ , ...,θ , use θ ,θ , ...,θ where k is chosen large enough to make the sample values approximately independent.
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Aside from the problem of how to choose k , subsampling obviously requires a greater number of iterations to produce a final sample of the same size. Although the resulting independent sample will tend to give more efficient estimates than a dependent sample, this gain in efficiency will not compensate for the kfold increase in computation. Moreover, if we run the algorithm for km iterations then we might as well summarize the results using all km iterates, because the average of all km iterates is more precise than the average of every kth iterate (Geyer, 1992; MacEachern and Berliner, 1994). Aside from issues of data storage, if the goal is to obtain direct summaries of a posterior distribution then subsampling the chain is generally not advantageous. Subsampling and issues of Monte Carlo error will be taken up again in Section 4.4; for now we discuss various ways to extract inferentially meaningful summaries from a single, dependent parameter sample. 4.2.2 Summarizing a dependent sample Posterior moments Suppose that we are interested in a particular scalar component or function of the unknown parameter, denoted generically by ξ=ξ(θ) To the Bayesian, a useful point estimate of ξ is the posterior mean E(ξ.Yobs). From a decisiontheoretic standpoint, the posterior mean is the optimal estimate under squarederror loss (e.g. DeGroot, 1970). Even when no explicit loss function is available, the posterior mean is still often regarded as the most natural singlenumber summary. Another useful quantity is the posterior variance V( ξYobs), which measures uncertainty about the unknown ξ. If θ (m)
(1)
,θ
(2)
. . . , θ are values from P(θYobs) produced by a Markov chain Monte Carlo method, then consistent estimates of the posterior mean and variance of ξ are given by
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−
Eö ( ξ  Yobs) = ξ =
1 m
1
∑
ξ(t )
(4.4)
t =1
and
Fˆ ( a) =
m
∑ m −1I (ξ (t) ≤ a) t =1 (t)
(4.5)
(t)
respectively, where ξ =ξ(θ ). Higher moments, if desired, may be estimated in a similar fashion. Posterior distributions and densities (1) (2)
If θ ,θ ,...,θ (1) (2)
(m)
, are a sample from P(θYobs), then it follows
(m)
that ξ ,ξ ,...,ξ
are a sample from P(ξYobs), the observed
data marginal distribution of ξ By (4.3), the posterior cumulative distribution function
(
α
) ∫−∞ P(ξ  Yobs )dξ
P ξ ≤ α  Yobs =
(4.6)
can be consistently estimated for any a by the proportion of (1) (2) (m) sample values ξ ,ξ ,..., ξ that fall at or below a. Applying this estimate for every a gives the empirical distribution function,
Fˆ ( a) =
m
∑ m −1I (ξ (t) ≤ a)
(4.7)
t =1
where I(⋅) is an indicator function equal to one if the argument is true and zero otherwise. The empirical density associated with (4.7) is the discrete probability function that assigns mass (t) 1/m to each observed value ξ ,t=1,2, . . , m.. In most cases the true posterior distribution of ξ = ξ (θ ) is continuous, which suggests that we can improve the empirical density by smoothing it in some fashion. A histogram (1) (2) (m) partitions the range of sample values ξ ,ξ ,...,ξ into a
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small number of discrete intervals, typically of equal width, (t) and spreads the probability mass of each ξ uniformly over the interval into which it falls. Another important type of density estimator is the class of kernel estimators, which have the form
Fˆ ( a) =
m
∑ m −1I (ξ (t) ≤ a) t =1
The kernel K(α,ξ) is some nonnegative function centered at ξ with the property ∞
∫−∞ K (α , ξ )da = 1. The choice of kernel, which is always somewhat arbitrary, affects the shape of ^f and its degree of smoothness; popular choices are rectangular, triangular and Gaussian kernels with width determined by the number and range of sample values. For more information on kernel estimators see Silverman (1986) or Devroye (1987). Quantiles The p th quantile of a random variable with cumulative distribution function F is usually defined as the smallest x for which F(x)≥p. Substituting the empirical distribution function Fˆ into this definition gives the following method for estimating the quantiles of ξ based on a sample ξ (m).
...,ξ
(1)
(2)
,ξ
First, order the sample from smallest to largest; let ξ*
(t)
,t=1,2, ,m denote the order statistics, *(1) *(2) *(m) ξ ≤ ξ ≤…≤ξ If mp happens to be an integer, then the pth quantile is ∗(mp) ∗ ([mp]+1) estimated by ξ ; otherwise, it is estimated by ξ where [⋅]denotes the greatest integer function. In the above method, the estimate of any quantile is restricted to be one of the observed order statistics. A more common practice is to interpolate between the order statistics.
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Suppose we consider ξ
(i)
to be an estimate of the pth quantile;
there is no universally agreed upon value for p, but common choices include p=i/(m+1) and p =(i−1)/(m−1). Using the former, the estimated pth quantile is the ith order statistic when i = p(m + 1) happens to be an integer. When i = p(m + 1) is not an integer, we find i1 = [p(m + 1)] and i2 = i1 + 1 and use the estimate *i *i (4.8) Fˆ −1 ( p) = (1 − c)ξ ( 1 ) + cξ ( 2 ) , where c = p (m + 1) i1. Taking p = 0.5, this interpolation method gives the familiar form of a sample median: the middle value if m is odd and the average of the two middle values if m, is even. Interval estimates A 100 (1−α)% Bayesian posterior region is defined to be any set with posterior probability content at least 1α. That is, A is a 100(1−α)% posterior region for ξ=ξ(θ) if
P(ξ ∈ A  Yobs ) ≥ 1 − α ⋅ Unlike a frequentist confidence region, which has the property that the region will cover ξ with specified longrun frequency over repeated samples, the Bayesian region makes a probability statement about the parameter ξ given the current sample. Given the posterior distribution for a scalar ξ , there are various methods for constructing a Bayesian interval estimate. In the highest posterior density (HPD) method, the interval is chosen so that every value within the interval has posterior density at least as high as every value outside of it (e.g. Box and Tiao, 1992). The HPD method yields the shortest possible interval for ξ , but it is not invariant under nonlinear transformations of the parameter. Another simple technique is the equaltailed method, in which the endpoints are chosen so that the posterior probability of falling above the upper endpoint and the probability of falling below the lower endpoint are both equal to α/2. For example, a 95% equaltailed interval runs from the 2.5th percentile to the 97.5th
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percentile of the posterior distribution. Equaltailed Bayesian posterior intervals can be directly estimated from a sample (1) (2) ( m) ξ ,ξ ,...,ξ by using (4.8). Estimating an HPD interval is more complicated, requiring both a smooth estimate of the posterior density of ξ and its associated cumulative distribution function. With large datasets and under suitable regularity conditions, the posterior distribution of a parameter ξ tends in many cases to be approximately normally distributed (e.g. Cox and Hinkley, 1974). Under normality the HPD and equaltailed intervals coincide, and
Ε (ξ  Yobs ) ± z1−α / 2 V (ξ  Yobs ) ,
(4.9)
−1
where zp =Φ (p) denotes the pth quantile of the standard normal distribution, is a 100(1−α)% posterior interval for ξ. Just as evidence about a parameter in the frequentist case is often summarized by an MLE and an asymptotic Variance, Bayesian inferences can also be summarized by a posterior mean and posterior variance, and for large samples and relatively diffuse priors the two answers will tend to be very close. Unlike likelihoodbased asymptotic methods, however, Bayesian posterior simulation allows us to readily check the normal approximation, and form alternative interval estimates without recourse to a normality assumption, by examining the simulated density and posterior quantiles directly. Hypothesis tests Suppose that we are interested in examining the plausibility of the hypothesis ξ = ξ0 for some specific value ξ 0, versus the alternative that ξ>ξ0 or ξ < ξ0.. In the classical framework of hypothesis testing, one defines a test statistic that measures departures from the null hypothesis ξ = ξ0. The evidence against the null is typically measured by a pvalue, the probability of observing a test statistic as extreme or more extreme than the one actually observed, calculated under the assumption that the null hypothesis is true. A small pvalue
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indicates either that a rare event must have occurred, or that the null hypothesis must be false. In the Bayesian framework, the continuity of the posterior distribution for ξ makes ξ = ξ0 an event of zero posterior probability. The probability of a onesided alternative event ξ > ξ0 or ξ < ξ0,, however, is nonzero and is a direct measure of the plausibility of that alternative. The area under the posterior density Ρ(ξΥobs) to the left or to the right of ξ0 may be thought of as a Bayesian pvalue; a very small tail area on either side suggests that ξ0 is poorly supported and implausible given the observed data. A twosided Bayesian tail area may be defined as the α for which a 100(1−α)% Bayesian posterior interval just barely covers ξ ο . With Markov chain Monte Carlo, estimates of these tail areas are directly available from the empirical distribution function of the simulated values (4.7). In large samples, where Bayesian posterior intervals tend to closely agree with their frequentist counterparts, Bayesian pvalues will also tend to resemble frequentist pvalues. Beyond scalar quantities Until this point our discussion has been limited to inference about a single scalar summary of θ . In multiparameter problems (e.g. linear regression modeling), it is common to summarize the results of an analysis by presenting point and interval estimates for a number of scalar quantities (e.g. regression coefficients). It is important to remember, of course, that individual point and interval estimates do not immediately translate into joint inferences, because the quantities of interest are often correlated. Some of the methods above for summarizing a dependent sample generalize readily to higher dimensions, but others do not. For example, suppose that Τ ξ=(ξ1(θ),ξ2(θ),...,ξd(θ)) is a ddimensional function of θ . The posterior means, variances and covariances for ξ may be estimated by the
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obvious multivariate extensions of (4.4)(4.5). Obtaining a 100(1−α)% Bayesian posterior region, however, is more problematic. The HPD method does extend to d ≥ 2 dimensions, but estimating an HPD region using a samplebased estimate of the joint density would be difficult at best. Under the simplifying assumption that the posterior is multivariate normal, however, an assumption that may be reasonable if the data sample is large and the parameters are examined on an appropriate scale), the HPD method can be implemented rather easily. Denoting the observeddata posterior mean vector and covariance matrix of ξ by ξˆ and V, respectively, it follows from wellknown properties of the multivariate normal distribution that Τ ξ − ξˆ V −1 ξ − ξˆ  Y ~ χ 2 (4.10)
( )
( )
obs
d,
and a 100(1−α)% posterior region is the set of all vectors ξ ο f or which Τ ξ − ξˆ V −1 ξ − ξˆ ≤ χ 2
(
where
χ d2, p
o
)
(
o
)
d ,,1−α
denotes the pth quantile of the pth distribution. This region is a ddimensional ellipsoid centered at ξˆ . The Bayesian pvalue for testing ξ = ξο is the choice of a for which the ellipsoid just barely covers ξο,
(
)
(
)
Τ P χ 2d ≥ ξ o − ξˆ V −1 ξ o − ξˆ . Substituting simulationbased estimates of ξˆ and V into these expressions yields simulated posterior regions and pvalues. The assumption of multivariate normality may be checked by applying standard multivariate diagnostics to the simulated values of ξ, and, if necessary, transformations may be applied to the individual components of ξ to make normality more plausible.
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4.2.3 RaoBlackwellized estimates Under certain conditions, it is possible to greatly improve the precision of Monte Carlo estimates by RaoBlackwellization (Gelfand and Smith, 1990; Liu, Wong and Kong, 1994). The name of this method is derived from the wellknown RaoBlackwell theorem of mathematical statistics, which states that if S is an unbiased estimate of a scalar parameter and T is a sufficient statistic, then S* = E(ST) is also unbiased and has a smaller variance than S (unless S is already a function of T, in which case S* = S and the two variances are equal). Suppose that we are interested in estimating the posterior mean of ξ = ξ(θ). Recall that the output of a data augmentation algorithm is a sequence 2 1 t Υ ( ) , θ (1) , Υ ( ) , θ ( 2 ) ,..., Υ ( ) , θ ( t ) ,.... mis
mis
mis
If this sequence is preceded by a sufficiently long burnin ( t ) and θ(t) are distributed according to period, then Υmis
P(YmisYobs) and Ρ (θΥobs),, respectively for all t. The direct estimate
ξ=
1 m
m
∑ ξ ( t) t =1
will be unbiased for E(ξ  Yobs ) But notice that if an expression
for the completedata posterior mean E(ξΥobs,Υmis) is available in closed form, then we can get another estimate by averaging over the draws of Ymis,
ξ =
1 m
( t) )⋅ ∑ E(ξ  Yobs , Ymis m
(4.11)
t =1
The RaoBlackwellized estimate ξ is unbiased because
∫ E(ξ  Yobs , Ymis )P(Ymis  Yobs )dYmis = E(ξ  Yobs )
Moreover, it is at least as efficient as the direct estimate ξ , because by the key idea of the RaoBlackwell theorem, ( t) V E ξ ( t )  Yobs , Ymis  Yobs ≤ V ξ ( t )  Yobs .
(
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)
(
)
When the completedata posterior mean Ε(ξΥobs,Υmis) is easy to compute, it pays to use the RaoBlackwellized estimate rather than the direct estimate. RaoBlackwellized estimates may also be available for quantities other than the posterior mean. For example, because the posterior density of ξ may be written
P(ξ  Yobs ) = ∫ P(ξ  Yobs , Ymis )P(Ymis  Yobs )dYmis , a RaoBlackwellized density estimate is 1 m
( t) ), ∑ P(ξ  Yobs , Ymis m
(4.12)
t =1
a mixture of the completedata densities over the simulated values of Ymis. It can be shown that (4.12) is superior to direct (1) (2) (m ) estimates based on ξ ,ξ , , ξ , including kernel estimates (Gelfand and Smith, 1990). RaoBlackwellized density estimates tend to have a smooth appearance even for small m. Mixtures of completedata densities were also used by Tanner and Wong (1987) as an essential part of their data augmentation algorithm. The key idea of RaoBlackwellization is to make full use of the functional form of completedata posterior summaries and rely on simulation to solve only the missingdata aspect of the problem. Notice that when there is no missing information about ξ , the completedata and observeddata posteriors coincide, i.e. P(ξ  Yobs , Ymis ) = P(ξ  Yobs ) ⋅ When this happens, RaoBlackwellized estimates of the posterior moments, posterior density, etc. of ξ do not depend 1 2 m at all on the sample values Υ ( ) , Υ ( ) ,..., Υ ( ) , and Monte mis
mis
mis
Carlo error is entirely eliminated. Direct estimates based on (1) (2) (m ) ξ ,ξ , , ξ , however, would still contain random error for finite m even with no missing information. The relative efficiency of the two estimates is closely related to the fraction of missing information for ξ. This relationship is illustrated by the following simple example.
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Example: the efficiency of RaoBlackwellizationRecall Example 1 of Section 3.4.3 in which Y = (y1, y2,..., yn) is an iid sample from N(µ,ψ ), the first n 1 elements of Y are observed and the remaining no = n  n1 elements are missing. When ψ is known, the prior π(µ)∞ c (a constant) leads to the observed
(
)
data posterior µ  Υobs ~ N yobs , n1−1ψ , ...,where yobs is the mean of the observed data. Let µ
(1)
,µ
(2)
(m)
, . . . ,µ
be a sample
of successive values of µ from a run of data augmentation following a burnin period. Assuming the burnin is sufficiently long, the marginal distribution of each member of the sample is the stationary distribution, µ ( t ) ~ Ν yobs , n1−1ψ (4.13)
(
)
for t = 1, 2,..., m, where conditioning on Yobs is implicit and has been suppressed in the notation. Using (3.37), we can also (1) find the correlation structure of the dependent sample µ , (2)
µ ,...,µ
(
( m)
. The lagk autocovariance is
Cov µ ( t ) , µ ( t + k )
)
(
)
= Ε Cov µ ( t ) , µ ( t + k )  µ ( t ) t t t +Cov Ε µ ( )  µ ( ) , Ε µ ( + k )  µ ( t ) = 0 + Cov µ ( t ) , yobs + λk µ ( t ) − yobs t) ( k = λ V µ
) (
(
( )
(
) )
= λ n1−1ψ , where λ=no/n is the fraction of missing information. It follows k
(1)
(2)
that the joint distribution of µ ,µ ,..., µ
( m)
is multivariate
normal with all means equal to y obs, and covariance matrix n1 ψΛ,, where
—1
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1 λ λ2 L λm m −1) ( 1 λ L λ λ ∆ = λ( 2 ) λ (4.14) 1 L λ( m − 2 ) M M M M λm λm −1 λm − 2 L 1 The direct estimate of the posterior mean Ε(µΥobs) is
µ=
1 m
which has variance
V (µ ) = Τ
m
∑ µ ( t), t =1
ψ T 1 Λ1, n1m 2
(4.15)
where 1=(1,1, ,1 ) . Let us now investigate the precision of the RaoBlackwellized estimate. Recall that the Pstep of data (t) augmentation draws µ from a normal distribution with mean
(
() E µ  Yobs , Ymis t
)
and variance V(µΥobs,Υ () ymis = t
(
( t) = n −1 n1 yobs + n0 ymis ( t)
= (1 − λ ) yobs + λy mis (t)
mis)=n
1 no
1
)
ψ, where
n
∑ yi
(t)
i= n1 +1
is the average of the n0 responses imputed at the previous Istep. The RaoBlackwellized estimate of Ε(µΥobs) is therefore
µ
=
1 m
( t) ) ∑ E(µ  Yobs , Ymis m
t =1
1 m t () ymis = (1 − λ ) yobs + λ m t =1 To find the variance of µ , we need to know the covariance
∑
(1 )
(2)
( m)
structure of the sequence ymis , ymis ,..., ymis . From (3.36) it follows that
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(
)
( t +1) ( t ) ymis  µ ~ Ν µ ( t ) , n0 −1ψ ,
(4.16)
where conditioning on Y obs is to be understood. Together, t (4.16) and (4.13) imply that V y ( ) → n −1 + n −1 µ .as
( )
(1
mis
0
)
t→∞. To derive the lagk autocovariance, notice that
(
( t) ( t + k ) Cov ymis , ymis
)
(
)
( t) ( t + k ) ( t) = E Cov ymis , ymis  ymis ( t + k ) ( t) ( t) ( t) + Cov E ymis  ymis , E ymis  ymis ( t + k ) ( t) ( t) = Cov ymis , E ymi s  ymis . By repeated application of the conditional expectation rule E(U) = E[E(UV) ], one can show that t+k t t E y ( )  y ( ) = 1 − λk y + λk y ( ) ,
(
and thus
mis
mis
(
(
) (
(
) ( ) ) ) = λ V ( y( ) ) obs
( t) ( t + k Cov ymis , ymis
k
(
t mis
)
)
mis
)
= λk n1−1 + no −1 ψ
After a sufficient burnin period, the covariance matrix for the 1 1 1 2 m sample y ( ) , y ( ) ,..., y ( ) is thus (n +n ) ψΛwhere Λ is mis
mis
1
mis
o
the patterned matrix shown in (4.14), and the variance of the RaoBlackwellized estimate is λ2ψ (4.17) V ( µ˜ ) = 2 n1−1 + n0 −1 1T Λ1 m Comparing (4.17) with (4.15), we see that the relative efficiency of ~µ to −µ is
(
V (µ ) Vµ˜
)
n1−1 = 2 λ n1−1 + n0 −1
(
)
= λ−1, the inverse of the fraction of missing information. The advantage of the RaoBlackwellized estimate over the direct estimate is greatest when the fraction of missing information is small, and diminishes for λ near 1. With 10% missing information, RaoBlackwellizing the estimate without
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changing the number of iterations increases precision by a factor of 10. In other words, RaoBlackwellization allows us to achieve the same precision as with the direct estimate while using only λ times as many iterations. Now consider density estimation in the twoparameter case where µ and ψ are both unknown. Under the diffuse prior 1
π(µ,ψ)∝ ψ , the completedata posterior (3.40) implies that the marginal density for ψ is that of a scaled inverseχ
2
distribution, 2 /2 ( n − 1)S Ρ (ψ  Υobs , Υmis ) = κ −1ψ − ( n +1) exp − , 2ψ where the normalizing constant is ( n −1) / 2 n − 1 2 k =Γ 2 ( n − 1)S 2 The marginal distribution for µ can be shown to be
(
)
µ  Υobs , Υmis ~ y + S / n t n −1, for which the density is
P( µ  Yobs , Ymis ) =
()
n( µ − y )2 1 + ( n −1) S 2 n − 1)S 2 ( π n
Γ 2n
( )
Γ n2−1
−n / 2
(Section 5.2.2). In this problem, of course, the observeddata marginal posterior densities are also available in closed form; 2 we merely replace n, y and S in the completedata marginals 2 by n 1, yobs and S obs, respectively. If we simulate the observeddata posterior using data augmentation, however, we can also estimate the observeddata marginal posterior densities, either directly from the iterates of µ and ψ or by RaoBlackwellization, and compare the estimates with the known true density functions. To illustrate this, a single run of data augmentation was performed for the univariate sample of size n1 = 10 shown in Table 3.1 (a), assuming that an additional n0 = 3 observations were missing. Beginning with arbitrary starting values for µ
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and ψ, the chain was run for m = 500 iterations following an initial burnin period of 100 iterations. The true observeddata marginal densities for µ and ψ are displayed in Figure 4.1, along with three simulationbased estimates: histograms of the iterates of µ and ψ , kernel estimates based on the same and RaoBlackwellized estimates obtained by averaging the expressions for the completedata marginal densities over the iterates of Ymis. The kernel estimates are based on Gaussian kernels for µ and ψ with standard deviations of 1 and 10, respectively. In this problem, for which the fraction of missing information is 3/13=0.23, the RaoBlackwellized estimates are nearly indistinguishable from the true densities. The histograms and kernel estimates, however, show a greater amount of random error. Alth ou gh th ese un iv ar iate examp les are simp listic, we can ex pect th is ty pe o f r esult to hold tru e in general: RaoBlackw ellization can gr eatly incr ease th e eff icien cy o f simu latio n based estimates, particu lar ly w hen f raction s o f missin g inf ormatio n are small. Alth oug h a RaoBlackw ellized estimate may r eq uir e some ad ditio nal analytic
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Figure 4.1. Histograms of m = 500 consecutive iterates of µ and ψ with true marginal densities (solid lines), kernel density estimates (dotted lines) and RaoBlackwellized density estimates (dashed lines)
work to find a closedform expression for the completedata posterior summary, the extra effort is often worthwhile.
4.3 Multiple imputation Like parameter simulation, multiple imputation is a Monte Carlo approach to the analysis of incomplete data. Described by Rubin (1987) in the context of nonresponse in sample surveys, the technique is quite general and can readily be used in many nonsurvey applications as well. Multiple imputation shares the same underlying philosophy as EM and data augmentation: solving an incompletedata problem by repeatedly solving the completedata version. In multiple imputation, the unknown missing data Ymis, are replaced by
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(1) (2) ( m) . Each of the m completed simulated values Υmis , Υmis ,..., Υmis datasets is analyzed by standard completedata methods. The variability among the results of the m analyses provides a measure of the uncertainty due to missing data, which, when combined with measures of ordinary sample variation, lead to a single inferential statement about the parameters of interest.
4.3.1 Bayesianly proper multiple imputations If multiple imputation is to yield valid inferences, the simulated values of Y mis must possess certain properties. Multiple imputations drawn from a distribution possessing these properties are said to be proper. Rubin (1987) gives a technical definition for proper multiple imputations; his definition is tied to the frequentist properties of estimators over repeated realizations of a posited response mechanism. For the most part, a thorough understanding of Rubin’s definition is not crucial for the purposes of this book, for the following two reasons. First, our statistical procedures are derived primarily from perspectives of likelihoodbased or Bayesian inference; we are assuming that valid inferential statements can, be obtained through summaries of a likelihood function or posterior distribution arising from a parametric model. Second, because of our ignorability assumption, we will never need to specify a nonresponse mechanism in our analyses. Indeed, with the complicated patterns of missingness often encountered in multivariate datasets,’ it may be quite difficult to specify any realistic mechanism for the nonresponse, ignorable or otherwise. Rubin’s definition is important for discussing the statistical validity of multiple imputation from frequentist and designbased perspectives. For now, however, let us consider a different concept of what it means for multiple imputations to be proper, which is more suited to the purposes of this book. We will say that multiple imputations are Bayesianly proper if they are independent realizations of P(YmisYobs), the posterior predictive distribution of the missing data under some completedata model and prior. Notice that P(YmisYobs) may be written as
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P(Ymis  Yobs ) = P(Ymis  Yobs , θ )P(θ  Yobs )dθ , the conditional predictive distribution of Y mis, given θ averaged over the observeddata posterior of θ . Bayesianly proper multiple imputations thus reflect uncertainty about Ymis given the parameters of the completedata model, as well as uncertainty about the unknown model parameters. The fact that P(YmisYobs) does not rely on the observed response pattern R (Section 2.3) indicates that the resulting multiple imputations are appropriate under an assumption of ignorability. We will discuss Rubin’s definition at the end of this chapter. Except for that brief digression, however, the concept of proper imputations used in the remainder of this book will be that of Bayesianly proper imputations. For brevity, we will usually omit the term Bayesian and refer to the imputations simply as proper; the reader should understand that our usage of the term is not the same as Rubin’s.
∫
Proper multiple imputations and data augmentation It is convenient to create multiple imputations using data augmentation and related algorithms, because the simulated values of Ymis, created by these algorithms have P(YmisYobs) as their stationary distribution. Because proper multiple imputations must be independent, however, we will not in general be able to use successive iterates of Ymis because they tend to be correlated. Rather, we will have to subsample the chain, e.g. take every kth iterate, where k is chosen large enough so that the dependence will be negligible. Alternatively, we can create proper imputations by simulating m independent chains of length k and retaining the final values of Ymis from each chain, where k is large enough to ensure that the imputations are essentially independent of the starting values or starting distribution. Although this means that creating m imputations requires km iterations, the computational burden will not necessarily be severe, because only a small number of imputations are usually required, in typical applications, we can obtain good results with m as small as 35.
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Why only a few imputations are needed For the reader who is unfamiliar with multiple imputation, the claim that m = 3 is often adequate may be very surprising; in other applications of Monte Carlo, hundreds or thousands of draws are often needed to achieve an acceptable level of accuracy. In multiple imputation, however, a very small value of m will usually suffice. There are two fundamental reasons for this. First, like RaoBlackwellization, multiple imputation relies on simulation to solve only the missingdata aspect of the problem. As with any simulation method, one could effectively eliminate Monte Carlo error by choosing m to be very large, but with multiple imputation the resulting gain in efficiency would typically be unimportant because the Monte Carlo error is a relatively small portion of the overall inferential uncertainty. If the fraction of missing information about a scalar estimand is λ, the relative efficiency (on the variance scale) of a point estimate based on m imputations to one based on an infinite number of imputations is 1 approximately (1+λ/m) (Rubin, 1987, p. 114). When λ = 0.2, for example, an estimate based on m = 3 imputations will tend to have a standard error only 1 + 0.2 / 3 =1.033 times as large as the estimate with m = ∞. With λ = 0.5, an estimate based on m = 5 imputations will tend to have a standard error only 1 + 0.5 / 5 =1.049 times as large. In most applications, the additional resources that would be required to create and store more than a few imputations would not be well spent. The second reason why we can often obtain valid inferences with a very small m is that the rules for combining the m completedata analyses explicitly account for Monte Carlo error. A multipleimputation interval estimate makes provisions for the fact that both the point and variance estimates contain a predictable amount of simulation error due to the finiteness of m , and the width of the interval is accordingly adjusted to maintain the appropriate probability of coverage.
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4.3.2 Inference for a scalar quantity In Section 4.2 we assumed that the scalar quantity to be estimated was an explicit function of the parameters of the completedata model, denoting it by ξ = ξ(θ). Switching now to a notation more consistent with that of Rubin (1987), we denote a generic scalar estimand by Q. In multipleimputation inference, Q may be an explicit function of the parameters of the imputation model, the completedata model under which the multiple versions of Ymis were created. When this is the case, multipleimputation estimates of Q are simply RaoBlackwellized estimates, and the rules for inference given below can be interpreted as RaoBlackwellized methods that make special provisions for Monte Carlo error incurred by using a small m. In many other cases, however, Q will not be a parameter of the imputation model. In sample surveys, Q may be a function (a mean, a proportion, a ratio of means, etc.) of data from a finite population. In classical samplesurvey methods (e.g. Cochran, 1977), the population data are not modeled but regarded as fixed, and inferences are based purely on the randomization used to draw the sample. Some theoretical justification for using proper multiple imputations from a parametric model in finitepopulation survey inference is given by Rubin (1987). In other nonsurvey applications, Q is often a function of parameters of a model tailored to the specific goals of the analysis. This model, which we call the analyst’s model, may be somewhat different from the imputation model. When the imputation model and analyst’s model differ, questions naturally arise about the validity of the inference; these questions will be addressed at the end of this chapter. Completedata estimators To use multiple imputation, we must have a rule for inference about Q in the completedata case. Let Qˆ be the completedata point estimate for Q, the estimate that we would use if no data were missing. Let U be the variance estimate associated with Qˆ , so that U is the completedata standard error.
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Because Qˆ and U are both functions of Υ=(Υobs,Υmis), we will sometimes write them as Qˆ ( Υ , Υ ) and U(Υobs,Υmis) obs
mis
respectively. Multipleimputation inference assumes (a) that Qˆ and U are firstorder approximations to a posterior mean and variance of Q, (4.18) Qˆ (Yobs , Ymis ) ≈ E(Q  Yobs , Ymis ),
U (Yobs , Ymis ) ≈ V (Q  Yobs , Ymis ), (4.19) under a reasonable completedata model and prior; and (b) that the completedata problem is sufficiently regular and the sample size sufficiently large for the asymptotic normal approximation U −1 / 2 Q − Qˆ ~ N (0, 1) (4.20)
(
)
to work well. The approximation (4.20) can be justified either from a frequentist or a Bayesian perspective. To the frequentist, it is a statement about the repeatedsampling properties of Qˆ and U for a fixed value of Q; to the Bayesian, it is a statement about the posterior distribution of Q with Y (and hence Qˆ and U) held fixed. Many, but not all, commonly used estimators can be regaxded as approximate posterior means, and their variance estimates as approximate posterior variances. MLEs and their asymptotic variances, derived from the curvature of the observed or expected loglikelihood at the mode, typically satisfy (4.18)(4.20) (Cox and Hinkley, 1974). Estimators that are clearly inefficient, e.g. a sample mean based on only half of the sample, are definitely ruled out, as they do not use all the available information in Y. Certain classes of nonparametric procedures (e.g. methods based only on ranks) should also be ruled out, as they tend to sacrifice some efficiency to avoid specification of a full parametric model. With multiple imputation, just as with complete data, it is good practice to perform the analysis on a scale for which the asymptotic normal approximation is likely to work well; for example, with a correlation coefficient, it is advisable to apply Fisher’s transformation (3.14).
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Rule for combining completedata inferences With m imputations, we can calculate m different versions of Qˆ and U. Let ( t) Qˆ ( t ) = Qˆ Yobs , Ymis
(
and
)
(
() U ( t ) = U Υobs , Υmis t
)
be the point and variance estimates using the tth set of imputed data, t = 1,2,...,m. Rubin (1987, Chap. 3) gives the following rule for combining them. The multipleimputation point estimate for Q is simply the average of the completedata point estimates,
1 Qˆ = m
m
∑ Qˆ (t) .
(4.21)
t =1
The variance estimate associated with Q has two components. The withinimputation variance is the average of the completedata variance estimates,
U =
1 m
m
∑ U ( t)
(4.22)
t =1
The betweenimputation variance is the variance of the complete data point estimates, in
B=
1 m −1
∑ (Qˆ (t) − Q ) . m
2
(4.23)
t =1
The total variance is defined as Τ = U + 1 + m −1 Β,
(4.24)
and inferences are based on the approximation Τ −1 / 2 Q − Q ~ t v ,
(4.25)
(
)
(
)
where the degrees of freedom are given by 2
U . v = ( m − 1)1 + 1 + m −1 B Thus a 100(1−α)% interval estimate for Q is
(
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)
(4.26)
(4.27) Q ± t v,1−α / 2 T , and a pvalue for testing the null hypothesis Q=Q against a twosided alternative is 2P t v ≥ T −1 / 2  Q − Q ′ 
(
)
or, equivalently,
2 P F1, v ≥ T −1(Q − Q ′)
(4.28)
Missing information Notice that the degrees of freedom (4.26) depend not only on m, but also on the ratio 1 + m −1 B (4.29) r= . U Rubin (1987) calls r the relative increase in variance due to nonresponse, because U represents the estimated total variance when there is no missing information about Q (i.e. when B = 0). When m, is large and/or r is small, the degrees of freedom will be large and (4.25) will be approximately normal. If we define information as minus one times the average second derivative of the logposterior density of Q , the information in the approximate posterior (4.25) is (ν+1)(ν+3)
(
)
1 1
Τ . With no missing information, the posterior would become normal with mean Qˆ and variance U , for which the
information is U −1 . It follows that λˆ = U −1( v + 1)( v + 3) −1T −1 U
(
)
(4.30) r + 2 / ( v + 3) = r +1 is an estimate of the fraction of missing information about Q. In applications, calculation of r and λˆ is highly recommended, as they are interesting and useful diagnostics for assessing how the missing data contribute to inferential uncertainty about Q.
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Heuristic justification An imprecise but intuitive justification for this procedure is the following. Let us assume that the observeddata posterior for Q is approximately normal, so that if the observeddata posterior moments could be calculated we would use the interval
(4.31) E(Q  Yobs ) ± z1−α / 2 V (Q  Yobs ) . Because E(QYobs) and V (QYobs) are not readily available, however, we use simulationbased estimates of them provided by the multiple imputations. Notice that by (4.18) and (4.19) we can write (4.32) Ε (Q  Υobs ) ≈ Ε (U  Υobs ) and V (Q  Yobs ) ≈ E(U  Yobs ) + V Qˆ  Yobs , (4.33)
(
)
where the moments on the righthand sides of (4.32) and (4.33) are calculated over the distribution P(YmisYobs) from which the multiple imputations are drawn. By the law of large numbers, Q , U and B approach E Qˆ  Yobs , E(U  Yobs ), and
(
)
(
)
V Qˆ  Yobs , respectively, as m→ ∞ . Thus, with an infinite number of imputations, (4.34) Q ± z1−α / 2 U + B would be identical to (4.31). Because m is typically small, however, we need to make two adjustments to (4.34). First, the interval must be widened to reflect the fact that Q is . .With proper randomly different from E Qˆ  Y
(
obs
)
imputations, B/m is an unbiased estimate of the variance of Q , so to account for the error in Q we must increase the estimate of the total variance by this amount. Second, because the estimated variance components U and B are also estimated with error, we need to widen (4.34) further by replacing the normal quantile with one from a t distribution.
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Further justification Rubin (1987) derives the procedure more formally by Bayesian arguments, showing that (4.25) is an approximate observeddata posterior distribution for Q based on the reduced information in and Qˆ (1) , Qˆ (2) ,..., Qˆ ( m) 1) 2) m) ( ( ( rather than on the infinite number of U , U ,..., U imputations that one would ideally have. The expression (4.26) for the degrees of freedom v are obtained by approximately matching the first two moments of the reducedinformation posterior to those of a t distribution. Despite the Bayesian derivation, evaluations have shown that this method leads to inferences that are well calibrated from a frequentist standpoint. Rubin and Schenker (1986) report that multipleimputation interval estimates tend to have at least the nominal coverage (i.e. a 95% interval covers the true parameter at least 95% of the time) in a variety of scenarios even for m as small as 2. When the actual coverage falls below the nominal coverage, it tends to be either because (a) the fraction of missing information is unusually large, or (b) the completedata normal approximation (4.20) works poorly. In the former case, we can obtain better results by choosing a larger m. In the latter case, the poor results should be regarded as a inherent shortcoming of the asymptotic approximation for complete data rather than as a failure of the multiple imputation methodology. In many cases, the quality of the completedata normal approximation can be improved by a suitable reparameterization. In addition to simulation studies, further theoretical justification for this method is provided by Schenker and Welsh (1988), who established frequentist consistency of multipleimputation inferences for linear regression analysis with an incomplete response variable. The result was later extended by Brownstone (1991) to incomplete predictors. Additional references supporting the use of (4.25) are given by Rubin (1996).
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4.3.3 Inference for multidimensional estimands Several extensions of the above method have been developed for estimands that are multidimensional. Suppose now that Q 1is a k x 1 vector. Rather than finding confidence regions for Q, which are often difficult to interpret (especially when k is large), we will focus on finding a pvalue for testing the hypothesis that Q equals a particular value of interest Q 0. In practice, this typically arises because one is interested in comparing two models for the data, M 0 and M1, where M 1 is more general than M0 and reduces to M 0 when Q = Q0. We now discuss three alternative rules for calculating a pvalue from multiplyimputed data. Combining point estimates and covariance matrices Let Qˆ be a completedata point estimate of Q, and U an asymptotic covariance matrix associated with Qˆ . The following method assumes that, with complete data, the distribution of ( Qˆ  Q) is sufficiently close to N(0, U) that an accurate pvalue may be obtained from the multivariate Wald statistic and a chisquare reference distribution, Τ (4.35) Ρ χκ2 ≥ Qˆ − Qo U −1 Qˆ − Qo With large samples, (4.35) is a valid pvalue both in the frequentist and the Bayesian sense. With incomplete data we cannot calculate (4.35) and need a new test statistic that is a function only of Yobs. As in the scalar case, with m imputations we calculate m estimates covariance matrices Qˆ (1) , Qˆ (2) ,..., Qˆ ( m) a n d m 1) 2) m) ( ( ( U , U ,..., U . The multivariate analogues of (4.2l)
(
(4.24) are
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)
(
)
Q= U = B=
m
1 m
∑ Qˆ (t) ,
1 m
∑ U ( t) ,
t =1 m
t =1
1 m −1
∑ (Qˆ (t) − Q )(Qˆ (t) − Q )
(
m
t =1
Τ
,
)
T = U + 1 + m −1 B. Using the natural multivariate extension of (4.28), one might suppose that T P Fk ,υ ≥ Q − Qo T −1 Q − Qo / k would be an appropriate pvalue, where v depends on the precision with which T estimates the observeddata posterior variance of Q, V (Q  Yobs ) ≈ E(U  Yobs ) + V Qˆ  Yobs .
(
)
(
)
(
)
It turns out, however, that finding an adequate reference distribution for the statistic
(Q − Q0 )T T −1(Q − Q0 ) / k
is not a simple matter. The main problem is that for small m, the betweenimputation covariance matrix B is a very noisy estimate of V Qˆ  Yobs , , and does not even have full rank if m
(
)
≤ k. One way out of this difficulty is to make the simplifying assumption that the population between and withinimputation covariance matrices are proportional to one another, V Qˆ  Yobs ∝ E(U  Yobs ),
(
)
which is equivalent to assuming that the fractions of missing information for all components of Q are equal. Under this assumption, a more stable estimate of total variance is T˜ = (1 − r1 )U where
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(
)(
r1 = 1 + m −1 tr BU
−1
)/ k
(4.36)
is the average relative increase in variance due to nonresponse across the components of Q. Using T˜ rather than T, the test statistic becomes T (4.37) D = Q − Q T˜ −1 Q − Q / k , 1
(
0
)
(
0
)
and a pvalue for testing Q = Q0 is p = Ρ Fk , v1 ≥ D1 .
(
)
The best approximation to date for the degrees of freedom v1 is given by Li, Raghunathan and Rubin (1991),
[ (
) ]
2
υ1 = 4 + ( t − 4) 1 + 1 − 2t −1 r1−1 ,
(4.38)
where t = k(m  1). This procedure requires t > 4; when t ≤ 4, we may use an alternative expression given by Rubin (1987),
(
)(
υ1 = t 1 + k −1 1 + r1−1
)2 / 2
Although (4.37) and 4.38) are derived under the strong assumption that the fractions of missing information for all components of Q are equal, Li, Raghunathan and Rubin (1991) report encouraging results even when this assumption is violated. In a simulation study, they examined the performance of this procedure using values of m between 2 and 10 and average fractions of missing information up to 0.5, in problems with up to k = 35 parameters. At worst the procedure tends to be somewhat conservative (overstating the pvalues), and there is a small loss of power relative to the ideal case with m=∞. These simulations support the use of this procedure in many situations of practical interest. It is important to note, however, that the simulations assume the appropriateness of (4.35), the chisquare approximation for the completedata Wald statistic. For the procedure to work well in practice, we need a large sample and an appropriate scale for Q to ensure validity of the usual completedata asymptotic approximations.
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Combining pvalues Calculation of the statistic D, and its associated degrees of freedom requires access to the point estimates and covariance matrices from the m completedata analyses. Software packages for common procedures, e.g. linear or logistic regression, typically allow the user to examine and save the covariance matrices for further analysis. When k is large, however, this procedure may be somewhat cumbersome, particularly when a large number of tests are to be performed. One might ask whether it is possible to obtain a valid inference using only the m completedata pvalues, or, equivalently, the m, completedata Wald statistics −1 T t d ( ) = Q ( t ) − Q0 U ( t ) Q( t) − Q , (4.39) W
(
)( ) (
0
)
t = 1, 2,...,m. Such a procedure is described by Li et al. (1991), who propose the statistic
D2 =
dW k
−1
− ( m + 1)( m − 1) −1r2 , 1 + r2
(4.40)
where
dW =
1 m
m
∑ dW(t) t =1
is the average of the Wald statistics, and 2 m ( t) −1 1 r2 = 1 + m dW − dW m −1 t =1 1 is (1 + m ) times the sample variance of their square roots. The quantity r2 is a clever estimate of r1, the average relative crease in variance due to nonresponse, based only on the Wald statistics. Notice that with no missing information, r2 = 0 and (4.40) reduces to the average of the Wald statistics divided by k. The combined pvalue for testing Q = Q0 is p = P(Fk ,υ 2 ≥ D2 ), where the degrees of freedom are
(
)
∑
(
)2
ν 2 = κ −3 / m ( m − 1) 1 + r2 −1 .
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(4.41)
This procedure was developed partly using theoretical arguments and partly through the results of simulation studies for m = 3, so it should be expected to work best with m = 3 imputations. Li et al. (1991) examined the behavior of this procedure in problems with up to k = 25 parameters, with m ranging from 2 to.10 and the average fraction of missing information λ up to 0.5 both when the individual fractions of missing information are equal and unequal. For a nominal 5%level test, the procedure tends to be conservative for λ < 0.2 and anticonservative for λ > 0.2. In what we might expect to be one of the worst cases (k = 25, m = 2 and λ_ = 0.5) the actual level of the 5% test is about 8%. Overall, the results seem to be best with m = 3, which is not surprising because the procedure was developed with m = 3 in mind. In simulations D2 was not highly correlated with the more nearly optimal statistic D1, so there appears to be a substantial loss in power when using D2 rather than D1. Li et al. (1991) suggest that this procedure be used only as a rough guide, and that the analyst should interpret it as providing a range of pvalues between one half and twice the calculated value. Combining likelihoodratio test statistics A third procedure for multivariate estimands, which may be regarded as intermediate between the previous two, is described by Meng and Rubin (1992b). Making use of the well known fact that the Wald statistic is asymptotically equivalent to that of the likelihoodratio test, they propose a method for combining the completedata likelihoodratio test statistics. The resulting statistic, D3, is typically easier to compute than D, although not quite as convenient as D2 It is, however, asymptotically equivalent to D, for any m, so it should retain the good performance of D, in a wide variety of scenarios. Let ψ denote the vector of unknown parameters in the analyst’s model, and Q=Q(ψ) a kdimensional function of ψ that is of interest; specifically, we wish to test the hypothesis that Q = Q0 for a given Q0. Let ι (ψΥobs,Υmisι) denote the completedata loglikelihood function, ψˆ the MLE or
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maximizer of ι(ψΥobs,Υmis), and ψˆ
the maximizer of
ι(ψΥobs,Υmis) subject to the constraint Q(ψ)=QΟ . In regular problems, the completedata likelihood ratio test statistic dL = dL (ψˆ , ψˆ 0  Yobs , Ymis ) = 2 l(ψˆ  Yobs , Ymis ) − l(ψˆ 0  Yobs , Ymis )
[
]
χ k2
is asymptotically distributed as under the null hypothesis, and k is asymptotically equivalent to the Wald statistic (4.35). Let t ( t) dL( ) = dL ψˆ ( t ) , ψˆ 0( t )  Yobs , Ymis
(
)
be the likelihoodratio test statistic from the tth imputed dataset, t = 1, 2,...,m , where ψˆ ( t ) is the maximizer of (τ) ) and ψˆ ( t ) is the maximizer of ι(ψΥ ,Υ
(
obs
mis
)
( t ) subject to Q(ψ)=Qo. Let l ψ  Υobs , Υmis
dL =
1 m
m
∑ dL(t) t =1
be the average of these likelihoodratio statistics, and m
1 ψ = m
∑ ψˆ (t)
(4.42)
1 m
∑ ψˆ 0(t)
(4.43)
ψ0 =
t =1 m
t =1
the averages of the completedata estimates of ψ across imputations. Finally, let
1 d˜L = m
( t) ) ∑ dL (ψ 0 ,ψ  Yobs , Ymis m
t =1
be the average of the likelihoodratio statistics evaluated at ψ 0 and ψ rather than at the imputationspecific parameter estimates. The test statistic proposed by Meng and Rubin (1992b) is
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D3 =
d˜L , k (1 + r3 )
where
(4.44)
(
)
m +1 d − d˜L (4.45) k ( m − 1) L is an alternative estimate of the average relative increase due to nonresponse that is asymptotically equivalent to (4.36). The pvalue associated with D3 is p = P Fκ , v 3 ≥ D3 (4.46) r3 =
(
)
with degrees of freedom calculated in the same manner as for D1, 2 −1 −1 if t = k ( m − 1) > 4 4 + ( t − 4) 1 + 1 − 2t r3 v3 = 2 otherwise t 1 + k −1 1 + r3−1 / 2 In addition to the usual likelihoodratio test statistics for each imputed dataset, this procedure also requires evaluation of the completedata likelihood ratio at (ψ , ψ 0 ) for each dataset. Implementation of this procedure thus requires code for evaluating the completedata loglikelihood at userspecified values of the parameter, something which is not typically provided in standard statistical software. For many commonly used models, however, the completedata loglikelihood is straightforward to derive and compute, and with a little effort on the part of the analyst the procedure can often be implemented without difficulty. Because this method is asymptotically equivalent for any m to the one that uses D1, the properties of the two methods should be very similar. Notice that a Wald test depends on the particular choice of scale for the unknown quantity Q, whereas a likelihoodratio test is invariant to changes in scale. For this reason, some prefer the likelihoodratio test in certain cases, believing it to be somewhat more trustworthy than the Wald test when the normality of ( Qˆ  Q ) is in doubt. The likelihoodratio procedure for multiplyimputed datasets described above may, at first glance, appear to be scaleinvariant, but it is not; in
(
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[ (
)(
)
) ]
particular, the averaging of the parameter estimates (4.42)(443) will lead to somewhat different results under nonlinear transformations of ψ . The derivation of this procedure does assume the approximate completedata normality of (ψˆ − ψ ), , so for best results the parameter estimates should probably be averaged on a scale for which the normality assumption is reasonable. Care must be taken, however, to ensure that the averages of parameter estimates lie within the parameter space, which will not necessarily happen if the averaging is done on an arbitrary scale. The sensitivity of this procedure to alternative parameterizations is not entirely clear and is worthy of further investigation.
4.4 Assessing convergence In the last two sections, we examined techniques for extracting inferentially meaningful quantities from the output of Markov chain Monte Carlo. Responsible use of these methods requires some formal or informal assessment of the simulation algorithm’s convergence properties; we need to know whether the algorithm has run ’long enough’ for the results to be reliable. The meaning of convergence, and the diagnostic tools for assessing it, will vary according to the method of inference being used. In parameter simulation, we must choose a number of iterations to ensure that the resulting summaries (sample moments, quantiles, etc.) are sufficiently close to the posterior quantities they estimate; in this case, we need to assess convergence in the sense given by the law of large numbers (4.3). With multiple imputation, however, the goal is to simulate approximately independent draws from P(YmisYobs); in that case, we need to assess convergence of the distribution of the iterates to their stationary distribution. Because these two concepts of convergence are quite different, the relevant diagnostic tools for assessing them are necessarily different. Convergence to stationarity, the weaker of the two concepts, is discussed in Sections 4.4.14.4.3; convergence of estimated posterior summaries is discussed in Section 4.4.4.
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4.4.1 Monitoring convergence in a single chain We now address the question, ’How long do I have to run my algorithm before it converges to the stationary distribution?’. The answer, of course, is that it depends. The rate of convergence depends on the fractions of missing information (Section 3.5.3) which vary from application to application. Even within a single application, the number of iterations required to achieve approximate stationarity depends on the starting value or starting distribution. For example, convergence will be faster from a starting value near the center of the observeddata posterior than from a starting value in the tails. In practice, it is helpful to know roughly how large (t+k) a value k is needed for θ to be essentially independent of
θ
(τ)
for any θ
(t)
within the range of appreciable posterior
density. If such a value were known, then a burnin period of length k would be sufficient to achieve stationarity provided that the starting value was not highly unusual with respect to P(θΥobs). Moreover, after the burnin period, every kth iterate of Υmis could then be taken as an independent draw from P(θΥobs)., and every kth iterate of Ymis could be used for proper multiple imputation. Various methods for approximating k have appeared in the literature. The most accessible of these involve output analysis, examining the iterates of θ from one or more simulation runs. When θ is multidimensional, we can monitor the behavior of various components or scalar functions of θ . The marginal distributions of the components will often converge at different rates, however, and convergence by the kth iteration for every component or function that we examine does not necessarily imply that the joint posterior has converged; there is always a possibility that the distribution of some unknown function has not yet converged. Several methods have been proposed for choosing a value of k pertinent to the convergence of the entire joint distribution (Ritter and Tanner, 1992; Roberts, 1992; Liu and Liu, 1993), but these can be difficult to implement in practice. In typical
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missingdata scenariosaddressed by this book, fractions of missing information are moderate and data augmentation algorithms tend to converge quickly. Pathological behavior such as slow convergence or nonexistence of a stationary distribution usually means that the model is too complicated (i.e. has too many parameters) to be supported by the observed data, and the problem should probably be reformulated. For our purposes, the most sensible diagnostics are those that can be implemented quickly and easily, providing an informal but reliable assessment of whether the situation is normal or pathological. Convergence diagnostics have been and will probably continue to be the subject of vigorous research efforts, and improved methods may be available soon. Further discussion and references on convergence are given by Smith and Roberts (1993); Tanner (1993); and Gilks, Richardson and Spiegelhalter (1996). Timeseries plots and autocorrelation For an individual component or function ξ=ξ(θ), plotting the iterates of ξ from a single run can be a quick and easy way to assess convergence for that component. Recall Example 1 of Section 3.4.3 in which the first n1 values of a univariate normal sample are observed and the remaining n0 = n  n 1 are missing, and consider data augmentation for the twoparameter case in which θ = ( µ, ψ ) is unknown. Using the n1 = 10 data values in Table 3.1 (a), we performed runs of data augmentation for two different cases: n0 = 3, corresponding to 23% missing information, and n0 = 90, corresponding to 90% missing information. In each case, we used the starting value µ (0) , ψ (0) = 30, 70 and ran a single chain. Timeseries plots
(
)
(
)
of ψ and ψ over the first 100 iterations are shown in Figure 4.2. The variance ψ is plotted on a log scale, for which the posterior distribution is more nearly symmetric. ( Ο) = 30 is located in the distant tails of the Because µ observed data posterior (see Figure 4.1), Figure 4.2 (a) and (c) show initial trends as µ wanders back into the region of high
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posterior density. Had th e starting v alu e been lo cated near the center of the distrib ution ( e.g. at an o bserv ed data MLE or po ster ior mode) th is tr end w ou ld no t h av e been evident. O n ce th e p ar ameter s ar e in th e r eg io n of ap p reciab le d ensity , ser ial co r r elatio n p r o v id es evid en ce ab o u t h o w fast th e alg o rith m co n v er ges. For 2 3 % missing in f or mation ,
Figure 4.2. First 100 parameter iterates from single runs of data augmentation: (a) µ and (b) log ψ with no = 3, corresponding to 23% missing information; (c) µ and (d) log ψ with no 90, corresponding to 90% missing information.
(a) and (b) reveal no discernible trends; the plots resemble horizontal bands, indicating a low ratio of signal to noise. For 90% missing information, however, (c) and (d) reveal important trends lasting for 25 iterations or more, indicating that successive iterates are highly correlated. The plots in (a) and (b) are typical of situations in which the fractions of missing information are low to moderate, for which data augmentation is known to converge rather quickly. Longterm trends and high serial correlation, as in (c) and (d), are typical when the fractions of missing information are high and data augmentation converges slowly.
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To investigate relationships among successive iterates, we (t) (t+k) (t) could examine scatterplots of µ versus µ and ψ versus
ψ
(t+k)
for various choices of k. A more concise way to
represent these relationships, however, is through the autocorrelation function (ACF). The lagk autocorrelation for (t) a stationary series {ξ :=1,2, ,m} is defined to be Cov ξ ( t ) , ξ ( t + k )
ρk =
(
).
( ) Notice that by stationarity V (ξ ( ) ) = V (ξ ( ) ). . A sample estimate V ξ ( t) t
(4.47)
t+k
Figure 4.3. Sample ACFs for the series in Figure 4.2 (a)(d) estimated from iterations 11 to 100, with dashes indicating approximate 0.05level critical values for testing ρk = ρk+1= ρk+2= = 0.
of Pk is given by
ξ ( t ) − ξ )(ξ ( t + k ) − ξ ) ∑ t =1 ( rκ = , m ( t) − ξ 2 ξ ) ∑ t =1( m−k
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(4.48)
where ξ is the mean of the series (e.g. Box and Jenkins, 1976). A plot of rk versus k for relevant values of k, known as a sample ACF plot or correlogram, provides a useful summary of linear serial dependence. Sample ACFs for the four series in Figure 4.2 are shown in Figure 4.3. To prevent the estimates from being unduly influenced by initial trends due to the implausible starting value, the first 10 values from each series were omitted from the calculation of the sample ACFs. Because the four series in Figure 4.2 are actually two bivariate series, we could also have estimated crosscorrelation (t) ( t+k) functions to assess the relationships between µ and ψ and between µ
(t+k)
(t)
and ψ but for brevity these are omitted.
Variability of the sample autocorrelation The sample ACFs in Figure 4.3 (a) and (b) show that serial dependence in the first series (n0 = 3) dies out very quickly; the estimated correlations are below 0.2 even at lag 1. The second series, represented by (c) and (d), however, exhibits a high degree of serial dependence, and the correlations are still large beyond lag 10. Notice that in (c) and (d), as k increases rk drops below zero. In general, one would not expect negative autocorrelations; the negative estimates are fluctuations due to the small sample size. Sample ACFs can be quite noisy, especially when the true serial correlation is high, and adjacent autocorrelation estimates are themselves highly correlated. For this reason, it is helpful to calculate estimates of variability associated with a sample ACF. For a stationary normal process that dies out after lag k’ (i.e. ρk=0 for all k >k′) the variance of rk for k >k′ is approximately
V (rκ ) ≈
1 1 + 2 m
k′
∑ ρt2 ,
(4.49)
t =1
where m is the sample size or length of the series (Bartlett, 1946). Moreover, when ρ k=0 the distribution of r k is approximately normal (Anderson, 1942). Therefore, an
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approximate alevel test of the null hypothesis of no correlation at lag k or beyond, ρk=ρk +1 =ρk+2=…=0, (4.50) versus the alternative hypothesis ρk≠0, rejects the null if 1/ 2
k −1 1 rk ≥ z1−α / 2 1 + 2 rt2 m t =1 Critical values for 0.05level tests of (4.50) for each k are shown in Figure 4.3 as dashed lines. In (a) and (b), none of the correlations for lag I or beyond are significantly different from zero. In (c) and (d), the correlations do not differ significantly from zero beyond lag 6, but the large standard errors indicate that the estimates are very noisy. To accurately estimate the true correlation structure for (c) and (d), we obviously need a much larger sample size. Figure 4.4 shows sample ACFs based on simulation runs of length m = 10 000, for which the standard errors are negligibly small. From these plots, it is apparent that the autocorrelations are effectively zero by lag 3 for (a) and (b), and by lag 40 for (c) and (d). A s d e m o n s t r a t e d b y th i s e x a m p l e , l o n g  t e r m tr e n d s o r d r i f ts in s c a la r s u m m a r i e s o f θ in d ic a te s l o w c o n v e r g e n c e t o s ta t io n a r i ty , w h e r e a s th e a b s e n c e o f s u c h t r e n d s s u g g e s ts r a p id c o n v e r g e n c e . T i m e  s e r ie s p l o t s m a y a l s o h e l p d i a g n o s e p a th o lo g ic a l s it u a t i o n s i n w h ic h t h e a l g o r i th m d o e s n o t c o n v e r g e a t a l l b e c a u s e t h e p o s t er i o r
∑
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Figure 4.4. Sample autocorrelation functions for the series in Figure 4.2 (a)(d) estimated from 10 000 iterations, with dashes indicating approximate 0.05level critical values for testing ρk=ρk+1=ρk+2=…=0.
distribution does not exist. Recall the example of Section 3.5.2 in which a single value y 1 from N ( µ, ψ ) is observed but a second value y2 is missing, and we apply the improper prior −1 π(µ,ψ)∝ψ . The I and Psteps of data augmentation (3.46)(3.47) are well defined, but the observeddata posterior is not proper because y1 alone provides no information about ψ . Figure 4.5 shows timeseries plots from a single run of data augmentation for this example with 0) 0) ( ( y = 0, µ = 1 and ψ = 1. In the first 100 iterations, the 1
range of the observed values of log ψ exceeds 35, which means that ψ itself varies over more than 15 orders of magnitude. With more iterations, ψ continues to drift unless the program halts due to numeric overflow or underflow. Whenever ψ wanders close to zero, µ is constrained to be very close to y 1 = 0, but when ψ is large µ can become large in either the positive or negative direction. Such highly erratic
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behavior in timeseries plots suggests that one or more components of θ are nearly or entirely inestimable from Yobs. Warnings about time series and autocorrelation Timeseries plots and autocorrelation are easy to understand and implement, but they are not foolproof. Suppose that for all the
Figure 4.5. First 100 iterates of (a) _ and (b) log ψ from a single run of data augmentation with a nonexistent posterior.
scalar summaries of θ we examine, the autocorrelations have effectively died out after lag k. Should we conclude that the algorithm effectively converges to stationarity after k steps? The answer is no, for several reasons. First, zero correlation is not precisely the same as independence, and nonlinear associations may exist beyond lag k. In practice this is probably not a major concern, particularly when the components or functions of θ have been scaled to approximate normality. More importantly, the possibility always exists that the algorithm has not converged with respect to some component or function of θ that we have not examined. For this reason, it is always wise to monitor scalar functions that are suspected to converge slowly, i.e. functions for which the fractions of missing information are thought to be high. Practical suggestions for finding such functions are given in Section 4.4.3. A final reason why timeseries plots and sample ACFs can mislead is that the observeddata posterior distribution may be oddly shaped, and the algorithm may have inadequate opportunity to visit certain regions of the parameter space for
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reasonable choices of m. For example, if the posterior is multimodal and the modes are separated by regions of low density, an algorithm could ’get stuck’ near one of the modes for a large number of iterations. If we had the misfortune of starting near a local mode that was far from the others, we could be misled into thinking that the algorithm had converged when, in fact, it had never left the vicinity of the local mode. Multiple modes and oddlyshaped posteriors are typically associated with datasets that are sparse, i.e. having a small sample size, high rates of missingness, and a large number of parameters to be estimated. In many cases these difficulties can be detected a priori by thoughtful examination of the data and missingness patterns. Multiple modes and high fractions of missing information can also be detected by the behavior of EM (Section 3.3), or by repeated simulation runs from a variety of starting values. 4.4.2 Monitoring convergence with parallel chains Another group of methods for diagnosing convergence to stationarity involves running multiple independent chains from a common starting value or starting distribution (Tanner and Wong, 1987; Gelfand et al., 1990; Gelman and Rubin, 1992a). Suppose that we choose R starting values of ⁄ from a distribution ⁄ If we simulate a single chain of length m from each starting value, then the iterates of θ form an array, θ (1:0) , θ (1:1) , θ (1:2) , L θ (1: m) ,
θ (2:0) , θ (2:1) , θ (2:2) , L θ (2: m) , M R:0) R:1) R:2) ( ( ( θ , θ , θ , L θ ( R: m) , r:t where value t from run r is denoted by θ ( ) . If f0 assigns all r:o its mass to a single point, then the starting values θ ( ) will be identical; otherwise they may be different. Denote the replicate values of θ at iteration t collectively by
{
}
θ (∗: t ) = θ ( r: t ) : r = 1, 2,..., R .
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∗: t If stationarity has been achieved by step t, then θ ( ) will be an iid sample from the target distribution P ( θΥobs).. ∗: t Examination of summaries of θ ( ) for t = 1, 2,... thus provide
evidence about how rapidly the process converges to stationarityfrom f0. As with previous methods, one needs to decide which summaries of the distribution of θ to monitor at each iteration. Some obvious choices are sample moments, quantiles and density estimates for scalar functions of θ . Note that even when stationarity is achieved, these sample quantities could vary considerably across iterations simply due to the finiteness of R. Unless R is very large we may have difficulty in deciding ∗: t whether the discrepancy between summaries of θ ( ) and
θ (∗: t +1) is due to nonstationarity or ordinary sampling fluctuation. It may be possible to reduce the sampling fluctuation in the estimates by RaoBlackwellization. Suppose that rather than retaining the iterates of θ from the multiple runs, we retain iterates of Ymis. Let
{
}
( ) ( ) Υmis = Υmis : r = 1, 2,..., R , ∗: t
r:t
( ) denotes the tth value of Y from simulation run where Υmis mis r. A comparison of RaoBlackwellized moment or marginal ∗: t +1) ∗: t suggests density estimates based on Υ ( ) and Υ ( r:t
mis
mis
whether convergence has been attained by iteration t. Use of this technique in a univariate normal example is illustrated in Figure 3.2. Overdispersed starting values Multiplechain methods help diagnose how many iterations are required for convergence from a particular starting distribution. This is somewhat different from our working notion of convergence given at the beginning of Section 4.4.1, t: k however, which requires a k large enough so that θ ( ) is
t t essentially independent of θ ( ) for any θ ( ) within the region of appreciable posterior density. To check convergence in the
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latter sense, one would need to try a variety of starting values over the region where Ρ(θΥobs) is appreciable. In general, one would expect to achieve stationarity more rapidly from a starting value near the center of the posterior (e.g. an MLE or posterior mode) than from a starting value in the tails. For this reason, Gelman and Rubin (1992a) recommend multiple runs from starting values that are overdispersed relative to (i.e. exhibiting greater variability than) the target distribution Ρ(θΥobs), because this will result in a conservative estimate of the number of iterations needed to achieve stationarity. Moreover, it will greatly reduce our chance of being misled if the posterior is so oddly shaped that single runs tend to get stuck in small regions. Obtaining starting values that are overdispersed relative to the target distribution Ρ(θΥobs), may not be a simple matter, because in applications Ρ(θΥobs),is the intractable distribution that the algorithm is intended to simulate. Gelman and Rubin (1992a) recommend drawing starting values from a multivariate tdistribution with tails heavier than the normal (e.g. a multivariate t with few degrees of freedom) centered at the posterior mode, with covariances determined by the second derivative matrix of the logposterior at the mode. If multiple modes are found, they recommend using a mixture of multivariate distributions centered at each mode. In practice this method would be tedious to implement for many of the problems in this book, because the observeddata loglikelihoods are often complicated and difficult to differentiate. Numerical estimates of a secondderivative matrix can be obtained with the SEM algorithm (Section 3.3.4), but when the dimension of θ is high this can be computationally prohibitive as well. If the prior distribution being applied to θ is proper, then the prior may serve as a handy source of starting values, particularly if it is easy to simulate. When the prior is improper, however, this will not be possible. One simple method for obtaining an overdispersed starting distribution that may work well in a variety of problems is bootstrap resampling (Efron and Tibshirani, 1993). Suppose
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that the observed multivariate data matrix Y obs has n rows corresponding to n sample units, some of which have missing values on one or more variables. Let θˆ = θˆ( Υobs ) denote the
MLE or posterior mode of θ , which can be found, for example, via the EM algorithm. Suppose that we construct a ∗ , by drawing a simple random new observed data matrix Yobs sample of n* rows from Yobs with replacement, and calculate * θˆ* = θˆ Υobs , e.g. by applying the EM algorithm to Yobs. If we
(
)
take n* = n, then θˆ ∗ will be an approximate draw from the sampling distribution of θˆ and in wellbehaved problems for which the observeddata posterior is approximately normal, the distribution of θˆ ∗ will not be far from the observeddata posterior. If we use a value of n* considerably smaller than n, say n/2, then θˆ ∗ may tend to be overdispersed relative to Ρ(θΥobs).. Approximating the posterior distribution of a parameter by the sampling distribution of its MLE is not, in general, a practice to be recommended for purposes of inference (e.g. Hill, 1987). For the mere purpose of finding starting values for a Markov chain Monte Carlo algorithm, however, a high degree of accuracy is not required, and bootstrap resampling with n* < n may be perfectly adequate. 4.4.3 Choosing scalar functions of the parameter The methods we have discussed for diagnosing stationarity pertain to individual components or scalar functions of θ. When the dimension of θ is very high, it may not be feasible to monitor convergence for every component of θ, much less all the functions of θ , that seem relevant. If the goal of the analysis is to draw inference about one particular function ξ=ξ(θ), then we may not need to worry about convergence in the global sense; convergence with respect to the marginal distribution of ξ may be good enough. In the an alysis of real datasets, ho wever ( and p ar ticularly wh en we ar e tr yin g to cr eate mu ltiple imputation s fo r a v ariety of fu tur e an aly ses)
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it may be d iff icult to prespecif y all o f the f unction s o f θ that ar e relev an t, an d ach ievin g statio nar ity w ith respect to the en tire jo in t d istr ibu tion of θ b ecomes mo re impo rtant.
Figure 4.6. Incomplete multivariate dataset with four variables.
In highdimensional situations, we should pay particular Attention to components or functions of θ for which convergence is likely to be slow. Because convergence rates are closely related to missing information, it makes sense to focus on aspects of θ for which the fractions of missing information are high. Often some of these functions of θ can be identified a priori by examining the observed data and missingness patterns. For example, an incomplete multivariate dataset with four variables is depicted in Figure 4.6, with shaded areas indicating missing data. Because Y1 is fully observed and Y2 is nearly so, we expect the parameters governing the joint distribution of Y1 and Y2 to converge rapidly. The parameters governing the marginal distribution of Y4, however, will probably converge more slowly, as will the parameters describing the relationships of Y4 to the other variables (e.g. correlations and partial correlations). A high rate of missing observations for a variable does not automatically translate into high rates of missingness for its marginal parameters, because the variable may be highly correlated with other variables that are more fully observed. In
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practice, however, rates of missing observations are usually suggestive of rates of missing information. Worst linear function of the parameter If we could find a scalar function of θ that is ’worst’ in the sense that its marginal distribution converges, most slowly, then convergence with respect to this function would strengthen the evidence for global convergence. Locating such a function would be difficult, and in general that function would depend on the starting value or starting distribution. If we restrict attention to linear functions, however, functions of the form υ Τθ for some constant vector v, then a plausible choice for v can often be found from the convergence behavior of EM. Recall that in the vicinity of the mode the iterations of EM are approximately linear, with rate governed by the largest eigenvalue of the asymptotic rate matrix (Section 3.3.2). Suppose that we rotate the axes of the parameter space to form a new orthogonal coordinate system whose first axis is v1, the eigenvector of the rate matrix corresponding to this largest eigenvalue. The first coordinate of a point θ in this new system would then be proportional to the inner product of v1 with θ. From a standpoint of convergence, it makes sense to regard Τ υ1 θ as the worst linear function of θ , because among all linear functions its asymptotic rate of missing information is the highest. Moreover, use of this function is attractive from a computational standpoint because a numerical estimate of v, is readily obtained from the trajectory of EM, It follows from t (3.23) that near the mode, θ ( ) − θˆ is approximately proportional to v1. Therefore, an estimate of v, can be obtained simply by taking the difference between the convergent value 6 and any of the final iterates of EM, e.g. the estimate of θ one step prior to convergence. The suggested worst linear function of θ is then ξ (θ ) = νˆ Τ θ − θˆ , (4.51) 1
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(
)
where v1 is the numerical estimate of v1. Although it is not necessary to subtract θˆ from θ in (4.51), it seems useful to do so because the sign of ξ(θ) then indicates whether we are positioned to the left or to the right of the mode with respect to vˆ1 . We can interpret ξ ( θ) as a weighted sum of the components of θ , where the weights are equal to the perturbations from the mode in the final iterations of EM. Any components with no missing information will drop out of this sum, because for them EM converges immediately. Some limited experience with (4.51) in realdata problems suggests that this function is among the slowest to approach stationarity when the observeddata posterior is nearly normal. When the posterior is very nonnormal, however (e.g. if it has multiple modes) then other functions may converge more slowly. With some oddlyshaped posteriors, we have found functions other than (4.51) for which the ACFs take twice as many iterations to die out than for (4.51). We should be careful, therefore, not to attach undue significance to apparent stationarity for this function, particularly when some parameters are very poorly estimated. In problems for which the posterior is nearly normal, however, monitoring this function can be very helpful. Observeddata loglikelihood Another useful scalar function of θ to monitor is the observeddata loglikelihood function l(θ  Υobs ) or, for easier interpretation, the likelihoodratio statistic d (θ ) = 2 l θˆ  Υ − l(θ  Υ ) , L
[(
obs
)
obs
]
where θˆ is the MLE or posterior mode. For large samples, the observeddata posterior distribution of this function tends to be approximately χ 2 with degrees of freedom equal to the dimension of θ . If a single chain is started at the mode, and after a number of iterations the value of this function has not yet risen above the dimension of θ , then there is powerful evidence that stationarity has not yet been achieved. Unless
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evaluation of the observeddata loglikelihood is computationally burdensome, monitoring the behavior of this statistic is also highly recommended. 4.4.4 Convergence of posterior summaries Thus far we have discussed methods for diagnosing convergence of the distribution of the iterates of θ. to Ρ(θΥobs) This is the type of convergence necessary for generating proper multiple imputations of Y mis. If the goal is to make direct inferences about functions of θ , however, we need to assess convergence in a different sense; we need to ensure that our Monte Carlo estimates of summaries of the posterior distribution (moments, quantiles, densities, etc.) are sufficiently close to the targets they estimate. More generally, we need methods for measuring the Monte Carlo error in these summaries, and perhaps even adjusting interval estimates and pvalues to account for the error. Methods based on a single chain Let θ (1) , θ (2) ,..., θ ( M ) denote the output from a single simulation run of length M , possibly after discarding the iterates from an initial burnin period, and let
ξ=
1 M
M
∑ ξ ( t) t =1
(t)
denote the sample average of ξ
(t)
=ξ(θ ) for a scalar function
ξ.. Notice that many of the singlerun estimators described in Section 4.2, e.g. marginal moments, cumulative distribution and density estimates, can be written in this form. RaoBlackwellized estimates can also be written in this form if we let Y mis play the role of θ.. Estimating the variance of ξ from the sequence ξ (1) , ξ (2) ,..., ξ ( M ) is not a trivial matter, because members of the sequence are correlated; the sample variance of ξ (1) , ξ (2) ,..., ξ ( M ) divided by M may grossly underestimate V( ξ ).
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In most cases, a conservative upper bound for V( ξ ) can be estimated by subsampling the chain. Suppose that we average over every bth iterate,
1 ξ( b) = m
m
∑ ξ (tb) , t =1
where m = M/b. If ξ ( b) , ξ (2b) ,..., ξ ( mb) are uncorrelated, which can often be ascertained by inspection of the sample ACF,
( )

then an unbiased estimate of V ξ( b) is m 1, times the sample variance of ξ ( b) , ξ (2b) ,..., ξ ( mb) . This estimate is also a crude upper bound for the variance of ξ = ξ(1) ,
()
( )
because V ξ( b)
generally exceeds V ξ . More efficient techniques are available for obtaining consistent estimates of V ξ based on ξ (1) , ξ (2) ,..., ξ ( M ) .
()
Methods involving autocovariance and spectral analysis are described by Geweke (1992) and Geyer (1992), and an overview is also given by Ripley (1987). Care must be taken when applying these variance estimates based on a single chain; although they are theoretically consistent, some of them may grossly underestimate the actual variance in problems for which convergence to stationarity is slow (Raftery and Lewis, 1992a). Methods based on multiple chains Simple and reliable estimates of Monte Carlo error can also be obtained through the use of multiple chains (Gelman and Rubin, 1992a). Suppose that we perform R replicate runs from a common starting value or starting distribution, and, perhaps after a burnin period, obtain a sample of size m from each r:t run. Denote the tth value of θ from the rth run by θ ( ) . Let
( )
ξ ( r: t ) = ξ θ ( r: t ) be some scalar function,
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ξ (r) =
1 m
m
∑ξ
( r :t )
t =1
the sample average of ξ within the rth run, and
ξ=
1 Rm
R
m
∑ ∑ξ
( r :t )
r =1 t =1
the pooled average from all runs. Then the betweenrun variance
B=
1 R −1
∑ (ξ (r) − ξ ) R
2
r =1
unbiasedly estimates the variance of a single ξ (r ) , and B/R estimates the variance of the pooled estimate ξ . These variances being estimated are conditional upon the starting value or starting distribution. If the starting distribution is equal to P(θΥobs), or if there is a burnin period long enough to guarantee stationarity, then B/R will also be an unbiased estimate of the unconditional variance of ξ . If the burnin period is not long enough, then B /R will tend to be conservative (i.e. upwardly biased) if the starting values are overdispersed relative to P(θΥobs), otherwise it could be biased downward. Unless one is relatively certain that stationarity will be achieved by the end of the burnin period, it would be safer to start the chains from R different, preferably overdispersed starting values than from a single value. Interval estimation for a scalar summary Gelman and Rubin (1992a) propose that this estimate of Monte Carlo error be formally incorporated into a Bayesian interval estimate for the unknown ξ=ξ(θ). Let
W=
1 R( m − 1)
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∑ ∑ (ξ (r:t) − ξ (r) ) R
m
r =1 t =1
2
be the average of the R withinrun variances. Viewing the data as a balanced oneway layout, the analysis of variance for the randomeffects model leads to m −1 σˆ 2 = W+B m as an estimate of the posterior variance V(ξΥobs).Unlike the moment estimator (4.5) based on a single run, σˆ 2 is an unbiased estimate of V(ξΥobs), assuming the burnin period for each run is sufficient to ensure stationarity. Combining σˆ 2 with the estimated Monte Carlo error associated with ξ leads to the estimated total variance Tˆ = σˆ 2 + B / R and a 100(⁄)% posterior interval for ξ is
ξ ± tυ ,1−α / 2 Tˆ , (4.52) where tv,p denotes the pth quantile of the t distribution with v degrees of freedom. The assumption underlying (4.52) is that the posterior distribution of ξ is normal, and the use of a t distribution accounts for the fact that the components of variance are estimated rather than known. Gelman and Rubin (1992a) provide an expression for v based on an estimated variance of Tˆ , using a method similar to that of Satterthwaite (1946). When m is large enough so that the withinrun estimates ξ ( r ) are very close, v is large and tυ ,1−α / 2 becomes essentially a normal quantile. The use of (4.52) for inference has limitations, most notably the assumption of normality of the posterior distribution for ξ. When m is large and the distribution of the iterates appears to be nonnormal, it may be more reasonable to combine the runs and base an interval estimate on quantiles of the pooled sample of Rm observations. Whether or not one formally incorporates estimates of Monte Carlo error into the inference, however, the use of multiple runs and evaluation of the betweenrun variance can be very useful. Note also that multiple runs can be used to estimate the Monte Carlo variance of an estimator that does not have the form of a sample average, e.g. a sample quantile, for which a variance
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estimate based on a single run would be difficult to derive. Whenever a simulation run is replicated R times, there will automatically be R  1 degrees of freedom available for estimating the variance of any statistic calculated from a single run, regardless of its functional form.
4.5 Practical guidelines Sections 4.24.4 may leave the uninitiated reader with a bewildering array of choices and questions regarding almost every aspect of analyzing an incomplete dataset: whether to use parameter simulation or multiple imputation, which convergence diagnostics to monitor, and how to carry out the simulation in an efficient manner while avoiding potential pitfalls. We conclude this chapter with some suggestions on how to choose between methods and implement them in realdata problems. 4.5.1 Choosing a method of inference In many incompletedata problems, it will be possible to conduct inference either by (a) calculating appropriate summaries of the iterates of θ or by (b) generating multiple imputations of Ymis and combining the results of repeated completedata analyses. A third set of techniques, briefly discussed in Chapter 3, is based on direct evaluation of the observeddata likelihood function, 6.9’. likelihoodratio tests (Section 3.2.4). For a single, well defined inferential question, likelihood methods are in principle the most efficient because they do not involve simulation at all. They do require, however (as does multiple imputation), a sample sufficiently large for the usual asymptotic approximations to work well. Moreover, likelihood methods can be somewhat less versatile than Simulation in that special computational algorithms may be needed to answer specific questions. Likelihood methods can readily yield a pvalue for testing nested hypotheses, when maxima can be found under both the null and alternative hypotheses. In some problems, however, finding the maximum under one or both hypotheses may require a
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specially designed EM algorithm. Obtaining interval estimates can also be difficult, requiring analytic or numerical differentiation of the loglikelihood. A likelihoodbased pvalue, when available, is a useful benchmark against which to compare the results of a simulationbased method, and in largesample problems it may be regarded as the best answer. When the likelihood answer is less reliable or difficult to obtain, however, we will need to depend on parameter simulation or multiple imputation. Below are some important considerations in choosing among the simulationbased methods. Nature of the inferential question. Is there a well defined parameter or group of parameters of interest, or is the analysis more exploratory in nature? In the latter case, multiple imputation may be best; one good set of, say, m = 5 imputations may suffice for a variety of analyses, whereas parameter simulation could require hundreds or thousands of draws per analysis. Care must be taken, however, to ensure that the model used to create the imputations is general enough to encompass all of the analyses being contemplated (Section 4.5.4). Particularly when the dataset is large and simulation runs are expensive, multiple imputation may be the most practical approach, because generating and storing, say, five versions of Y mis, will tend to be cheaper than generating and storing a thousand or more iterates of a highdimensional θ . On the other hand, if interest truly does focus on a single parameter or small group of parameters, then parameter simulation may be quite reasonable, especially in smaller problems for which the simulations run quickly and the relevant summaries of θ are easily stored. Asymptotic considerations. In multiple imputation, the rules for combining completedata inferences all assume that the sample is large enough for the usual asymptotic approximations to hold. For smaller samples, when the asymptotic methods break down, simulationbased summaries of the posterior distribution of θ may be preferable, provided that one keeps in mind their Bayesian interpretation and
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dependence upon a prior. Rates of missing information. When fractions of missing information are low, methods that average over simulated values of Ymis (i.e. RaoBlackwellized and multipleimputation estimates) can be much more efficient than methods that average over simulated values of θ With low rates of missing information, multipleimputation estimates based on, say, m = 5 imputations may be nearly as precise as averages over hundreds of draws of θ . With high rates of missing information, however, a larger number of imputations may be necessary. Robustness. With parameter simulation, the form of the parametric completedata model often plays a crucial role in the inference. If the model’s assumptions are seriously violated, then the observeddata likelihood or posterior may be a poor summary of the data’s evidence about θ; indeed, if the modeling assumptions do not hold, then the interpretation of θ itself may be questionable. In multipleimputation inferences, however, the impact of the completedata model may be less crucial; rather than being applied to the complete data, the model is used only to predict the missing part. Imputations created under a false model may not have a disastrous effect on the final inference, provided that the analyses of the imputed datasets are carried out under more plausible assumptions; this is particularly true when the amounts of missing data are not large. For this reason, in many realistic scenarios multiple imputation may tend to be more robust to departures from the data model than parameter simulation. 4.5.2 Implementing a parametersimulation experiment Exploration. The best way to avoid pitfalls is to gain some basic understanding of the problem at the outset. How much information is missing? Do the observeddata likelihood and posterior seem well behaved, or are they oddly shaped with multiple modes, ridges, or suprema on the boundary?
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Sometimes these questions can be nearly answered through previous experience with similar datasets, and through prior examination of the observed data and patterns of missingness. The EM algorithm can also be an excellent diagnostic tool. Running EM from a variety of starting values and evaluating the loglikelihood function or logposterior density at the stationary points can reveal multiple modes, ridges and boundary suprema. The iterations of EM give quick estimates of the largest rates of missing information and the worst linear functions of θ. If the highest rates of missing information are extremely high, say 90% or more, we should expect a data augmentation algorithm to converge very slowly. If the rates are high, say 50  80%, convergence may not be too difficult to attain, but we should remember that inferences about certain aspects of θ may rely heavily upon unverifiable assumptions about the missingness mechanism (see Section 3.3.4). V the likelihood and posterior seem well behaved and rates of missing information are moderate, say 40% or less, we may expect the simulations to proceed without much difficulty. Preliminary runs. Preliminary simulation runs are necessary to give us an idea of how many iterations are needed to achieve stationarity. In these runs, important functions of θ should be saved and plotted. Unfortunately, the singlechain diagnostic methods of Section 4.4.1 tend to work best when they are actually needed the least, when algorithms converge reliably and rapidly from any reasonable starting value. Performing a variety of exploratory runs from different starting values, preferably overdispersed relative to P( θΥobs) is highly recommended to avoid the pitfalls associated with oddly shaped posteriors. Timeseries plots that overlay the output of multiple runs on the same set of axes are useful; they help us to identify pathological situations in which the algorithm appears to ’converge’ quickly from each starting value, but the convergence is illusory because the iterates from the different runs do not overlap.
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Single versus multiple chains. Whether the simulation should be carried out using a single chain or multiple chains has been the subject of lively debate in the Markov chain Monte Carlo literature (Gelman and Rubin, 1992a; Geyer, 1992; Raftery and Lewis, 1992b; Smith and Roberts, 1993). Given that Rm iterations are to be performed, is it better to use a single run of length Rm, or R parallel runs of length m? If the cost of the two methods is the same, then the singlechain method may be somewhat more precise for estimating a single quantity such as posterior mean, because fewer burnin iterations will be discarded. On the other hand, the cost of the two methods may not be the same; multiple chains might be convenient and perhaps even less expensive if multiple computers or parallel processing are available. If we are confident that the algorithm converges reliably and in a reasonable amount of time from a particular starting value (e.g. a mode), then running a single chain from this starting value is not a bad strategy. Such confidence, however, can probably be gained only through multiple runs from a variety of starting values at the preliminary stage. Moreover, the use of multiple chains is arguably the simplest and most reliable way to assess the Monte Carlo error of an estimate. The importance of reproducibility. When analyzing data by simulation, one must recognize that a simulation is an experiment and should be conducted according to well accepted practices of scientific inquiry. Before considering a result to be reliable, we should have confidence that another knowledgeable analyst, carrying out an independent analysis with the same data and the same model, would be led to essentially the same conclusions. Short of finding another analyst to reproduce the results, the best way to gain such confidence is through replication. Before completely trusting the results from a single run, we should, if at all possible, try to verify them by repeating the experiment with a new random number generator seed and a new starting value of θ.
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4.5.3 Generating multiple imputations If the immediate goal of the simulation is to create proper multiple imputations of Y mis then the comments above regarding exploration of the observeddata likelihood or posterior, preliminary runs and convergence diagnostics still apply. If the imputations are to be used in a wide variety of analyses, we should strive for global convergence to the joint posterior distribution of θ , rather than convergence of the marginal distributions of scalar functions of θ , because we may not know which functions of θ will be the subject of future analyses. In preliminary runs, we should pay close attention to those functions for which the rates of missing information are high. In judging how many iterations are needed to achieve approximate global stationarity, we should choose a number k large enough that the ACFs of the worst functions of θ are effectively zero by lag k, and perhaps even double or triple that number, if possible, to provide an extra margin of safety. Once we have decided on the number of iterations k needed for stationarity, we can proceed to generate the m, multiple imputations. One way to do this is to run a single chain for mk iterations, taking every kth iterate of Ymis. With a single chain, there is a danger in choosing a value of k that is too small; the multiple imputations could be correlated and understate the missingdata uncertainty. This danger can be avoided by running m parallel chains of length k from overdispersed starting values, and taking the final value of Ymis from each chain. If the starting values are truly overdispersed, then choosing k too small will overstate the missingdata uncertainty and cause inferences to be conservative. If overdispersed starting values are difficult to obtain, then running a single chain of length mk, or m parallel chains of length k emanating from a common starting value (e.g. a mode), are acceptable provided that we are relatively certain that the choice of k is large enough. 4.5.4 Choosing an imputation model Inference by multiple imputation proceeds in two distinct phases: first, the missing data are filled in m times; second, the
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m versions of the complete data are analyzed and the results are combined into a single inferential statement. These two phases may be carried out on different occasions and even by different persons. This temporal separation of the phases is one of the most important advantages of multipleimputation inference, because the missingdata aspects of the problem are confined entirely to the first phase; after imputation, no special incompletedata techniques are needed to complete the second phase. Consequently, one good set of m, imputations can effectively solve the missingdata problems for a large number of future analyses. Multiple imputation is especially attractive for large datasets (e.g. publicuse files from censuses or sample surveys) that will be analyzed in a variety of ways by a variety of people, many of whom may not have the technical knowledge or resources needed to analyze the incomplete version of the data. Multiple imputations can be created by a person or organization having special expertise in missingdata techniques; in many cases, the imputer will have detailed knowledge or even additional data that cannot be made available to the analysts but which may be relevant to the prediction of Ymis (Rubin, 1987). Because the imputation and analysis phases are distinct, it is natural to ask whether multiple imputation leads to valid inferences when the imputer’s model and the analyst’s model differ. The rules for combining completedata inferences were derived under some implicit assumptions of agreement between the two models. For example, the validity of (4.32)(4.33) requires that the imputation and analysis phases condition on the same set of observed data Yobs. If the imputed datasets are distributed to a variety of users, however, it is possible or even likely that inconsistencies will arise between the imputer’s and analyst’s models. The validity of inultipleimputation inferences when the imputer’s and analyst’s models differ has been the subject of recent controversy (Fay, 1992; Kott, 1992; Meng, 1995; Rubin, 1996). A basic understanding of the implications of discrepant models is important, even for the imputer who produces imputations solely for personal use, because discrepancies are common and can impact the multipleimputation inferences either positively or negatively.
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We now discuss in broad terms some types of discrepancies and their potential impact on multipleimputation inferences. When the analyst assumes more than the imputer One possible inconsistency is that the analyst’s and imputer’s models differ, but that the analyst’s model can be regarded as a special case of the imputer’s. For example, suppose that a dataset contains three variables, Y1, Y2 and Y3, that only Y3 has missing values and that proper multiple imputations are simulated under a linear regression of Y3 on Y1 and Y2, (4.53) Y3 = β0 + β1Y1 + β2Y2 + ε , where the error ε is normally distributed. Furthermore, suppose that the analyst subsequently models Υ3 as a linear regression given only Y 1, omitting Y2 from the model, The analyst’s model is then a special case of the imputer’s with β2=0. The practical implication of the discrepancy depends on whether the analyst’s extra assumption is true. Note that β2=0 being true does not invalidate the imputer’s model at all; (4.53) still applies. Therefore, inferences derived from multiple imputation will be valid, although probably somewhat conservative, because the imputations will reflect an extra degree of uncertainty due to the fact that the imputer’s procedure estimates β 2 rather than setting it to zero. For example, predictions for future observations of Y3 at specified values of Y 1 will be unbiased, but interval estimates will be somewhat wider than they would have been if the imputer had assumed β2=0. On the other hand, suppose that in reality β2 is not zero. The predictions of Y 3 given by the analyst’s model will then be biased. This bias, however, will be the fault of the analyst rather than the imputer. In this case there is nothing wrong with the imputed datasets, and an analysis under an appropriate model will lead to appropriate conclusions. Biases and inappropriate conclusions that arise because an analyst uses an inappropriate model should not be regarded as a shortcoming of the imputation method, just as inappropriate
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analyses of a complete dataset are not the fault of the data collector. When the imputer assumes more than the analyst Another type of inconsistency arises when the analyst’s model is more general than the imputer’s; that is, the imputer applies assumptions to the complete data that the analyst does not. Once again, the practical implications of this inconsistency will depend on whether the extra assumptions are true, so we consider the two possibilities in turn. The case where the imputer’s additional assumptions are true has investigated by Fay (1992), Meng (1995) and Rubin (1996) and Fay (1992) shows by example that when m = ∝, the total variance T for a scalar estimand Q given by (4.24) may be larger than the variance of Q over repeated realizations of the sampling and imputation procedure. This does not, however, invalidate the method for combining completedata inferences ’about Q described in Section 4.3.2. In fact, as demonstrated by Meng (1995) and Rubin (1996), the point estimate Q is more efficient than an observeddata estimate derived purely from the analyst’s model, because it incorporates the imputer’s superior knowledge about the state of nature, a property that Rubin (1996) calls superefficiency. Moreover, the multipleimputation interval estimate (4.27) has an average width that is shorter than a confidence interval derived purely from the observed data and the analyst’s model, even though it is conservative, having frequency coverage greater than the nominal 100(1α )%. Meng (1995) demonstrates that under fairly general conditions, the addition of true prior information to an imputation model can only increase the efficiency of Q while, at the same time, decrease the width and increase the coverage of the multipleimputation interval estimate. Thus there is no real sense in which an imputer’s superior knowledge can invalidate the inference; on the contrary, additional information can only help. Reversing the threevariable example used above, suppose that the imputer creates imputations for Y3 under the reduced model
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(4.54) Y3 = β0 + β1Y1 + ε , which we assume to be true, but the analyst fits the more general model (4.53). Under ignorability, the analyst can obtain valid inferences without imputation by basing the regression analysis only on those units for which Y3 is observed. Alternatively, he or she can also perform a multipleimputation analysis using the imputations created under (4.54). If the imputation model is true, the latter approach will be superior to the former, because it will provide point and interval estimates for β 2 that are more tightly concentrated around the true value of zero; the extra information conveyed in the imputations results in a more efficient procedure than one based on the observed data alone. Now consider the situation where the imputer assumes more than the analyst, but the additional assumptions are false. Clearly, in this case multipleimputation inferences can be erroneous. In the regression example, imputations created under the mistaken assumption that β2=0 will bias the analyst’s estimates of β 2 toward zero. Multiple imputations created under an erroneous model can lead to erroneous conclusions, just as a faulty model for complete data can lead to faulty conclusions when no data are missing. Sometimes the nature of the analysis and the pattern of missing values force the imputer to make certain assumptions that the analyst apparently does not need to make. For example, consider a regression analysis with predictor variables that are partially missing. In order to impute the missing values, the imputer must posit a joint distribution for all variables in the dataset, including the predictors. The analyst, however, makes no distributional assumptions about the predictors in the completed datasets, and specifies only the conditional distribution of the response given the predictors. At first glance, it appears that a discrepancy exists between the imputer’s and analyst’s models, with the imputer’s model being the more restrictive of the two. This discrepancy is illusory, however, because if the analyst had been given only Yobs then he or she could not proceed to make an efficient inference without imposing some kind of similar distributional
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assumptions on the predictors. In situations of this type, the additional assumptions used by the imputer should not be viewed in a negative light, because the same kind of restrictions would have to be imposed by an analyst who did not have access to the imputed values. 4.5.5 Further comments on imputation modeling From the above discussion, we see that the major danger of inconsistency between the imputer’s and analyst’s models arises when the imputer makes poorly grounded assumptions but the analyst does not. For this reason, it is important that the imputation model does not impose restrictions on unknown parameters that will later be the subject of the analyst’s inquiry. For example, if the analyst is going to investigate the correlation between two variables Y1 and Y2, then both variables need to be present in the multivariate imputation model even if only one of them has missing values, and the correlation between them should be left unspecified. Design variables (see Section 2.6.2) should be included in the imputation model if at all possible. To produce highquality imputations for a particular variable Y1, the imputation model should include variables that are (a) potentially related to Y1, and (b) potentially related to missingness of Y 1. A general guideline is that the imputer should use a model that is general enough to preserve any associations among variables (two, three, or even higherway associations) that may be the target of subsequent analyses. Balanced against the theoretical advantages of a large, general imputation model are practical limitations in computing resources, or inherent limitations of the observed data Yobs, which may prevent us from using an imputation model as general as we would like. As the number of variables and parameters grows, we may find that the ideal model may be too large to implement in the available computing environment. Moreover, we may find that the model has more parameters than can be estimated from Yobs, particularly when the prior distribution is diffuse; we may not be able to use the model without an informative prior. When the imputer is producing imputations purely for personal use, he or she may
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be able to tailor an imputation model for the intended analyses. An organization that must impute a large publicuse data file, however, must try to anticipate the analyses of many future data users and build the imputation model accordingly. In some cases compromises will have to be made: the imputer may have to sacrifice some of the imputation model’s generality to stay within the constraints of what the computing environment and the observed data can support. Construction of imputation models that are appropriate for specific analyses will be illustrated by the realdata examples of Chapters 59. Further discussion of practical considerations in choosing an imputation model for a large multipurpose database is given by Schafer, Khare and EzzatiRice (1993). Robustness It is important to remember that failure of an imputation model does not damage the integrity of the entire dataset, but only the portion that is imputed. Unless large amounts of data are imputed, biases introduced by an inappropriate imputation method may not be disastrous because they can be mitigated by the nonimputed data. In contrast, however, inferences by methods that do not separate the imputation phase from the analysis phase (e.g. methods of parameter simulation described in Section 4.2) will suffer more greatly under model failure, because the erroneous modeling assumptions will then be applied to the entire dataset rather than just the missing part. Once a missingdata problem is solved through imputation, an analyst ’tends to have greater freedom to investigate alternative models than would otherwise be possible if he or she had access to the observed data alone. Analyses not based on full parametric models The basic methods of multipleimputation inference (Section 4.3) were derived under the assumption that the completedata estimators Qˆ and U are firstorder approximations to the posterior mean and variance, respectively, of the estimand Q. Some methods of statistical inference, however, are not readily interpretable as approximate Bayesian procedures
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under any known parametric model. Examples of this include: nonparametric methods such as those based on ranks or permutation distributions; some of the classical designbased estimators for complex sample surveys, and their associated variance estimates calculated by methods such as the jackknife and balanced repeated replication (Wolter, 1985); and estimates and standard errors from generalized linear models based on quasilikelihood (McCullagh and Nelder, 1989). Is it acceptable to use multiple imputation in the context of any of these procedures? A partial answer is provided by Rubin (1987, pp. 118199), who states conditions under which a multipleimputation procedure will yield inferences with frequentist validity without reference to any specific parametric model. Multiple imputations that possess this property are said to be proper. Rubin’s definition of proper basically means that the summary statistics Q , U and B, defined in (4.21)(4.23), yield approximately valid inferences for the completedata statistics Q and U over repeated realizations of the missingdata mechanism. The three conditions necessary for imputations to be proper are:
(
)
1. As the number of imputations becomes large, Q − Qˆ / B . should become approximately N(0, 1) over the distribution of the response indicators R with Y held fixed. 2. As the number of imputations becomes large, U should be a consistent estimate of U, with R regarded as random and Y regarded as fixed. 3. The true betweenimputation variance (i.e. the variance of Q over an infinite number of multiple imputations) should be stable over repeated samples of the complete data Y, with variability of a lower order than that of Qˆ . Rubin (1987) shows that if (a) the completedata inference based on Qˆ and U is valid over repeated samples, and (b) the imputation method is proper, then the multiple imputation will
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yield inferences that are valid from a purely frequentist standpoint. Except in trivial cases (e.g. univariate data missing completely at random), it can be extremely difficult to determine whether a multipleimputation method is proper according to this definition. The most elaborate examples to date are given by Binder and Sun (1996). These lend important insights into the behavior of multiple imputation in inferential settings that are nonparametric and nonBayesian. For the complicated multivariate situations described in this book, however, we have little hope of analytically demonstrating that Bayesian, modelbased imputation methods are proper. From a practical standpoint, knowing whether an imputation method is technically proper for a particular analysis is less important than knowing whether it actually behaves well or poorly over repeated samples. The latter question can be addressed through simulation studies with realistic completedata populations and realistic response mechanisms. Examples of simulation studies will be given in Sections 6.4 and 9.5.3.
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CHAPTER 5
Methods For Normal Data
5.1 Introduction The most common probability model for continuous multivariate data is the multivariate normal distribution. Many standard methods for analyzing multivariate data, including factor analysis, principal components and discriminant analysis, are based upon an assumption of multivariate normality. Moreover, the classical techniques of linear regression and analysis of variance assume conditional normality of the response variables given linear functions of the predictors, which is the conditional distribution implied by a multivariate normal model for all the variables. Because statistical methods motivated by assumptions of normality are in such widespread use, it is natural to seek general techniques for inference from incomplete normal data. Datasets encountered in the real world often deviate from multivariate normality, but in many cases the normal model will be useful even when the actual data are nonnormal. There are several important reasons for this. First, one can often make the normality assumption more tenable by applying suitable transformations to one or more of the variables. Second, if some variables in a dataset are clearly nonnormal (e.g. discrete) but are completely observed, then the multivariate normal model may still be used for inference provided that (a) it is plausible to model the incomplete variables as conditionally normal given a linear function of the complete ones, and (b) the parameters of inferential interest pertain only to this conditional distribution (Section 2.6.2). Finally, even if some of the incompletely observed variables are clearly nonnormal, it may still be reasonable to
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use the normal model as a convenient device for creating multiple imputations. As pointed out in Section 4.5.4, inference by multiple imputation may be robust to departures from the imputation model if the amounts of missing information are not large, because the imputation model is effectively applied not to the entire dataset but only to its missing part. For example, it may be quite reasonable to use normal model to impute a variable that is ordinal (consisting of small number of ordered categories), provided that the amount of missing data is not extensive and the marginal distribution is not too far from being unimodal and symmetric. When using the normal model to impute categorical data, however, the continuous imputes should be rounded off to the nearest category to preserve the distributional properties as fully as possible and to make them intelligible to the analyst. We have found that the normal model, when used in this fashion, can be an effective tool for imputing ordinal and even binary data in instances where constructing a more elaborate categoricaldata model would be impractical (Schafer, Khare and EzzatiRice, 1993).
5.2 Relevant properties of the completedata model
5.2.1 Basic notation We begin by establishing some notational conventions that will be used throughout the chapter. The dataset, as depicted in Figure 2.1, is assumed to be a matrix of n rows and p columns, with rows corresponding to observational units and columns corresponding to variables. Denote the complete data by Y = (Yobs, Y mis), where Yobs and Ymis, are the observed and missing portions of the matrix, respectively. Let yij denote an individual element of Y, i = 1.2,...,n, j = 1,2,...,p. The ith row of Y, expressed as a column vector (all vectors will be regarded as column vectors), is yi = (yi1, yi2,...,yip)T. We assume that y1, y2,...,yn are independent realizations of a random vector, denoted symbolically as (Y1, Y2,...,Yp)T which
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has a multivariate normal distribution with mean vector µ and covariance matrix Σ;; that is,
y1, y2 ,..., yn  θ ~ iidN ( µ, Σ ), where θ=(µ,Σ) is the unknown parameter. Throughout the chapter, we assume no prior restrictions on θ other than the positive definiteness of Σ(Σ>0); that is, we allow θ to lie anywhere within its natural parameter space. Because the density of a single row is
{
}
P( yi  θ ) = 2πΣ − 2 exp −12 ( yi − µ ) Σ −1( yi − µ ) , 1
T
the completedata likelihood is, discarding a proportionality constant, Τ n 1 n (5.1) L(θ  Υ )∞  Σ  − 2 exp − 2 ( yi − µ ) Σ −1( yi − µ ). i=1
∑
Maximumlikelihood estimates By expanding the exponent in (5.1) and using the fact that −1 yiΤ yi = tr yiΤ Σ −1 yi
∑
= tr Σ −1 yi yiT , it follows that the completedata loglikelihood can be written as l(θ  Υ ) = − 2n log Σ − 2n µ T Σ −1µ (5.2) + µ T Σ −1T1 − 12 tr Σ −1T2, where n
T1 =
∑ yi = Y T 1, Υ
(5.3)
∑ yi yiT = Y Τ
(5.4)
i=1 n
T2 =
i=1
are the completedata sufficient statistics, and 1 = (1, 1,..., 1)T. Note that T1 is the vector of column sums,
(
T1 = Σ ni =1 yil , Σ ni =1 yi2 ,..., Σ ni =1 yip
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)
T
,
and T 2 is the matrix of columnwise sums of squares and crossproducts, Σ in=1 yi21 Σ in=1 yi1 yi2 L Σ in=1 yi1 yip n Σ y y Σ in=1 yi22 L Σ in=1 yi2 yip T2 = i=1 i2 i1 M M M n n n 2 Σ i=1 yip yi1 Σ i=1 yip yi2 L Σ i=1 yip Because the multivariate normal is a regular exponential family and the loglikelihood is linear in the elements of T1 and T2, we can maximize the likelihood by equating the realized values of T1 and T 2 with their expectations, E(T1) = n µ and E(T2) = n(Σ+µµΤ). This leads immediately to the well known result that the MLEs for µ and Σ are the sample mean vector
y = n −1
n
∑ yi,
(5.5)
i=1
and the sample covariance matrix S = n −1Y T Y − yy T
= n −1
n
∑ ( yi − y)( yi − y)T ,
(5.6)
i=1
respectively. Note that S is a biased estimate of Σ, and in practice it is more common to use the unbiased version n (n− 1)1 S. Further details on estimation and frequents inference for the multivariate normal model can be found in standard texts on multivariate analysis (e.g. Anderson, 1984). 5.2.2 Bayesian inference under a conjugate prior The simplest way to conduct Bayesian inference in the completedata case is to apply a parametric family or class of prior distributions that is conjugate to the likelihood function (5.1). A conjugate class has the property that any prior π(θ) in the class leads to a posterior P(θΥ)∝π(θ)L(θΥ) that is also in the class. When both µ and Σ are unknown, the most natural
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conjugate class for the multivariate normal data model is the normal invertedWishart family. The invertedWishart distribution If X is an m x p data matrix whose rows are iid N(0,Λ),, then the matrix of sums of squares and crossproducts A = XTX is said to have a Wishart distribution, and we write A~W(m,Λ). (5.7) The parameters m and Λ are often called the degrees of freedom and scale, respectively. The dimension of A (p × p) is not explicitly reflected in the notation (5.7) because it is conveyed by the dimension of Λ. The Wishart distribution arises in frequents theory as the sampling distribution of S. For our purposes it will be more convenient to work with the invertedWishart distribution. If A~W(m,Λ) then B = A1 is said to be invertedWishart, and we write B ~ W −1( m, Λ ). Omitting normalizing constants, the invertedWishart density for m ≥ p can be shown to be
m + p + 1 1 (5.8) P( B  m, Λ ) ∝ B − exp − trΛ−1B −1 2 2 over the region where B > 0. For m < p, the matrix A is singular and B = A 1 does not exist. Notice that (5.8) is a proper density function for any choice of m ≥ p and Λ > 0; we need not restrict ourselves to integer values of m. The mean of the invertedWishart distribution is 1 (5.9) E( B  m, Λ ) = Λ−1. m − p −1 provided that m ≥ p + 2. In the special case of p = 1, the invertedWishart reduces to a scaled invertedchisquare, 1 −2 cχ m , , with c=Λ . These and other wellknown properties of the Wishart and invertedWishart distributions are discussed in many texts on multivariate analysis; an excellent reference is Muirhead (1982).
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For our purposes, it will also be useful to know that the mode of the invertedWishart density is 1 (5.10) mode( B  m, Λ ) = Λ−1 ⋅ m + p +1 Demonstrating this fact involves maximizing the logarithm of (5.8), an exercise which is nearly identical to deriving the ML estimates for the multivariate normal distribution by maximizing the loglikelihood (5.2). We omit details of this calculation, but for a thorough demonstration in the case of the loglikelihood the interested reader may refer to Mardia, Kent and Bibby (1979, pp. 103105). The normal invertedWishart prior and posterior Returning to the problem of Bayesian inference for θ =(µ,Σ) under a multivariate normal model, let us apply the following prior distribution. Suppose that, given Σ, µ, is assumed to be conditionally multivariate normal, µ  Σ ~ N µ0 , T −1Σ , (5.11)
(
)
where the hyperparameters µ o ∈ℜ p and T>0 are fixed and known. Moreover, suppose that Σ is invertedWishart, (5.12) Σ ~ W −1( m, Λ ) for fixed hyperparameters m ≥ p and Λ > 0. The prior density for θ is then
π (θ ) ∝
Σ −(
m+ p + 2 2
{
× exp
) exp{− 1 tr Λ−1Σ −1} 2
− τ2
( µ − µ0 )T Σ −1( µ − µ0 )}
(5.13)
Following some matrix algebra, the completedata likelihood function (5.1) can be rewritten as n L(θ  Y ) ∝ Σ − 2 exp − 2n tr Σ −1S (5.14) T × exp − 2n ( y − µ ) Σ −1( y − µ )
{
{
}
}
Multiplying this likelihood by (5.13) and performing some algebraic manipulation, it follows that Ρ (θΥ) has the same
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form as (5.13) but with new values for (τ;m,µ0,Λ) that is, the completedata posterior is normal invertedWishart,
(
−1
)
µ  Σ, Y ~ N µ0′ , (τ ′) Σ ,
(5.15)
Σ  Υ ~ W ( m ′, Λ ′), where the updated hyperparameters are τ ′ = τ + n, −1
(5.16)
m ′ = m + n, n τ µ0′ = y+ µ , τ + n ο τ + n and
−1
τn Λ ′ = Λ−1 + nS + ( y − µ0 )( y − µ0 )T . n τ + In the special case of p = 1, the posterior becomes
(
−1
)
µ  Σ, Y ~ N µ0′ , (τ ′) Σ , ΣY ~
−2 c ′χ m ′,
where n
c′ = c +
τn
∑ ( yi − y)2 + τ + n ( y − µ0 )2 i=1
and c = Λ−1 is the prior scale for Σ. Existence of the prior distribution requires τ > 0, m ≥ p and Λ > 0. Notice, however, that we may apply the updating formulas and still obtain acceptable values of τ′, m′, and Λ′ for certain τ ≤ 0 and m < p. Under ordinary circumstances it would not make sense to use a negative value for τ,, because
µ′0 would then become a weighted average of y and µ0 with negative weight for µ 0. Taking τ = 0, however, may be quite sensible when little or no prior information about p is available, because it results in a posterior distribution for µ centered about y . Moreover, in some cases a choice of m < p may be attractive as well: see Section 5.2.3 below.
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Inferences about the mean vector By integrating the normal invertedWishart density function (5.13) over Σ , one can show that the marginal prior distribution of µ implied by (5.11)(5.12) is a multivariate t distribution centered at µ0 with v = m − p + 1 degrees of freedom. The mean of this distribution is µ0 provided that v > 1, and the covariance matrix is ( v − 2) −1τ −1Λ−1 provided that v > 2. Other properties of this multivariate t distribution are discussed in many texts on multivariate analysis; a good reference is Press (1982). In particular, the marginal prior distribution of any scalar component or linear function of the components of µ is univariate t. Suppose that ξ = αΤµ, where a is a constant vector of length p. The marginal prior distribution of ξ implied by (5.11)(5.12) is then (ξ−ξ0)/σ∼τv, where v=m− p+1,ξ0 = αΤµ0, and
α Τ ∆−1α . τv The marginal prior density is σ=
( (ξ − ξ )2 ( ) o 1 + P(ξ ) = υσ 2 Γ ( 2v ) πυσ 2
Γ v2+1
− υ +1) / 2
(5.17)
where Γ (⋅) denotes the gamma function. After observing Y we can obtain P(ξ Υ ), the marginal posterior distribution of ξ , simply by replacing the hyperparameters (τ,m,µο,Λ) in the above expressions with their updated values (τ′,m′,µ0′ ,Λ′). Inferences about the covariance matrix In many problems the parameters of interest are functions of µ, and Σ is best regarded as a nuisance parameter. On occasion, however, an estimate of Σ is needed. From a Bayesian standpoint there is no universally accepted `best’ estimate of Σ. The optimal estimate depends on the choice of a
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loss function, and in practice it tends to be difficult or impossible to choose among the various loss functions. Bayesian estimation of a covariance matrix raises some interesting theoretical problems that have yet to be resolved (Dempster, 1969a). If the current state of knowledge about Σ is described by Σ~W1(m,Λ), then competing estimates include the mean (5.9) and the mode (5.10). To complicate matters further, suppose that the mean µ and the covariance matrix Σ are both of interest, and the current state of knowledge about θ = ( µ ,Σ) is represented by the normal invertedWishart distribution µ  Σ ~ N µ0 , τ −1Σ ,
(
)
−1
Σ ~ W ( m, Λ ). By a calculation that is essentially equivalent to maximizing the multivariatenormal loglikelihood function, one can then show that the joint mode is achieved at µ = µ0 and 1 Σ= Λ−1. m+ p+2 Note that maximizing the joint density for µ and Σ is not equivalent to maximizing the marginal densities for µ and Σ separately. When a Bayesian estimate of Σ is needed, we will adopt the following ruleofthumb: if the current state of knowledge about Σ is described by Σ~W 1(m,Λ) irrespective of µ , then estimate Σ by m1Λ 1. This represents a compromise between the mean (5.9) and the marginal mode (5.10). 5.2.3 Choosing the prior hyperparameters A noninformative prior When no strong prior information is available about θ , it is customary to apply Bayes’s theorem with the improper prior
π (θ ) ∝ Σ −(
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p +1 2
)
(5.18)
which is the limiting form of the normal invertedWishart density (5.11)(5.12) as τ→0, m→−1 and Λ−1→0. Notice that
µ does not appear on the righthand side of (5.18); the prior `distribution’ of µ is assumed to be uniform over the p dimensional real space. Under this improper prior, the completedata posterior becomes µ  Σ, Y ~ N y, n −1Σ . (5.19)
(
(
)
)
Σ  Y ~ W −1 n − 1, ( nS ) −1 .
(5.20)
A nonBayesian justification for the use of this prior is that the posterior distribution of the pivotal quantity T T 2 = ( n − 1)( y − µ ) S −1( y − µ ) becomes (n−1)p(n−p)−1F p , n − p , t h e s a m e a s i t s s a m p l i n g distribution conditionally upon θ (DeGroot, 1970). The ellipsoidal (1−α) 100% HPD region for µ under this prior is identical to the classical (1α)100% confidence region for µ from sampling theory, and for inferences about µ the Bayesian and frequents answers coincide. The improper prior (5.18) also arises by applying the Jeffreys invariance principle to µ and Σ (Box and Tiao, 1992). If our primary interest is not in µ but in Σ, then the frequents justification for using (5.18) as a noninformative prior is not as strong because of the ambiguities involved in estimation of Σ. Notice, however, that if we use our ruleofthumb that a reasonable estimate for Σ~W1(m,Λ),is m 1Λ1, then (5.20) leads to the point estimate (n1)1nS. This is the estimate of Σ that is most widely used in practice, because it is unbiased for fixed θ over repetitions of the sampling procedure. For these reasons, we will accept (5.18) as a reasonable prior distribution when prior information about θ is scanty.
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Informative priors When an informative prior distribution is needed, it is often possible to choose reasonable values for the hyperparameters by appealing to the device of imaginary results. Suppose that we regard the improper prior (5.18) as representing a state of complete ignorance about θ . After observing a sample of n observations with mean y and covariance matrix S, the new state of knowledge is represented by (5.19)(5.20). By this logic, we can interpret the hyperparameters in (5.1l)(5.12) as a summary of the information provided by an imaginary set of data: µ 0 represents our best guess as to what µ might be (the imaginary y ); τ represents the number of imaginary prior observations on which the guess µ0 is based; m1Λ1 represents our best guess as to what Σ might be (the imaginary S); and m = τ−1 represents the number of imaginary prior degrees of freedom on which the guess m1 Λ1 is based. A ridge prior It sometimes happens that the sample covariance matrix S is singular or nearly so, either because the data are sparse (e.g. n is not substantially larger than p), or because such strong relationships exist among the variables that certain linear combinations of the columns of Y exhibit little or no variability. When this happens, it may be difficult to obtain sensible inferences about µ unless we introduce some prior information about Σ . The following is a suggestion for choosing a prior distribution to stabilize the inference when little is known a priori about µ or Σ. Suppose that we adopt the limiting form of the normal invertedWishart prior (5.13) as τ→0 for some m and Λ. The posterior becomes µ  Σ, Y ~ N y, n −1Σ , (5.21)
(
[
)
]
−1 Σ  Y ~ W −1 m + n, Λ−1 + nS ,
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(5.22)
which is proper provided that m + n ≥ p and (Λ −1+nS) > 0. Notice that this posterior is very similar to the posterior distribution (5.19)(5.20) obtained under the standard noninformative prior, except that the covariance matrix Σ has been ’smoothed’ toward a matrix proportional to Λ−1. If we take m = ∈ for some ∈ > 0 and Λ−1 =∈S* for some covariance matrix S*, then our ruleofthumb estimate of Σ is
(
)
∈ 1 n Λ−1 + nS = S ∗ + S, n+ ∈ n+ ∈ m+n a weighted average of S and S* with weights determined by the relative sizes of n and ∈. When S is singular or nearly so, it makes sense to choose S* to move the weighted average of the two matrices away from the boundary of the parameter space. One effective way to do this is to set the diagonal elements of S* equal to those of S and the offdiagonal elements equal to zero, S* = Diag S. (5.23) The resulting `prior’, which is not really a prior in the Bayesian sense because it is partly determined by the data, has the practical effect of allowing the means and variances to be estimated from the data alone, but smooths the correlation matrix slightly toward the identity. The degree of smoothing is determined by the relative sizes of ∈ and n, and ∈ can be regarded as an imaginary number of prior degrees of freedom added to the inference. Note that ∈ need not be an integer, and in some cases even a small fractional value of ∈ may be sufficient to overcome computational difficulties associated with singular covariance matrices. Use of this prior is closely related to the technique of ridge regression (e.g. Draper and Smith, 1981), and can be regarded as a form of empirical Bayes inference (e.g. Berger, 1985). This prior can be very helpful for stabilizing inferences about µ when some aspects of Σ are poorly estimated.
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5.2.4 Alternative parameterizations and sweep Suppose that z is a p × 1 random vector distributed as N (µ, Σ),
(
which we partion as z Τ = z1Τ , z2Τ
)
.where
z1 and z2 are
subvectors of lengths p1 and p 2 = p  p 1 respectively. It is well known that the marginal distributions of z1 and z2 are N(µ1,Σ11)
(
)
and N(µ2,Σ22) where µΤ= µ1Τ , µ2Τ and
Σ11Σ12 Σ= Σ 21Σ 22 are the partitions of µ and Σ corresponding to z Τ = z1Τ , z2Τ .
(
)
Moreover, the conditional distributions are also normal; in particular, the distribution of z2 given z1 is normal with mean Ε ( z2  z1 ) = µ2 + B2.1( z1 − µ1 )
= α 2.1 + B2.1 z1 and covariance matrix Σ22Æ1, where
−1 α 2.1 = µ2 − Σ 21Σ11 µ1, −1 Β2.1 = − Σ 21Σ11 , (5.24) −1 Σ 22.1 = Σ 22 − Σ 21Σ11 Σ12 are the vector of intercepts, matrix of slopes and matrix of residual covariances, respectively, from the regression of z2 on z1 . Because specifying the joint distribution of z1 and z2 is equivalent to specifying the marginal distribution of z1 and the conditional distribution of z2 given z1, we can characterize the parameters of the distribution of z either by θ = ( µ ,Σ) or by φ=(φ1,φ2), where φ1=(µ1,Σ11) and φ2=(α2.1,Β2.1,Σ22.1) It is easy to show that the transformation φ=φ(θ) is onetoone, with the inverse transformation θ=φ−1(φ) given by µ2 = α 2.1 + B2.1µ1 Σ12 = Σ11B2T.1, (5.25) T Σ 22 = Σ 22.1 + B2.1Σ11B2.1. Moreover, the parameters φ1 and φ2 are distinct in the sense that the parameter space of φ is the Cartesian crossproduct of
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the individual parameter spaces of φ1 and φ 2 ; that is, any choice of α2.1,Β2.1 and Σ22.1 > 0 will produce a valid θ = (µ,Σ) with Σ > 0. When a probability distribution is applied to θ =(µ,Σ) .it is occasionally necessary to find the density function for φ. Let f (θ) be the density of θ and g(φ) the density of φ=φ(θ) induced by f. The relationship between g and f is
(
g(φ ) = f φ −1(φ ) J
)
−1
,
where J is the Jacobian or firstderivative matrix of the transformation from θ to φ, and Jmeans the absolute value of the determinant of J. Notice that α21, Β2.1 and Σ 22.1 are of the same dimension as µ2, Σ21 and Σ22, respectively, so J can be partitioned as ∂µ1 ∂µ1 ∂µ1 ∂µ1 ∂µ1 ∂µ ∂Σ11 ∂µ2 ∂Σ 21 ∂Σ 22 1 ∂Σ11 ∂Σ11 ∂Σ11 ∂Σ11 ∂Σ11 ∂µ1 ∂Σ11 ∂µ2 ∂Σ 21 ∂Σ 22 ∂α ∂α ∂α ∂α ∂α 2⋅1 2⋅1 2⋅1 2⋅1 , J = 2⋅1 ∂Σ11 ∂µ1 ∂Σ 21 ∂Σ 22 ∂µ1 ∂B2⋅1 ∂B2⋅1 ∂B2⋅1 ∂B2⋅1 ∂B2⋅1 ∂µ Σ Σ Σ 22 ∂µ ∂ ∂ ∂ 11 2 21 1 ∂Σ 22⋅1 ∂Σ 22⋅1 ∂Σ 22⋅1 ∂Σ 22⋅1 ∂Σ 22⋅1 ∂µ1 ∂Σ11 ∂µ2 ∂Σ 21 ∂Σ 22 where the submatrices along the diagonal are square. By inspection of (5.24), we see that this matrix has the pattern I 0 0 0 0 0 I 0 0 0 J = × × I × 0 , 0 × 0 × 0 0 × 0 × I where I denotes an identity matrix, 0 denotes a zero matrix and x denotes a matrix that is neither I nor 0. It is a wellknown property of determinants that
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A B = AC (5.26) 0 C for square A and C . Applying (5.26) repeatedly, the determinant of J reduces to ∂B J = 2⋅1 . (5.27) ∂Σ 21 With Σ11 held fixed, Β 2.1=Σ21Σ−111 is a linear transformation of Σ21. It can be shown that the Jacobian of the linear transformation from W (p × q) to Z = WB for nonsingular B (q × q) is Β p (e.g. Mardia, Kent and Bibby, 1979, Table 2.4.1), and thus
 J  = Σ11
− p2
.
(5.28)
The sweep operator The algorithms presented in this chapter will require repeated use of the transformations (5.24) and (5.25). To simplify both the notation and implementation of these algorithms, we will rely heavily on a device known as the sweep operator. First introduced by Beaton (1964), the sweep operator is commonly used in linear model computations and stepwise regression. Dempster (1969b) describes its relationship to methods of successive orthogonalization, and Little and Rubin (1987) demonstrate the usefulness of sweep in ML estimation for multivariate missingdata problems. Further information and references are given by Thisted (1988). Suppose that G is a p x p symmetric matrix with elements gij. The sweep operator SWP[k] operates on G by replacing it with another p x p symmetric matrix H, H = SWP[k]G, where the elements of H are given by hkk = −1 / gkk , (5.29) h jk = hkj = g jk / gkk for j ≠ k , h jl = hlj = g jl − g jk gkl / gkk for j ≠ k and l ≠ k . After application of (5.29), the matrix is said to have been swept on position k. In a computer program, sweep can be
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carried out as follows: first, replace gkk with hkk = −1/g kk; next, replace the remaining elements gjl=gjl in row and column k with hjk = gjlgjlhjk. and finally, replace the remaining elements gjl = glj in the other rows and columns by hjl = gjl  gklhjk. This method is efficient both in terms of computation time and memory, because no storage locations other than the matrix itself are necessary. Because both G and H are symmetric, further savings can be achieved by computing and retaining only the uppertriangular portion of the matrix. Suppose that a p × p matrix G is partitioned as G11 G12 G= , G21 G22
where G11 is p 1 × p 1. After sweeping on positions 1, 2,..., p1, the matrix becomes −1 −1 −G11 G11 G12 SWP[1, 2,..., p1 ]G = −1 −1 G21G11 G22 − G21G11 G12 which is recognizable as a matrix version of (5.29). The notation SWP[1,2,...,p1] indicates successive application of (5.29), SWP[1, 2,..., p1 ]G = SWP[ p1 ]L SWP[2] SWP[1]G. Sweeps on multiple positions need not be carried out in any particular order, because the sweep operator is commutative, SWP[ k2 ] SWP[ k1 ]G = SWP[ k1 ] SWP[ k2 ]G.
Sweeping a p × p matrix G on positions 1, 2,..., p has the effect of replacing G by −G−1. This inverse exists if and only if none of the attempted sweeps involve division by zero. When inverting a matrix with sweep, we can also readily obtain the determinant. Let ϒ k denote the kth diagonal element of the matrix after it is swept on positions 1, 2,..., k − 1, γ k = (SWP[1, 2,.., k − 1]G) kk . Then p
G=
∏γ k , k =1
(5.30)
where γ1 is taken to be g 11 , the first element of G. Thus the determinant can be found by computing the product of the
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pivots (i.e. the diagonal elements of the matrix) as they appear immediately before the matrix is swept on them (Dempster, 1969b). It is also convenient to define a reversesweep operator that returns a swept matrix to its original form. The reversesweep operator, denoted by H = RSW[k]G, replaces the elements of G with hkk = −1 / gkk , (5.31) h jk = hκl = g jk / gkk for j ≠ k , h jl = hlj = g jl − g jk gkl / gkk for j ≠ k and l ≠ k . Notice that reverse sweep is remarkably similar to sweep, with the only difference being a minus sign in the calculation of hjk = h kj . It is easy to verify that reverse sweep is indeed the inverse of sweep, RSW[k] SWP[k] G = G and that reverse sweep is commutative, RSW[k2] RSW[k1] G = RSW[k1] RSW[k2] G. Computing alternative parameterizations From a statistical viewpoint, the sweep operator is highly useful for the following reason: when applied to the parameters of the multivariate normal model, sweep converts a variable from a response to a predictor. Suppose that z is a p × 1 random vector distributed as N(µ,Σ), and we partition it as zΤ
(
= z1T , z2T
)
where z1 has length p 1. Let us arrange the
parameters θ = (µ,Σ) as a (p+1) × (p+1) matrix in the following manner, −1 µ1T µ2T −1 µ T (5.32) θ= = µ1 Σ11 Σ12 µ Σ Σ Σ µ 21 22 2 The reason for placing 1 in the upperleft corner will be explained shortly. To simplify bookkeeping, we will allow the row and column indices to run from 0 to p rather than from 1 to p + 1, so that the parameters pertaining to the jth variable
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will appear in row and column j. Suppose that we sweep this θmatrix on positions 1, 2,...,p 1; the result will be, by the matrix analogue of (5.29), −1 −1 −1 −1 − µ1Τ Σ11 µ1 µ1Τ Σ11 µ2Τ − µ1Τ Σ11 Σ12 −1 −1 −1 Σ11 µ1 − Σ11 Σ11 Σ12 . µ − Σ Σ −1µ Σ Σ −1 Σ − Σ Σ −1Σ 21 11 1 21 11 22 21 11 12 2 Comparing this to (5.24), we see that the last p  p1 rows and columns contain α 2.1, Β2.1, and Σ22.1, the parameters of the conditional distribution of z2 given z1, −1 −1 −1 − µ1T Σ11 µ1 µ1T Σ11 α 2T⋅1 −1 −1 Σ11 SWP[1,... p1 ]θ = µ1 − Σ11 B2T⋅1 Σ 22⋅1 α 2⋅1 B2⋅1 Moreover, the upperleft (p1 + 1) × (p1 + 1) submatrix contains in swept form the parameters of the marginal distribution of z1, −1 −1 −1 − µ1Τ Σ11 −1 µ1Τ µ1 µ1Τ Σ11 = RSW[1,..., p1 ] −1 −1 Σ11 µ1 − Σ11 µ1 Σ11 We have thus shown that φ=(µ1,Σ11,α2.1,Β2.1,Σ22.1), expressed in matrix form as −1 µ1T α 2T⋅1 (5.33) φ = µ1 Σ11 B2T⋅1 . α B Σ 2⋅1 22⋅1 2⋅1 can be computed from the θmatrix by first sweeping the full matrix on positions 1, 2,..., pl, and then reverse sweeping the upperleft (p1 + 1) × (p1 + 1) submatrix on the same positions. The reason for placing 1 in the upperleft corner of the θ matrix (5.32) is that this matrix can be considered to be already swept on position 0. Notice that if we reversesweep θ on position 0, we obtain −1 µ T 1 µT RSW[0] (5.34) = . Σ µ Σ + µµ T µ the parameters of the multivariate normal distribution expressed in terms of the first two moments of z about the
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origin. This unswept version of θ is quite useful because it is the natural representation for computing ML estimates. Suppose that Y is an n × p data matrix whose rows are independent realizations of the random vector z. If we arrange the sufficient statistics Τ1=ΥΤ1 and Τ2=ΥΤΥ into a (p + 1) × (p + 1) matrix n T1T T = [1, Y ]T [1, Y ] = (5.35) , T1 T2 then the moment equations for ML estimation set (5.34) equal to n −1T. Hence the ML estimate of θ may be computed from the sufficient statistics by θˆ = SWP[θ ]n −1T . Because ML estimates are invariant under transformations of the parameter, the MLE for an alternative parameterization φ can be obtained by sweeping θˆ on the appropriate positions.
Figure 5.1. Matrix of missingness patterns associated with Y with 1 denoting an observed variable and 0 denoting a missing variable.
5.3 The EM algorithm When portions of the data matrix Y are missing, ML estimates cannot in general be obtained in closed form; we must resort to iterative computation. The EM algorithm for a multivariate
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normal data matrix with an arbitrary pattern of missing values was described by Orchard and Woodbury (1972); Beale and Little (1975); Dempster, Laird and Rubin (1977); and Little and Rubin (1987). Because of its usefulness and its similarities to the simulation algorithms that follow, we describe in detail one possible implementation of EM for incomplete multivariate normal data. 5.3.1 Preliminary manipulations To simplify notation and facilitate computations, it is helpful at the outset to group the rows of Y by their missingness patterns. A matrix of missingness patterns corresponding to Y is shown in Figure 5.1. We will index the missingness patterns by s = 1, 2,..., S, where S is the number of unique patterns appearing in the data matrix. The trivial pattern with all variables missing should be omitted from consideration. Rows of Y that are completely missing contribute nothing to the observeddata likelihood and would only slow the convergence of EM by increasing the fractions of missing information (Section 3.3.2). For bookkeeping purposes it will be helpful to define the following quantities. Let R be an S × p matrix of binary indicators with typical element rsj, where 1 if Y j isobservedinpattern s, rsj = 0 if Y j ismissinginpattern s. The matrix R is shown in Figure 5.1. For each missingness pattern s, let Ο(s) and Μ (s) denote the subsets of the column labels {1, 2,..., p} corresponding to variables that are observed and missing, respectively,
{
}
O(s ) = j : rsj = 1 ,
{
}
M ( s ) = j : rsj = 0 . Finally, let Ι(s) denote the subset of {1, 2,..., n} corresponding to the rows of Y that exhibit pattern s. For example, suppose that the data matrix has ten rows with no missing values, and
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after sorting these rows are labeled 1,..., 10; the first row of R is then (1, 1,..., 1), and O(1) = {1, 2,..., p},
M (1) = ∅,
I (1) = {1, 2,..10}. 5.3.2 The Estep Recall that in the Estep of EM, one calculates the expectation of the completedata sufficient statistics over P(ΥmisΥobs,θ) for an assumed value of θ . These statistics are of the form Σ iyij and Σiy ijy ik , so to perform the Estep we need to find the expectations of yij and yijyik over Ρ(ΥmisΥobs,θ). Because the rows y1, y 2 ,...,yn of Y are independent given θ, we can write
P(Ymis  Yobs , θ ) =
∏ P( yi(mis)  yi(obs) ,θ ), n
i=1
where y i(obs) and yi(mis) denote the observed and missing subvectors of yi, respectively. The distribution Ρ(yi(mis)yi(obs) ,q) is a multivariate normal linear regression of yi(mis) on yi(obs), and the parameters of this regression can be calculated by sweeping the θ−matrix on the positions corresponding to the variables in yi(obs). If row i is in missingness pattern s, then the parameters of Ρ(yi(mis) y i(obs) ,θ) are contained in SWP[Ο (s)]θ in the rows and columns labeled M(s) Let A denote the swept parameter matrix A = SWP[O( s )]θ , and let ajk denote the (j, k)th element of A, j, k = 0, 1,..., p. Using the results of Section 5.2.4, the reader may verify that the first two moments of y i(mis) with respect to P(YmisΥobs,θ) are given by
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(
)
E yij  Yobs , θ = aoj +
(
)
∑ akj yik ,
κ ∈O( s )
Cov yij , yik  Yobs , θ = a jk for each i ∈Ι(s)and j, k∈M.(s) For any j∈Ο(s), of course, the moments are E yij  Yobs , θ = yij ,
(
)
(
)
Cov yij , yik  Yobs , θ = 0, because yij is regarded as fixed. Applying the relation E yij yiκ  Yobs , θ = Cov yij , yik  Yobs , θ
(
)
(
(
)
)
+ E yij  Yobs , θ E( yiκ  Yobs , θ )
it follows that
(
)
y for j ∈O( s ), E yij  Yobs , θ = y *ij for j ∈M s , ( ) ij
and
yij yik for j , k ∈ O( s ), E yij yiκ  Yobs , θ = yij* yik for j ∈ M ( s ), k ∈ Ο( s ), α + y∗ y∗ for j , k ∈ M ( s ), ij ik jk
(
where
)
yij∗ = α oj +
∑ α kj yik .
(5.36)
k ∈O( s )
The Estep consists of calculating and summing these expected values of y ij and y ijy ik over i for each j and k. The output of an Estep can then be written as Ε(ΤΥobs,θ) where T is the matrix of completedata sufficient statistics L yip n yi1 yi2 2 y y y i1 i1 i2 L yi1 yip n T n 1 Y yi22 L yi2 yip . = T= T T Y 1 Y Y i=1 O M 2 y ip
∑
The elements below the diagonal are not shown and may be omitted from the calculations because they are redundant.
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Notice that the matrix Α=SWP[O(s)]θ needed for the Estep depends on the missingness pattern s, and thus in practice the elements of Ε (ΤΥobs,θ) must be calculated by first summing expected values of y ij and yijyik for i∈ Ι(s), and then summing across patterns s = 1, 2,..., S, with a new Amatrix being calculated for each missingness pattern. 5.3.3 Implementation of the algorithm Once Ε(ΤΥobs,θ) has been found, carrying out the Mstep is relatively trivial. For a given value of T the completedata MLE is θˆ = SWP[0]n −1T , and the Mstep merely carries out this same operation on Ε(ΤΥobs,θ) rather than T. A single iteration of EM can thus be written succinctly as θ ( t +1) = SWP 0 n −1E T  Y , θ ( t ) . (5.37)
[]
(
obs
)
In principle the EM algorithm for incomplete multivariate normal data is completely defined by (5.37), but from a practical standpoint we should still consider how to implement the algorithm in an efficient manner. It is beneficial to keep both processing time and memory usage down, but tradeoffs between the two are inevitable; one can always reduce processing time at the expense of additional memory by storing rather than recomputing quantities that must be used repeatedly. The implementation suggested here stores rather than recomputes the portions of Ε(ΤΥobs,θ) that do not depend on θ and thus remain the same for every Estep. This method may not be optimal for any particular dataset, but it is not difficult to program and seems to perform well in a wide variety of situations. Observed and missing parts of the sufficient statistics We can express the matrix T as the sum of matrices corresponding to the individual missingness patterns. Let
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L Σyip ns Σyi1 Σyi2 2 Σyi1 Σyi1 yi2 L Σyi1 yip L Σyi2 yip , Σyi22 Τ( s ) = O M 2 y ip where all sums are taken over i∈I(s), and ns=∑i∈I(s) is the sample size in missingness pattern s; then S
T=
∑ T (s ). s =1
Each T(s) can be further partitioned into an observed part and a missing part. Notice that the elements of T(s) in the rows and columns labeled M(s) are functions of Y mis and perhaps Y obs whereas the remaining elements of T(s) are functions of Y obs only. Define a new matrix T mis (s) which has the same elements as T(s) in the rows and columns labeled M(s), but with all other elements set to zero, and define Tobs(s) to be T(s) −T mis(s). For example, consider a dataset with p = 3 variables, and suppose that missingness pattern s has Y 1 and Y3 observed but Y2 missing; then ns Σyi1 0 Σyi3 Σyi21 0 Σyi1 yi3 , Tobs ( s ) = 0 0 2 Σy13 0 0 0 Σyi2 0 Σy y i1 i2 0 , Tmis ( s ) = 2 Σyi2 Σyi2 yi3 0 where all sums are taken over i∈I(s). Finally, define S
Tobs =
∑
Tobs ( s ) and Tmis =
s =1
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S
∑ Tmis (s ), s =1
Figure 5.2. Single iteration of EM for incomplete multivariate normal data, written in pseudocode
so that T = Tobs + Tmis. The Estep may then be written E(T  Yobs , θ ) = Tobs + E(Tmis  Yobs , θ ) S
=
∑
s =1
Tobs ( s ) +
S
∑ E(Tmis (s )  Yobs ,θ ). s =1
The elements of Tobs can be calculated once at the outset of the program and stored for all future iterations of EM. An implementation in pseudocode One possible implementation of an iteration of EM is shown in Figure 5.2. It is written in pseudocode, a shorthand language that can be understood by anyone with programming experience and is easily converted into standard languages like Fortran or C. In this pseudocode, the symbol `: =’ indicates the operation of assignment; for example, `a: = b ’ means `set a equal to b.’ This implementation requires two (p + 1) × (p + 1) matrix workspaces: T, into which the expected sufficient statistics are accumulated, and q, which holds the current estimate of the parameter. For simplicity, the rows and columns of these matrices are labeled from 0 to p rather than
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from 1 to p + 1. In addition, a single vector of length p, denoted by c = (c1,...,cp), is needed as a temporary workspace to hold the values of yi∗ j given by (5.36). The iteration begins by setting T equal to T obs which we assume has already been computed. The expectations of y ij and yijyik that contribute to Tmis are then calculated and added into T, one missingness pattern at a time. In order to calculate these expectations within a missingness pattern s, the θmatrix must be put into the required SWP[Ο(s)] condition; for this, we use the convenient bookkeeping device that a diagonal element θ jj is negative if and only if θ has been swept on position j. Finally, after the expected sufficient statistics are fully accumulated into T, the new parameter estimate is calculated and stored in θ in preparation for the next iteration. For efficiency, the code in Figure 5.2 does not calculate the offdiagonal elements of T more than once. If θ and T are stored as twodimensional arrays, then only the uppertriangular portions should be used, and Tjk or θ jk should be interpreted as the (j, k)th element if ≤ k or the (k, j)th element if j > k. Memory requirements can be reduced by retaining only the uppertriangular parts of T and 0 in packed storage. To reduce the impact of rounding errors, T, θ , and c should be stored in double precision. Rounding errors can also be reduced by centering and scaling the columns of Y at the outset; for example, we could transform the observed data in each column of Y to have mean zero and unit variance before running EM. If the data are centered and scaled, however, we should remember that θ will be expressed on this transformed scale, and for interpretability we may need to transform the estimate of θ back to the original scale at the end of the program. Starting values EM requires a starting value θ(0)=(µ(0),∑(0))for the first iteration. Any starting value may be used provided that ∑(0) is
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positive definite, but in practice it helps to choose a value that is likely to be close to the mode. Several choices for starting values are described by Little and Rubin (1987). The mean vector and covariance matrix calculated only from the completely observed rows of Y may work well, provided that there are at least p + 1 such rows. Another easy method is to use the observed data from each variable to supply starting values for the means and variances, and set the initial correlations to zero; if the columns of Y have been centered and scaled at the outset to have mean 0 and variance 1, then this corresponds to taking µ(0)=(0,0,...,0)Τ and ∑(0)=Ι. Unless the fractions of missing information for some components of θ are very high, the choice of starting value is usually not crucial; when the missing information is low to moderate, the first few iterations of EM tend to bring θ to the vicinity of the mode from any sensible starting value. When writing a program for general use, it is helpful to give the user the option of supplying a starting value, because restarting EM from a variety of locations helps to diagnose unusual features of the observeddata likelihood, such as ridges and multiple modes. Estimates on the boundary It sometimes happens, particularly with sparse datasets, that the observeddata likelihood function increases without limit as θ approaches the boundary of the parameter space (i.e. as ∑ approaches a singular matrix). When this occurs, the EM algorithm may behave in a variety of ways. In some problems, the elements of θ stabilize and EM appears to converge to a solution on the boundary. In other problems, the program halts due to numeric overflow or attempted division by zero. In yet other problems, the sweeps required for the Estep become numerically unstable as the iterates approach the boundary, and substantial rounding errors are introduced. We have found that these rounding errors sometimes `deflect’ θ away from the boundary, causing a sudden large drop in likelihood from one iteration to the next. The iterates may approach the boundary
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for a number of steps, deflect away, approach again, and deflect away again in a recurring fashion. If the elements of θ do not appear to have converged after a large number of iterations, then it is advisable to monitor both the loglikelihood (Section 5.3.5) and some aspect of ∑ (e.g. the determinant, or the ratio of the largest eigenvalue to the smallest) to determine whether the iterates are approaching the boundary. When an ML estimate falls on the boundary, it is often helpful to apply a ridge prior and use EM to find the posterior mode as described below. 5.3.4 EM for posterior modes This EM algorithm can be easily altered to compute a mode of the observeddata posterior distribution rather than an MLE. As discussed in Section 3.2.3, the Estep is no different; only the Mstep needs to be modified. The exact form of this modification will depend on the prior distribution applied to θ. Priors for incomplete data At this point, it is worthwhile to consider what prior distributions may be appropriate for an incomplete dataset. Because a prior distribution by definition reflects one’s state of knowledge about θ before any data are observed, the fact that some data are missing should from a strictly Bayesian viewpoint have no effect whatsoever on the choice of a prior. To the Bayesian purist, any prior that is appropriate for complete data will be equally appropriate for incomplete data. Most statisticians would agree, however, that choosing a prior distribution (including its analytic form) purely by introspection can be difficult, and in practice most priors are chosen at least partly for computational convenience. The normal invertedWishart family of prior distributions, described in Sections 5.2.2 and 5.2.3, is computationally convenient for the EM and data augmentation algorithms in this chapter. In general, this family is not conjugate when data are incomplete; the observed data posterior P(θ  Yobs ) under a
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normal invertedWishart prior is tractable only in special cases. Yet EM and data augmentation are both easy to implement under this family of priors, because the simplicity of these algorithms depends upon the tractability of the completedata problem. When prior information about θ is scanty, we suggest that the customary diffuse prior for complete data,
π (θ ) ∝ Σ −( 2 ), may also be reasonable when some data are missing. Recall from Section 5.2.3 that one important justification for this prior with complete data is that Bayesian and frequents inferences about p coincide. This result does not immediately generalize to incomplete data, but limited experience suggests that Bayesian inferences under this prior may also be approximately valid from a frequents point of view. Little (1988) reports that in the case of bivariate datasets with missing values on one variable generated by an ignorable mechanism, this prior leads to Bayesian inferences about µ that are wellcalibrated; the HPD regions tend to have frequency coverage close to the nominal levels. Because this prior treats the variables Y1, Y2,..., Yp in a symmetric fashion, we conjecture that similar results may hold for more complicated multivariate scenarios as well. When data are sparse and certain aspects of ∑ are poorly estimated, we suggested in Section 5.2.3 that a useful prior for complete data was the limiting form of the normal invertedWishart with τ=0, m=∈ for some ∈ > 0, and Λ −1=∈Diag S, where S is the completedata sample covariance matrix. With incomplete data S cannot be calculated, but a useful substitute is the matrix with diagonal elements equal to the sample variances among the observed values in each column of Y. This prior effectively smooths the variances in ∑ toward the observeddata variances and the correlations toward zero. If the observed data in each column of Y have been scaled at the outset of the program to have unit variances, then this prior will simply take Λ−1=∈Ι. p +1
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Modications to the Mstep The joint mode of the normal invertedWishart distribution,
(
)
µ  Σ ~ N µ0 , τ −1Σ , −1
Σ ~ W ( m, Λ ), is achieved at µ 0 and (m+p+2)1Λ−1 for µ and ∑, respectively (Section 5.2.2). Thus the completedata posterior mode for θ = ( µ ,Σ) under the normal invertedWishart prior with hyperparameters (τ,m,µ0,Λ), denoted by θ˜ = µ˜ , Σ˜ , is
( )
1 µ˜ = µ0′ and Σ˜ = ( Λ ′)−1, m′ + p + 2 where µ 0, m ′ and Λ′ are the updated versions of the hyperparameters given in Section 5.2.2. By reversesweeping the mode on position 0 and equating the result to a matrix of modified sufficient statistics, ˜T −1 µ˜ T 1 µ˜ T −1 n T1 RSW[0] n = = ˜ ˜ ˜ ˜ T Σ˜ µ˜ Σ˜ + µµ µ˜ T1 T2 the mode can be computed as if it were an ML estimate based on T˜1 and T˜2 rather than T1 and T2. Solving for T˜1 and T˜2 and substituting expressions for the updated hyperparameters gives n T + τ nµ ~ T1 = n+τ 1 n+τ 0 and n T˜2 = T2 − 1n T1T1T + Λ−1 + A + 1n T˜1T˜1T n+m+ p+2 as the modified sufficient statistics, where τ A= (T − nµ0 )(T1 − nµ0 )T . n(τ + n) 1 To modify the EM algorithm shown in Figure 5.2 to compute a posterior mode rather than an MLE, we need only to replace the expected sufficient statistics T1 and T 2 in the workspace T by the modified versions T˜1 and T˜2 immediately before executing the final step θ : = SWP[0]n1Τ.
(
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)
5.3.5 Calculating the observeddata loglikelihood One of the great advantages of the EM algorithm is that it never requires calculation of the observeddata loglikelihood function or its derivatives. The observeddata likelihood for this problem, discussed in Example 3 of Section 2.3.2, or its logarithm l(θΥobs), would be very tedious to differentiate or maximize by gradientbased methods. Evaluation of l(θΥobs), at a specific value of θ , however, is not overwhelmingly difficult; the computations required for a single evaluation are comparable to those needed for a single iteration of EM. It follows from (2.10) that the observed dataloglikelihood function may be written as S T ∗−1 1 ∗ ∗ ∗ 1 − 2 log Σ s − 2 yi( obs ) − µ s Σ s yi( obs ) − µ s , s =1 i∈I ( s ) ∗ where yi(obs) denotes the observed part of y i and µ s and Σ ∗s denote the subvector of µ and the submatrix of ∑, respectively, that pertain to the variables that are observed in pattern s. An equivalent but computationally more convenient expression is
(
∑∑
l(θ  Yobs ) =
)
(
∑ {− 2 log Σ∗s − 12 trΣ*s −1M s }, S
ns
)
(5.38)
s =1
where ns is the number of observations in missingness pattern s and
Ms =
∑ ( yi(obs) − µ s∗ )( yi(obs) − µ s∗ )
T
i∈I ( s )
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.
Figure 5.3. Calculation of observeddata loglikelihood function.
Pseudocode for calculating l(θΥobs) is shown in Figure 5.3. This algorithm requires a p × p matrix workspace M to hold values of Ms, and a p × 1 vector c for temporary storage of µ . The constants d and t hold log Σ ∗s and trΣ *−1 s Ms, respectively, and after execution the loglikelihood value is contained in 1. This program modifies the parameter matrix θ; if necessary, however, the single line θ:=RSW[Ο(S)θ may be added at the end of the program, which will return θ to its original state except for rounding errors. Notice that the algorithm for evaluating L(θΥobs) bears a strong resemblance to a single step of EM. An obvious question to ask is whether the two sets of code can be combined, so that an evaluation of the loglikelihood is efficiently woven into EM itself This is certainly possible, but subject to the following caveats. First, the loglikelihood would have to be evaluated at the parameter estimate from the previous iteration; that is, we would have to evaluate l( θ(t) Υobs) as we computed θ ( t + 1 ) . Second, notice that a
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loglikelihood evaluation requires accumulation of the observed parts of the completedata sufficient statistics, rather than the expected values of the missing parts. Recall that the EM code in Figure 5.2 assumes that T obs the portion of the expected value of T that does not change over the iterations, has already been computed and stored at the outset of the program. Evaluation of the observeddata loglikelihood, however, requires access to the individual matrices Tobs(s) for s = 1,2,..., S which could be very cumbersome to store. If, as in Figure 5.3, the matrices Tobs(s) are not stored but effectively recomputed at each iteration, then the proportionate reductions in computing time achieved by combining the two algorithms over running them separately would not be overwhelming. When EM is used to find a posterior mode rather than an MLE, the function that is guaranteed to be nondecreasing at each iteration is no longer the observeddata likelihood but the observeddata posterior density. The logarithm of the observeddata posterior density is log P(θ  Yobs ) = l(θ  Yobs ) + log π (θ ), where unnecessary normalizing constants have been omitted. Thus the logposterior density may be evaluated by adding log π(θ) to the result of the algorithm in Figure 5.3. Under a normal invertedWishart prior with hyperparameters (τ,m,µ0,Λ), this additional term is m+ p+2 log π (θ ) = − log Σ − 12 trΣ −1 M 0 , 2 where
M 0 = Λ−1 + τ ( µ − µ0 )( µ − µ0 ) , and unnecessary constants have again been omitted. T
5.3.6 Example: serumcholesterol levels of heartattack patients Ryan and Joiner (1994, Table 9.1) report serumcholesterol levels for n = 28 patients treated for heart attacks at a Pennsylvania medical center. For all patients in the sample, cholesterol levels were measured 2 days and 4 days after the attack. For 19 of the 28 patients, an additional measurement
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was taken 14 days after the attack. The data are displayed in Table 5.1 (a), with readings at 2, 4 and 14 days denoted by Y1, Y2 and Y3, respectively. Regarding the complete data as a random sample from a trivariate normal distribution, we applied EM to find the observeddata
Table 5.1. EM algorithm applied to cholesterol levels for heartattack patients measured 2, 4 and 14 days after attack
ML estimates of the nine parameters in θ=(µ,Σ) (ML estimates for this dataset could also be calculated noniteratively; see Section 6.5). Denote the elements of µ and ∑ by µj and σjκ,
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respectively, for j, k = 1, 2, 3, and let ρjκ =σjκ(σjjσκκ)1/2 denote the correlations. From starting values chosen based on a crude guess, µ(ο)=(200,200,200)Τ and ∑ ( ο)=(50)2Ι, convergence within four significant digits to 2195 1455 835.4 253.9 2127 1515 µˆ = 230.6, Σˆ = 222.2 1953 was achieved in just three iterations. Because no data, e missing for Y1 or Y 2, the five parameters (µ1,µ2,σ11,σ22,ρ12) converge in a single step regardless of the starting value. Iterates of the four remaining parameters, expressed as µ3,
σ3=√σ33, ρ 13 and ρ 23, are displayed to six significant digits in Table 5.1 (b). For estimation of θ , the iterations beyond t = 4 are superfluous because precision beyond three or four digits is rarely necessary. As discussed in Section 3.3.4, however, these additional iterations can be used to estimate elementwise rates of convergence, which are typically equal to the largest fraction of missing information. Elementwise rates of convergence for the four parameters that do not converge in one step, estimated using (3.27), are displayed in Table 5.1 (c). These estimates, which are all close to 47%, do not measure the individual rates of missing information for the four parameters µ 3, σ3, ρ13 and ρ23; rather, they pertain to the function of θ for which the rate of missing information is highest. Notice that the 47% rate of missing information is somewhat higher than the 9/28 = 32% rate of missing observations for Y3. Because we know that the parameters pertaining to the joint distribution of (Y1, Y2) have no missing information, the 47% rate must pertain to some function of the parameters of the regression of Y 3 on Y1 and Y 2. It is instructive to consider why the largest rate of missing information exceeds the rate of missing observations for Y3. A hint is provided by the scatterplot of Y1 versus Y2 displayed in
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Figure 5.4 (a). The cases having missing values for Y3 tend to be slightly farther, on average, from the center of the (Y1, Y2) distribution than do the cases for which Y3 is observed. Because they are farther from the center, they exert more influence on the estimates of the regression parameters. A well known measure of influence in linear regression models is provided by the leverage values, the diagonal elements of the hat matrix (e.g. Draper and Smith, 1981).
Figure 5.4. (a) Scatterplot of Y1 versus Y2 for all cases, and boxplots of leverage values hii for cases having (b) Y3 observed and (c) Υ3 missing.
The hat matrix for linear regression is defined to be Η=Χ(ΧΤΧ)−1ΧΤ, where X is the matrix of predictor variables, in this case a 28 × 3 matrix containing the observed values of Y1 and Y2 and the column vector 1=(1,1,...,)Τ . Boxplots of the diagonal elements hii of H for the cases having Y3 observed and the cases having Y3 missing are shown in Figures 5.4 (b) and (c), respectively. The incomplete cases tend to have slightly higher values of hii and thus exert greater influence on an average, percase basis over the estimates of the regression parameters. The parameters of greatest interest in this problem appear to be functions of µ, such as comparisons or contrasts among µ1,
µ2 and µ 3. Although the rate of missing observations for Y3 is 32%, we might conjecture that the rate of missing information for µ3 or a contrast involving µ3 is substantially lower, because of the high correlations between Y3 and the completely
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observed variables Y1 and Y2. The rate of missing information for µ 3, a contrast involving µ 3 or any other function of θ may be estimated in a straightforward manner by multiple imputation; see Section 6.2.1. 5.3.7 Example: changes in heart rate due to marijuana use Weil et al. (1968) describe a pilot study to investigate the clinical and psychological effects of marijuana use in human subjects. Nine
Table 5.2. Change in heart rate recorded 15 and 90 minutes after marijuana use, measured in beats per minute above baseline
healthy male subjects, all of whom claimed never to have used marijuana before, received doses in the form of cigarettes of uniform size. Each subject received each of the three treatments (low dose, high dose and placebo) and the order of treatments within subjects was balanced in a replicated 3 × 3 Latin square. Changes in heart rate for the n = 9 subjects measured 15 and 90 minutes after the smoking session are displayed in Table 5.2. Because the article does not specify the order in which the treatments were given to the individual subjects, we will ignore this feature of the data and proceed as if the order effects are negligible. At first glance, it appears that missing data are only a minor problem here; only 5 of the 54 data values are missing. Yet,
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the EM algorithm converges very slowly. Depending on the starting values and convergence criterion, several hundred iterations may be needed to obtain convergence. The elementwise rates of convergence indicate that the largest fraction of missing information is approximately 97%. Moreover, the ML estimate of θ lies on the boundary of the parameter space. The ML estimates of the means, standard deviations and correlations are displayed in Table 5.3, along with the eigenvalues of the estimated correlation matrix. The smallest eigenvalue is zero to three decimal places, indicating that the estimated covariance matrix is singular or nearly so. Why do so few missing values create such difficulty in this example? Table 5.3. ML estimates of means, standard deviations and correlations for the columns of Table 5.2, with eigenvalues of the estimated correlation matrix
There are two primary reasons. First, the incomplete cases appear to be very influential. A comparison of the ML estimates of the means in Table 5.3 (a) with the means of the observed data in the columns of Table 5.2 is quite revealing. The large discrepancy for the fourth column (10.6 versus 1.0) demonstrates that a disproportionate amount of information about the mean for that column is provided by subjects 4 and 5. Further examination of Table 5.2 reveals that these two
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subjects have rather extreme values in some of the other columns, which gives them high leverage. When these two subjects are deleted, EM converges rapidly and the estimated largest fraction of missing information drops to 45%. A second reason why this example is problematic is that the completedata estimation problem is poorly conditioned. The number of subjects n = 9 is not much greater than the number of variables p = 6. When n and p are nearly equal, it becomes likely that certain linear combinations of the columns of Y will show little or no variability, particularly when the columns are correlated. The multivariate normal model for this example has 27 parameters, too many to be estimated well from a dataset of this size even with complete data. Although certain aspects of θ are poorly estimated, however, we can still make reasonable inferences about the parameters of interest; see Section 5.4.4.
5.4 Data augmentation
5.4.1 The Istep Data augmentation for incomplete multivariate normal data is remarkably similar to the EM algorithm. The deterministic Eand Msteps are replaced by stochastic I and Psteps, respectively, where the Istep simulates t +1 Y ( ) ~ P Y  Y , θ ( t) ,
(
mis
mis
obs
and the Pstep simulates θ ( t +1) ~ P θ  Y
(
( t +1) obs , Ymis
)
).
Because the rows y 1 , y2,...,yn of Y are conditionally independent given θ, the Istep is carried out by drawing t +1 yi(( mis)) ~ P yi( mis )  yi( obs ) , θ ( t ) independently for i = 1, 2,..., n. As discussed in Section 5.3.2, if row i is in missingness pattern s then the conditional
(
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)
distribution of yi(mis) given yi(obs) and θ is multivariate normal with means (5.39) E yij  Yobs , θ = a0 j + akj yik k ∈O( s ) and covariances Cov yij , yik  Yobs , θ = a jk (5.40)
(
)
(
∑
)
for j,k∈M(s), where ajk denotes an element of the matrix (5.41) A=SWP[Ο(s)]θ. Thus the Istep of data augmentation involves nothing more than the independent simulation of random normal vectors for each row of the data matrix, with means and covariances given by (5.39) and (5.40). A convenient way to simulate random normal vectors within the Istep is to create a Cholesky factorization routine that operates
Figure 5.5. Calculation of A:= CholsA.
on square submatrices of (5.41). The Cholesky factor of a positive definite matrix A, denoted by C=CholA, is an uppertriangular matrix of the same dimension of A having the property that C T C = A. To simulate a random vector z from N(b, A), we may take z=b+(CholA)Τz0, where z 0 is a vector of the same length as z containing independent standard normal variates. A typical Cholesky factorization routine operates on the uppertriangular portion of a symmetric matrix, overwriting it with its Cholesky factor, To draw from the distribution of y i(mis) given yi(obs) and θ,
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however, we need to calculate the Cholesky factor of only the square submatrix of (5.41) corresponding to the rows and columns in M(s). For a set S of row labels of a matrix A, let us use Α:=CholsA (5.42) to indicate the operation that overwrites (the upper triangular portion of) the square submatrix {ajκ : j,k∈S} with its Cholesky factor, while leaving the remaining elements of A unchanged. A simple algorithm for this operation, adapted from pseudocode given by Thisted (1988, p. 83), is shown in Figure 5.5. Once the Cholesky factorization is available, the Istep becomes a simple matter of cycling through the missingness patterns s = 1,..., S, calculating CholM(s)SWP[Ο(s)]θ for each s, and simulating yi(mis) for each i∈Ι(s). An implementation of the Istep is shown in Figure 5.6. The code simulates the
Figure 5.6. Istep for incomplete multivariate normal data.
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missing values in Y mis and stores them in the appropriate elements of Y. In addition, the code contains four lines preceded by the single character ’C’ which accumulate the simulated completedata sufficient statistics and store them in a (p + 1) × (p + 1) matrix workspace T. If the Istep is to be followed by a Pstep, then these sufficient statistics will be needed to describe the completedata posterior distribution of θ. If the Istep will not be followed by a Pstep (e.g. if it is the final step of a chain for producing an imputation of Ymis) then these four lines may be omitted. The code in Figure 5.5 requires two temporary workspaces: a p × p matrix C for storing Cholesky factors, and a p × 1 vector z for holding simulated N (0, 1) variates. 5.4.2 The Pstep Under the prior distributions discussed in Sections 5.2.2 and 5.2.3, the complete data posterior Ρ(θΥobs,Υmis) is a normal invertedWishart distribution. The Pstep of data augmentation, therefore, is merely a simulation of the normal invertedWishart distribution,
(
)
µ  Σ ~ N µ0 , τ −1Σ , Σ ~ W −1( m, Λ ), for some (τ,m,µο,Λ)determined by the prior, the observed data ( t ) imputed at the last Istep. The Υobs, and the missing data Ymis specific values of ( τ ,m,µο,Λ)) are calculated using the formulas for updating hyperparameters given in Section 5.2.2. The most obvious way to generate ∑~W1(m,Λ) is to take ∑=(ΧΤΧ)1, where X is an m × p random matrix whose rows are independent draws from N(0,Λ). This method cannot be used for noninteger values of m, however, and may be cumbersome for large m because it requires mp random variates. More efficient methods for generating random Wishart matrices are available that require simulation of only p (p + 1)/2 random variates. One such method relies on a characterization of the Wishart distribution known as the
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Bartlett decomposition (e.g. Muirhead, 1982). If Α~W(m,I) where I is a p × p identity matrix and m ≥ p, then we can write A = BTB where B is an upper triangular matrix whose elements are independently distributed as 2 b jj ~ X m − j +1, j = 1,..., p,
(5.43)
b jk ~ N (0, 1), j < k . (5.44) Suppose that we generate an uppertriangular matrix B according to (5.43)(5.44), so that ΒΤΒ~W(m,I), and take M=(ΒΤ)1C, where C is the Cholesky factor of Λ −1(i.e.CΤC=Λ−1). Then ∑=MΤM will be distributed as W1(m,Λ) because
(M T M )
−1
( ) −1 ~ W m, (C T C ) . = C −1BT B C T
−1
(Here we have made use of the property that D ~W(n,Γ) implies C ΤDC~W(n,CΤΓC) which follows immediately from the definition of the Wishart distribution.) Moreover, taking µ = µ0 + τ −1 / 2 M Τ z. where z~N(0,Ι) is a p × 1 vector of independent standard normal variates, results in µ ∑~N(µ0,τ−1Σ)This method requires the inversion of only the triangular matrix BT, which can be accomplished via a simple backsolving operation. Note that with the exception of M, all matrices used here are either symmetric or triangular, so memory requirements can be reduced by retaining only their uppertriangular portions in packed storage. 5.4.3 Example: cholesterol levels of heartattack patients Recall the example of Section 5.3.6 in which cholesterol measurements were recorded for patients 2, 4 and 14 days after heart attack. The EM algorithm converged rapidly with an estimated largest fraction of missing information equal to 47%. We applied data augmentation to this example under the
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noninformative prior (5.18). Output analysis from preliminary runs suggested that the data augmentation algorithm also converged rapidly. For illustration, we ran a single chain for 1100 iterations starting from the ML estimate of θ, discarded the first 100 iterations, and estimated ACFs for a variety of scalar functions of θ over the remaining 1000 iterations. We deliberately chose functions of θ for which the rates of missing information were thought to be high, including: 1. µ 3 and σ 3, the mean and standard deviation of Y3, respectively; 2. the parameters of the linear regression of Y3 on Y 1 and Y2, including the slopes −1 β31⋅12 [σ 31σ 32 ] σ11σ12 , = σ σ 21 22 β32⋅12 the intercept −1 σ σ ] σ 11σ 12 µ1 β30⋅12 = µ3 −[ 31 32 , σ 21σ 22 µ2
T
and the residual standard deviation σ 3⋅12 = σ 33⋅12 , , where
σ 33⋅12 = σ 33 −[
σ 31σ 32 ]
−1
σ 11σ 12 σ 13 ; σ 21σ 22 σ 23
and 3. the worst linear function ξ = ξ(θ) estimated from the final iterations of EM, as described in Section 4.4.3. This is the inner product of θ and the estimated eigenvector corresponding to the largest eigenvalue of EM’s asymptotic rate matrix. Because there are no missing values on Y1 or Y2, ξ is a weighted sum of µ3, σ 13, σ23 and σ33, where the weights are the perturbations from the ML estimates in the final iterations of EM.
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Table 5.4 Sample ACFs of selected scalar parameters estimated over iterations of data augmentation
Sample ACFs for these functions of θ up to lag 20 are displayed in Table 5.4. Correlations that are significantly different from zero at the 0.05 level, as determined by Bartlett’s formula (4.49), are marked with an asterisk. Because the series is so long and the serial dependence is not high, the standard errors are small and even very small correlations are deemed significant. Even for the worst functions examined, however, the correlations are effectively zero by lag 10, and definitely negligible by lag 20. Timeseries plots of these functions showed no unusual features and resembled those of the rapidlyconverging series displayed in Figure 4.2 (a) and (b). Based on this evidence, we feel safe in concluding that the algorithm effectively achieves stationarity by 20 iterations. The parameters of greatest interest in this problem are functions of µ=(µ1,µ2µ3)Τ . For illustration, we will focus
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attention on three quantities: µ 3, the average cholesterol level at 14 days;
Figure 5.7. Histograms of sample values of (a)µ 3, (b) δ 13, (C) τ13 and (d) dL from 5000 consecutive iterations of data augmentation.
δ13 = µ1 − µ3 , the average decrease in cholesterol level from day 2 to day 14; and τ13=100(µ1−µ3)⁄µ1, the relative percentage decrease in average cholesterol level from day 2 to day 14. To draw inferences about these quantities, we simulated another single chain of 5100 iterations starting from the ML estimate, discarded the first 100, and saved the 5000 remaining values of µ 3, δ13 and τ13. Histograms of the sample values for these three quantities are shown in Figure 5.7 (a)(c). Because µ 3 and δ13 are linear combinations of the elements of µ, obtaining RaoBlackwellized estimates of the marginal densities of these quantities is straightforward. Under the prior (5.18), the completedata posterior is given by (519)(5.20). Using (5.17), it follows that the completedata posterior density of a linear combination η−aTµ is −( n − p +1) / 2 T 2 η − a y P(η  Yobs , Ymis ) = k 1 + , (5.45) n − p)σ 2 (
(
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)
where n = 28 and p = 3 are the number of observations and −1 variables, respectively; σ 2 = ( n − p) aT Sa; y and S are the sample mean vector (5.5) and covariance matrix (5.6) computed from Υ=(Υobs,Υmis); and
k=
( ) Γ( ) π (n − p)σ Γ
n − p +1 2
n− p 2
2
.
RaoBlackwellized density estimates for µ3=(0,0,1)µ and
δ13=(1,0,−1)µ estimated from the first 1000 iterations after the τ13 initial burnin period are shown superimposed over the histograms in Figure 5.7 (a) and (b). Because τ13 is nonlinear its density is somewhat less easy to find, and RaoBlackwellized estimates for this quantity are not shown. In addition to µ3, δ13 and τ13, we also calculated and stored values of the likelihoodratio statistic d = d (θ ) = 2 l θˆ  Y − l(θ  Y ) L
L
[(
obs
)
obs
]
over the 5000 iterations, where θˆ is the ML estimate. For large samples, the posterior distribution of dL is approximately χ d2, , where d is the dimension of θ (in this case, 9). A histogram of the sample values of dL is displayed in Figure 5.7 (d) with the χ 92 density function superimposed over it, showing that the actual posterior matches the theoretical approximation quite closely. Simulated posterior means for µ3, δ13 and τ13 were found by averaging the 5000 iterates of each parameter. Simulated 95% posterior intervals were found by calculating the 2.5 and 97.5 percentiles of each sample using (4.8). To obtain a rough assessment of the random error in these estimates, a second chain was generated in an identical fashion with a different randomnumber generator seed. The simulated posterior means and 95% intervals (in parentheses) for the two replicate runs are shown below.
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µ3 222.2 (201.6, 244.0) 222.4
δ13 31.8 (8.9, 55.4) 31.4
τ13 12.4 (3.7, 20.9) 12.3
(201.7, 242.6) (8.9, 53.3) (3.7, 20.3) Inferences about µ3, δ13 and τ13 can also be conducted through multiple imputation. This will be demonstrated in Section 6.2.1. 5.4.4 Example: changes in heart rate due to marijuana use Returning to the data in Table 5.2, let µj denote the population mean corresponding to column j, and let δjκ=µj−µκ,j,k=1,...,6. Following the original article be Weil et al. (1968), we will focus attention on the six treatment comparisons below. 15 min utes 90 min utes Low vs. Placebo δ 21 Low vs. Placebo δ 54
High vs. Placebo δ 31 High vs. Placebo δ 64 High vs. Low δ 32 High vs. Low δ 65 Data augmentation under the usual noninformative prior (5.18) does not work for this problem; the iterates of θ quickly wander to the boundary of the parameter space, causing numeric overflow. This pathological behavior suggests that the posterior is not proper. To stabilize the inference, we applied a ridge prior as described in Sections 5.2.3 and 5.3.4. After centering and scaling the columns of Y so that the observed data in each column have mean zero and unit variance, we set the hyperparameters of the normal invertedWishart prior to τ=0, m=∈, and Λ−1=∈Ι for ∈ =0.5. Under this weak prior, EM converges slowly but reliably to a posterior mode in the interior of the parameter space, with the largest fraction of missing information estimated at 95%. The slow convergence of EM in this example suggests that data augmentation will also converge slowly, and output analysis from a preliminary run confirmed this. Using the same ridge prior, we simulated a single chain beginning at the
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posterior mode and monitored a variety of scalar summaries of θ. Timeseries plots for δ21 and δ54 (on the original scale) from the first 100 iterations are shown in Figure 5.8 (a) and (b), respectively. The iterates of δ21 appear to approach stationarity quickly, whereas the series for δ54 shows longrange dependence. This is not surprising, because δ54 is a function of
µ4, and our earlier analysis led us to conjecture that the rate of missing information for µ4 was very high. Sample ACFs for
δ21 and δ54 estimated from 10 000 iterations are displayed in Figure 5.8 (c) and (d), respectively. Figure 5.8 (d) is typical of the ACFs for other slowly converging functions of θ . For all the functions we examined, the serial correlations effectively died out by lag 50. The slow convergence in this example should lead us to use extra caution in designing the simulation experiment. Running independent chains from overdispersed starting values would be attractive,
Figure 5.8. Timeseries plots of (a) δ21 and (b) δ54 over the first 100 iterations of data augmentation, and sample AM for (C) δ 21 and (d) δ54 estimated from 10 000 iterations, with dashes indicating approximate 0.05level critical values for testing ρκ=ρκ+1=…=0.
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but obtaining overdispersed starting values is not easy. Bootstrap resampling is unlikely to work well, because n is not much larger than p, so the distribution of θ over bootstrap samples will probably bear little resemblance to the observeddata posterior. Sampling from the prior is not possible, because the prior is not a proper probability distribution. Because convergence to stationarity tends to be fastest when the starting value is near the center of the observeddata posterior, we decided to run ten independent chains of 5500 iterations each, starting each chain at the posterior mode. After discarding the first 500 values from each chain, the pth sample quantile for each contrast δ jκ, was calculated for p = 0.025, 0.25, 0.5, 0.75 and 0.975 from the remaining 5000 values. Finally, the sample quantiles were averaged across the ten chains. For each of these averages, the variance of the quantiles across chains was used to estimate a standard error with nine degrees of freedom. The estimated quantiles for all six parameters are displayed in Figure 5.9. All of the simulated 95% posterior intervals cover zero, indicating that there is no strong evidence that any of the contrasts is different from zero. Standard errors for the simulated quantiles
Figure 5.9. Simulated posterior medians, quantiles and 951% equaltailed intervals for six contrasts.
ranged from 0.02 to 0.72, which is quite small relative to the width of the intervals displayed in Figure 5.9, so these simulation results are sharp enough for our purposes.
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One could very well argue that the unrestricted multivariate normal model has too many parameters to be estimated from a dataset of this size, and that the unnecessarily large number of nuisance parameters hinders us from making clear inferences about the parameters of interest. Indeed, the long tails exhibited in the marginal posteriors of Figure 5.9, particularly for the two contrasts involving µ4, suggest that some of the nuisance parameters are very poorly estimated, and we might do well to simplify the model. One possible simplification is to reduce the number of free parameters by applying a priori constraints to ∑. For example, we could require ∑ to satisfy the condition of compound symmetry (i.e. equal diagonal elements and equal offdiagonal elements). Simulation algorithms for incomplete multivariate normal data with constrained covariance structure are possible, but they are beyond the scope of this book. A slightly different approach would be to specify fixed, additive effects for the rows and columns of the data matrix, and define the parameters of interest to be contrasts among the column effects (Chapter 9). Yet another possibility is to perform a simple bivariate analysis for each contrast, making inferences about δjκ using only the data in columns j and k . Under this bivariate approach, it is no longer possible to make joint inferences about the contrasts. Moreover, ignoring the data in columns other than j and k when making inferences about δ jk m a y t e n d t o i n t r o d u c e n o n r e s p o n s e b i a s e s ;
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Figure 5.10. Simulated posterior medians, quantiles and 951% equaltailed intervals for six contrasts using a bivariate approach.
the MAR assumption tends to be less plausible for the bivariate dataset than for the one with six variables. The decision whether to include additional variables in an analysis is not always an easy one, particularly for small datasets, and is an important topic worthy of further research. Simulated posterior quantiles from a bivariate analysis are shown in Figure 5.10. For each contrast, data augmentation was applied to the bivariate dataset under the standard noninformative prior (5.18). Output analyses suggested that convergence to stationarity was rapid. For each contrast, 10 100 steps of a single Markov chain were simulated, beginning from the ML estimate. The first 100 values of the simulated contrast were discarded, and sample quantiles were calculated from the remaining 10 000. The distributions in Figure 5.10 are much narrower than those in Figure 5.9, and there is now a fair amount of evidence that the three contrasts δ21, δ31, and δ65 are nonzero.
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CHAPTER 6
More On The Normal Model
6.1 Introduction In the last chapter, we introduced EM and data augmentation algorithms for the multivariate normal model. In this chapter, we illustrate how to effectively apply these algorithms with more realdata examples, and discuss modifications to the algorithms that can help to increase their efficiency. Sections 6.2 and 6.3 present two examples of analysis by multiple imputation. The first, which was previously analyzed in Chapter 5 by parameter simulation, is straightforward and illustrates some of the basic properties of multipleimputation point and interval estimates. The second is more complicated, involving categorical variables and inestimable parameters. By working through this second example, the reader will come to understand some of the complications and subtle issues that often arise with real data, and learn strategies for effectively dealing with these issues. Real data often do not conform to normality, and it is important to know whether the multipleimputation procedures advocated in this book are robust to departures from the modeling assumptions. Section 6.4 presents a simulation experiment to demonstrate the robustness of multiple imputation in a realistic setting. When rates of missing information are high, EM and data augmentation tend to converge slowly. Section 6.5 presents a new class of simulation algorithms, called monotone data augmentation, that tend to converge quickly under certain types of missingness.
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6.2 Multiple imputation: example 1 6.2.1 Cholesterol levels of heartattack patients Recall the example introduced in Section 5.3.6 in which serum cholesterol levels for heartattack patients were recorded 2 days (Y1), 4 days (Y2) and 14 days (Y3) after attack. Nine of the n = 28 values of Y3 were missing. In Section 5.4.3, we used data augmentation to simulate posterior distributions for three parameters of interest: 1. µ3 the mean cholesterol level at 14 days; 2. δ 13 = µ1−µ3, the average decrease in cholesterol level from day 2 to day 14; and 3. τ13 = 100(µ1−µ3)/µ1, the percentage decrease in cholesterol level from day 2 to day 14. We now demonstrate how inferences for these same quantities can be conducted by multiple imputation. 6.2.2 Generating the imputations Recall that proper multiple imputations are independent draws of Ymis from the posterior predictive distribution of the missing data, P(YmisYobs). The exploratory run of data augmentation revealed no discernible autocorrelations in scalar functions of θ beyond lag 10. Thus we can probably obtain acceptable imputations by (a) running data augmentation in a single chain starting from the MLE, and taking every tenth iterate of Ymis as an imputation; or (b) running independent, parallel chains of ten iterations each starting from the MLE, and taking the final value of Ymis from each chain as an imputation. Because of the small size of this dataset, however, iterations are computationally inexpensive, and we can easily afford to increase the number of steps. To illustrate a conservative approach, we generated m =5 multiple imputations by simulating five independent chains of 50 steps each.
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Independent starting values for the chains were obtained by running EM on independent bootstrap samples of size n/2 = 14 (Section 4.4.2). These starting values are probably overdispersed relative to the observeddata posterior P(θYobs), so that in the unlikely event that stationarity has not been achieved by 50 steps, the resulting inferences will tend to be conservative. The m = 5 sets of imputed values for Y3, rounded to integers, are displayed in Table 6.1. 6.2.3 Completedata point and variance estimates Multiple imputation r equires that for each estimand Q we specify a completedata point estimate Q Qˆ and a completedata variance Table 6.1. Cholesterol levels for heartattack patients measured 2, 4 and 14 days after attack, with m = 5 multiple imputations
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estimate U. It also requires a sample size large enough for the approximation
Qˆ − Q ~ N (0, 1) U
(6.1)
to work well with complete data. Let
yj =
1 n
n
n
∑ yij and S jk = n − 1 ∑ ( yij − y j )( yik − yk ) 1
i=1
i=1
for j, k = 1, 2, 3 denote the completedata sample means and covariances. For µ3, the obvious completedata estimates are Qˆ = y3 and U=S33/n. For δ13=µ1−µ3, the obvious choices are
Qˆ = y1 − y3
U = ( S11 − 2S13 + S33 ) / n. Asymptotic normality of y3 and y1 − y3 is guaranteed by the Central Limit Theorem, and a sample of size n = 28 should be large enough for the normal approximations to work well. For the nonlinear parameter τ13 = 100(µ1−µ3)/µ1, a firstorder Taylor expansion of the function
( y1 − y3 ) / y1
about
(µ1,µ3),
y1 − y3 µ1 − µ3 µ3 1 − ≈ 2 ( y1 − µ1 ) − ( y3 − µ3 ), y1 µ1 µ µ1 1 suggests that the completedata point estimate
Qˆ = 100( y1 − y3 ) / y1 will be approximately unbiased for τ13 , with approximate variance
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µ 1 1002 µ32 V Qˆ ≈ 4 σ 11 − 2 33 σ 13 + 2 σ 33 . n µ1 µ1 µ1
()
A reasonable completedata variance estimate is thus y 1 1002 y32 U = 4 S11 − 2 33 S13 + 2 S33 . n y1 y1 y1 A handy ruleofthumb used by survey statisticians is that a ratio of sample means will be approximately unbiased and normally distributed if the coefficient of variation (the standard deviation divided by the mean) of the denominator is 10% or less (e.g. Cochran, 1977, p. 166). The observed values of Y1 in Table 6.1 have a mean and standard deviation of 253.9 and 47.7, respectively, so the estimated coefficient of variation for y1 is 47.7 / 28 /253.9=0.036, suggesting that the
(
)
normal approximation should work well. Table 6.2. Completedata point estimates and standard errors for µ3, δ13 and
τ13 from m=5 multiplyimputed datasets
Following the notation of Section 4.3.2, let Qˆ ( t ) and U(t) denote the completedata point and variance estimates from the tth imputed dataset. Point and variance estimates for µ1,
δ13 and τ13 over the five imputations are displayed in Table 6.2. 6.2.4 Combining the estimates Combining the completedata point and interval estimates is a straightforward application of the formulas in Section 4.3.2 for
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inference with a scalar estimand. The overall estimates Q , standard errors T , degrees of freedom v for the tapproximation and 95% interval estimates are displayed in Table 6.3. The values of v are large, suggesting that the total variance estimates T are stable even though they are based on only m = 5 imputations. The point and interval estimates in Table 6.3 differ somewhat from those obtained by parameter simulation in Section 5.4.3, but the differences are mild relative to the sizes of the standard errors. Table 6.3 also displays two diagnostics described in Section 4.3.2: the relative increase in variance due to nonresponse r, and the estimated fraction of missing information λˆ . Although 32% of the Y3 values are missing, the estimated rates of missing information for µ3, δ 13 and τ13 are under 10%, due undoubtedly to the correlations between Y3 and the two variables that are never missing. 6.2.5 Alternative choices for the number of imputations For this analysis we chose m=5 imputations, because we knew that the fractions of missing information would not be severe. Recall that if the fraction of missing information for a parameter is λ, the relative efficiency of an estimate based on m imputations to one
Table 6.3. Results of multipleimputation inference for µ3, δ13 and τ13
based on an infinite number is approximately (1+λ /m) (Section 4.3.1). From EM we learned that the worst fraction of missing information for this problem was about 47% (Section 5.3.6). Thus in the worst case, m = 5 would lead to a point 1 estimate that is about (1+0.47/5) = 91% as efficient as one with m = ∞ . In fact, the estimated fractions of missing 1
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information for the parameters of interest were about 10%, so the estimates from m = 5 imputations appear to be about 1 (1+0.1/5) = 98% efficient. To those unaccustomed to multiple imputation, basing any conclusion on a Monte Carlo simulation with only m = 5 draws might seem risky. A critic might argue that with only five imputations, one or more ’bad’ (i.e. highly unusual) imputations could exert an undue influence on the results. To illustrate the effect of increasing the size of m, we generated an additional 95 imputations in the manner described above, for a total of 100 imputations. We then calculated point and interval estimates based on m − 3, 5, 10, 20, and 100. For m = 3 we used the first 3 imputations; for m = 5 we used the first 5 imputations; and so on. Finally, to get a rough idea of the amount of random variation in the estimates, we replicated the entire experiment, generating another 100 imputations from a different randomnumber generator seed and calculating another set of estimates for m = 3, 5, 10, 20, and 100. The point and interval estimates for the various values of m are displayed graphically in Figure 6.1. For comparison, Figure 6.1 also displays the results of the two parametersimulation runs of length 5000 described in Section 5.4.3. The multipleimputation (MI) intervals for m = 3 and m = 5 appear to have more random variation than the parametersimulation (PS) intervals. By m = 10, however, the MI intervals appear to remarkably stable, and there is little random variation (relative to the widths of the intervals) in any of the results for m = 10, 20 or 100. The variability for m = 3 and m = 5 does not mean that these
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Figure 6.1. Point and 95% interval estimates for µ 3 , δ 1 3 and τ 1 3 from parameter simulation (PS) and multiple imputation (MI).
intervals are unreliable. The intervals explicitly include simulation error as a component of uncertainty, and over repeated application they should still cover the true values of the parameters at least 95% of the time. To reduce random variation, one might consider increasing m, particularly if generating and storing imputations is not expensive. Based on Figure 6.1, however, there appears to be little reason to use more than m = 10 imputations for this problem. Advantages of multiple imputation over parameter simulation The PS estimates based on 5000 iterates of θ appear to be about as stable as MI estimates based on only m=10 imputations of Y mis. Notice, however, that the latter required only onetenth as much
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Table 6.4. Estimated fractions of missing information from m=3, 5, 10, 20 and 100 imputations
computation (500 steps of data augmentation versus 5000) and 0.6% as much storage (10 × 9 = 90 locations to hold imputations of Ymis, versus 5000 × 3 = 15 000 to hold values of
µ3, δ13 and τ13). A further advantage of MI is that it provides an estimated fraction of missing information for each estimand. For small m, however, these estimates can be noisy. To illustrate, estimated fractions of missing information for µ3, δ 13 and τ13 based on m = 3, 5, 10, 20, and 100 imputations (both replicates) are shown in Table 6.4. For small m, the estimates vary substantially between replicates. This is to be expected, because they depend on the betweenimputation components of variance which are estimated with only m1 degrees of freedom. Recall that our initial estimates of A based on m = 5 imputations were all under 10% (Table 6.3); after increasing the value of m to 100, the estimates rose to 1318%. Additional replications (not shown) demonstrate that even for m = 100, the estimates λˆ still have standard errors of approximately 0.02. Thus for small values of m, λˆ should be used only as a rough guide.
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6.3 Multiple imputation: example 2 6.3.1 Predicting achievement in foreign language study Raymond (1987) describes data that were collected to investigate the usefulness of a newly developed instrument, the Foreign Language
Table 6.5. Variables in foreign language achievement study, with number of missing values
Attitude Scale (FLAS), for predicting success in the study of foreign languages. In particular, the investigators wanted to determine whether the FLAS had substantial predictive ability beyond that already provided by other wellestablished instruments such as the Modern Language Aptitude Test (MLAT). Twelve variables were collected for a sample of n = 279 students enrolled in foreign language courses at The Pennsylvania State University in the early 1980s (Raymond and Roberts, 1983). Descriptions of the variables, along with the number of missing values for each one, appear in Table 6.5. The raw data, kindly provided by Dr. Mark Raymond, are reproduced in Appendix A.
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In this example, only 8% of all the values in the 279 × 12 data matrix are missing, and missingness rates per variable range from 0% to 18%. Only 62% of the cases (174 out of 279) have complete data for all twelve variables, however, so the casedeletion methods used by most statistical software packages would discard over one third of the
Figure 6.2. Histograms of observed data for variables in foreign language achievement study.
entire dataset. Imputing for the missing values makes more efficient use of the available data. 6.3.2 Applying the normal model Histograms of the observed values for each variable are displayed in Figure 6.2. Although these data clearly do not follow a multivariate normal distribution, we will still use the normal model for imputation. For the dichotomous and ordinal variables, we will impute under an assumption of normality and round off the continuous imputes to the nearest category. Examination of Figure 6.2 suggests that this strategy might not work well for AGE, PRI or GRD, because these variables are far from being symmetric and unimodal. To make the variables AGE, PRI and GRD less troublesome, we recoded them by collapsing some adjacent categories. (In Chapter 9, when we are able to explicitly model
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mixed continuous and categorical data, we will analyze these data again without recoding.) An overwhelming majority of students received final grades of A or B; very few received C or below; the data provide relatively little information to characterize the Corbelow group, so we recoded final grade as a simple dichotomy (A, B or below). Similarly, the three highest age groups had very few students in them, so age was collapsed to a dichotomy as well (less than 20, 20+). Prior experience was reduced from five categories to three. Histograms of the recoded versions of AGE, PRI, and GRD and the revised definitions of these variables appear in Figure 6.3 and Table 6.6, respectively. Notice that the variable LAN is nominal and should not be handled as a normal variable; the four language groups have no intrinsic ordering. To address this issue, LAN was replaced by a set of three dummy variables to distinguish among the four language groups: LAN2 = 1 if Spanish and 0 otherwise, LAN3 = 1 if German and 0 otherwise, and LAN4 = 1 if Russian and 0 otherwise. Including LAN2, LAN3 and LAN4 effectively treats the eleven remaining variables as multivariate normal within each of the four language groups, with a separate mean vector for each group and a common covariance matrix. The multivariate normal model clearly misspecifies the marginal distribution of the dummy variables, but this misspecification is of no consequence because the dummies are completely observed and do not need to be imputed (Section 2.6.2). Finally, it is important to remember that a normal distribution has support on the whole real line, but the continuous variables in this dataset have a limited range of possible values. For example, SAT scores may not exceed 800, and grade point averages may not exceed 4.0. Imputing under normality might occasionally result in an imputed value that is out of range. To handle this problem, we included a consistency check in our imputation routine. After performing the final Istep of data augmentation to create an imputation of Ymis, each row of the imputed dataset was examined to see whether any of the imputed values were out of range; if so, the missing data for that row were redrawn until the necessary
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constraints were satisfied. The final values of Ymis created by this procedure approximate proper multiple imputations under a truncated multivariate normal model. 6.3.3 Exploring the observeddata likelihood and posterior When LAN is replaced by three dummy variables, the dataset has p, = 14 variables. The EM algorithm applied to these 14 variables converged rapidly; the parameter estimates stabilized to four significant digits after only ten iterations. When EM converges so quickly, estimating the largest fraction of missing information from the iterations can be difficult, because the estimated elementwise rates of convergence (3.27) tend to become numerically unstable after only a few iterations. Moreover, the iterations at which instability begins vary from component to component. The multivariate normal model for 14 variables has 119 parameters. With so many parameters, it is not easy to estimate the fraction of missing information by visually inspecting the elementwise rates. In situations like this it is helpful to apply graphical techniques. To estimate the worst fraction of missing information, we first calculated elementwise rates (3.27) for each of the 119 parameters over the first 20 iterations of EM. After trimming away any values outside the interval (0, 1), we formed boxplots of the remaining values for each parameter, displaying them sidebyside. Boxplots for 50 randomly selected elements of µ and Σ are shown in Figure 6.4. Although a large number of outliers are present, all of the boxplots tend to be centered around 0.4. The median of the values in Figure 6.4 is 0.42, so a reasonable estimate of the worst fraction of missing information is 42%. Inestimability of parameters The moderate rates of missing information and the rapid convergence of EM might lead one to believe that the observeddata
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Figure 6.4. Boxplots of estimated elementwise rates of convergence for 50 randomly selected parameters.
likelihood function for this problem is well behaved. It turns out, however, that the likelihood is pathological. We performed a long exploratory run of data augmentation under the usual noninformative prior (5.18) and constructed timeseries plots for selected elements of µ and Σ. For most parameters, the algorithm appeared to achieve stationarity very quickly. For a few parameters, however, the simulated values drifted into implausible regions of the parameter space. Time series plots for the means of the two variables with the highest rates of missingness, MLAT and GRD, are shown in Figure 6.5. Figure 6.5 (a) is typical of the plots for most parameters, with no discernible trends. Figure 6.5 (b), however, shows extreme longrange dependence. The mean of the dichotomous variable GRD is known to lie between 1 and 2, but by the 900th iteration the series has drifted above 2. This unusual behavior suggests that one or more components of θ are nearly or entirely inestimable from the observed data. Additional runs of EM confirmed the presence of inestimable parameters. Using various simulated values of θ from the dataaugmentation series as starting values, we reran EM and found that in each case it converged to a different stationary value. Moreover, when we evaluated the observeddata loglikelihood function at these stationary values, the loglikelihood was exactly the same in each case. Thus it appears that the stationary values are not distinct modes, but form a ridge of constant likelihood. The pathological behavior in Figure 6.5 (b) arises because the observeddata posterior
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distribution is not proper; although the I and Psteps of data augmentation are both well defined, the algorithm is not
Figure 6.5. Time series plots of (a) mean MLAT and (b) mean GRD over 1000 iterations of data augmentation. Table 6.7. Crosstabulation of LAN with GRD
converging to any stationary distribution (Section 3.5.2). With a little exploration it is easy to detect the source of difficulty. Figure 6.5 (b) suggests that the inestimable part of θ pertains to the distribution of GRD. A crosstabulation of GRD with LAN, shown in Table 6.7, reveals that GRD is missing for all cases with LAN = 4. Because no values of GRD are available for any students enrolled in Russian courses, it is impossible to estimate the parameters of the conditional distribution of GRD given LAN= 4 from this dataset. 6.3.4 Overcoming the problem of inestimability One way to solve the problem of inestimability is to simply exclude the Russian language group and the variable LAN4
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from the analysis. Because GRD is missing for all 20 of these cases, they contribute little or no information about the main question of scientific interest, which pertains to the quality of FLAS as a predictor of GRD. Another way to handle the problem is to introduce a small amount of information about the inestimable portions of θ through a mildly informative prior distribution. Although excluding the Russian language group is certainly reasonable, we will adopt the latter approach to illustrate the use of an informative prior distribution. After centering and scaling the observed data for each variable to have mean 0 and variance 1, we applied the ridge prior described in Section 5.2.3 with τ=0, m= ∈ and Λ−1 =∈ I for ∈ =3. This prior adds the equivalent of three degrees of freedom to the estimation of Σ and smooths the estimated correlation matrix toward 1. With a sample size of n = 279 the degree of smoothing is slight, and the effect on those portions of θ that are already well estimated is almost negligible. For portions of θ that are poorly estimated, however, this prior smooths the estimates toward a model of mutual independence among all variables. Inferences under this prior will thus tend to be conservative in the sense that we will be less likely to conclude that associations among variables are present when in fact they are not. Under this prior, EM was found to converge reliably from a variety of starting values to a single posterior mode. The convergence was slower than before, requiring about 30 iterations, and the largest fraction of missing information was estimated at 92%. It may seem somewhat counterintuitive that the introduction of prior information appears to raise the worst fraction of missing information rather than lower it. This fraction, however, pertains only to those directions or functions of θ for which the function being maximized (i.e. the observeddata likelihood or posterior) is not flat. The elementwise rates estimate the largest eigenvalue of the asymptotic rate matrix that is less than one (Section 3.3.2). A ridge in the function produces one or more eigenvalues equal to one, and thus the inestimable functions of θ do not
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contribute to the estimated worst fraction of missing information when EM is used to maximize the likelihood. When an informative prior is introduced, however, the posterior is no longer precisely flat in any direction, and every function of θ then contributes to the estimated worst fraction of missing information. Under the informative prior, data augmentation also appears to converge reliably. Starting at the mode, we ran a single chain for 1000 iterations and monitored a variety of functions of q. Sample
Figure 6.6. Sample ACFs for (a) mean GRD and (b) the worst linear function of θ , estimated from 1000 iterations of data augmentation, with dashed lines indicating approximate critical values for testing ρk=ρk+1=⋅⋅⋅=0.
ACFs for two functions are shown in Figure 6.6. The mean of GRD, which behaved pathologically under the noninformative prior, now shows no appreciable dependence after lag 20. The worst linear function of θ , as estimated by the trajectory of EM in the vicinity of the posterior mode (Section 4.4.3), appears to achieve stationarity in about 25 steps. 6.3.5 Analysis by multiple imputation Following the preliminary run, we created m = 20 multiple imputations of the missing data by running 20 independent chains for 100 steps each. Starting values for the chains were obtained by finding posterior modes from independent bootstrap samples of 140 subjects each. Inferences for logisticregression coefficients Because the response variable GRD was collapsed to a dichotomy, we decided to measure the predictive ability of
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FLAS and the other variables by logistic regression (e.g. McCullagh and Nelder, 1989). Let πi denote the probability of GRD = 2 for subject i. We examined the model
log
πi = xiT β , 1− πi
(6.2)
where xi is a vector of covariates for subject i and β a vector of unknown coefficients. Covariates in xi included a term for the intercept; three dummy indicators for language (LAN2, LAN3 and LAN4); an indicator for age (AGE2 = 1 if 20+ and 0 otherwise); an indicator for sex (SEX2 = 1 if female and 0 otherwise); linear and quadratic contrasts for PRI (PRI L = −1,0,1 and PRI Q = 1,−2,1 for PRI = 1,2,3, respectively); Table 6.8. Multipleimputation inferences for logisticregression coefficients, full model
and the variables FLAS, MLAT, SATV, SATM, ENG, HGPA and CGPA. For each of the 20 imputed datasets, we computed ML estimates and asymptotic standard errors for the elements of β, and then combined the 20 sets using the formulas for multipleimputation inference for scalar estimands (Section 4.3.2).
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The results of the analysis are summarized in Table 6.8. For each coefficient, Table 6.8 displays the point estimate Q and standard error T , the tstatistic Q / T , the degrees of freedom v for the Student’s tapproximation, and the pvalue for testing the hypothesis Q = 0 against a twosided alternative. Also shown are the relative increase in variance due to nonresponse r and the estimated fraction of missing information λˆ . The pvalue for FLAS (0.021) suggests that this variable is useful for predicting GRD. Increasing FLAS by ten points multiplies the odds π i/(1−πi) by an estimated factor e × = 1.47 in other words, every tenpoint increase in FLAS makes a student 47% more likely (on the odds scale) to receive a grade of A, if other covariates are held constant. The most powerful predictor of final grade appears to be highschool GPA; a oneunit increase in HGPA causes the 2.27 predicted odds to be multiplied by e = 9.68. The only significant language effect is the coefficient of LAN3, which distinguishes between the German and French groups; a 1.12 student taking German appears to be about e = 3.06 times as likely to receive an A as a student taking French. Notice that LAN4, which contrasts Russian with French, has a nonsignificant effect (p = 0.979) and a high fraction of missing information (50%). This is to be expected, because essentially all information about this parameter comes from the prior distribution which tends to pull the estimated coefficient toward zero. 10 0.0386
Joint inferences for groups of coefficients The inferences in Table 6.8 pertain to the logisticregression coefficients individually. To make joint inferences about groups of coefficients, we need the methods for multidimensional estimands presented in Section 4.3.3. Of the three methods described there, we will demonstrate the procedure of Meng and Rubin (1992b) for combining likelihoodratio test statistics. With complete data, the loglikelihood function for the logistic model (6.2) may be written as
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T exi β 1 , z i log l( β  Yobs , Ymis ) = + (1 − z i ) log T T 1 + exi β 1 + e x i β i=1
n
∑
where zi = 1 if individual i has GRD = 2, and zi = 0 otherwise (e.g. McCullagh and Nelder, 1989). Suppose we want to test whether the coefficients for a group of variables (say, LAN2 and LAN4) are simultaneously zero. The usual likelihoodratio test with complete data requires us to fit (a) the full model with all variables, and (b) the reduced model with all variables except LAN2 and LAN4. Denote the ML estimates of β under the full and reduced models by βˆ and βˆ respectively. For notational convenience, we assume that βˆ and β˜ are of the same length, with the elements of β˜ corresponding to the omitted variables set to zero. The likelihoodratio test statistic is
(
) [(
) (
)]
dL βˆ , β˜  Yobs , Ymis = 2 l βˆ  Yobs , Ymis − l βˆ  Yobs , Ymis , which, under the reduced model, is approximately distributed as χ 22 because the reduced model differs from the full model by two parameters. The method of Meng and Rubin (1992b) requires two passes through the imputed data. Let βˆ ( t ) and β˜ ( t ) denote the ML estimates for the full and reduced models, respectively, fit to the tth Table 6.9. Multipleimputation likelihoodratio tests for eliminating groups of variables from the regression model
imputed dataset. In the first pass, we calculate the likelihoodratio statistic for each imputed dataset and find their average,
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1 d˜L = m
( t) ). ∑ dL (βˆ (t) , β˜ (t)  Yobs , Ymis m
t =1
In the second pass, we calculate the average of the likelihoodratio test statistics with βˆ ( t ) and β˜ ( t ) replaced by their averages,
1 d˜L = m
( t) ). ∑ dL (m −1 ∑ tm=1 βˆ (t) , m −1 ∑ tm=1 β˜ (t)  Yobs , Ymis m
t =1
The test statistic D3 and pvalue are then found by (4.44)(4.46). Using this technique, we tested three groups of variables and removed them from the model in turn after confirming that their pvalues were high. The three groups were (a) the language indicators LAN2 and LAN4; (b) the test scores SATV, SATM and ENG; and (c) the linear and quadratic contrasts for PRI. Results from each test are shown in Table 6.9, including the test statistic D 3, the degrees of freedom k and v3 for the Fapproximation, the pvalue, the relative increase in variance due to nonresponse r3, and the fraction of missing information λˆ calculated as λˆ = r3 / (1 − r3 ) . Notice that D3 for omitting LAN2 and LAN4 is slightly less than zero. With complete data, a likelihoodratio test statistic cannot be negative. With Meng and Rubin’s method, however, negative values do sometimes occur, particularly when the estimates of the coefficients in question are close to zero and their fractions of missing information are high. Multipleimputation inferences for the coefficients of the final regression model are shown in Table 6.10.
6.4 A simulation study We have claimed that it is often sensible to use a normal model to create multiple imputations even when the observed data are some
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Table 6.10. Multipleimputation inferences for logisticregression coefficients, final model
what nonnormal. A growing body of evidence supports this claim. The simulation results of Rubin and Schenker (1986), also reported by Rubin (1987, Chap. 4), demonstrate that for estimating the mean of a univariate population, imputations based on a normal model result in interval estimates with excellent repeatedsampling properties. Even for populations that are skewed or heavytailed, the actual coverage of multipleimputation intervals is very close to the nominal coverage, except when the fraction of missing information is high (in excess of 50%). A recent simulation study in the context of a large national health survey produced encouraging results for a wide variety of linear and nonlinear estimators under plausible nonnormal populations (Schafer et al., 1996). The study was designed to mimic the specific features of a health examination survey conducted by the U.S. National Center for Health Statistics, including a complex sampling plan with unequal selection probabilities and multiple phases of data collection. Results of that simulation, which involved a mixed model for continuous and categorical variables, will be discussed in Chapter 9. Here we present a miniature version of the simulation to convey the essential result: modelbased multiple imputation tends to work well for a wide variety of estimands, and is robust to moderate departures from the data model. 6.4.1 Simulation procedures Data for this simulation, provided by the National Center for Health Statistics (NCHS), were drawn from Phase 1 of the
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Third National Health and Nutrition Examination Survey (NHANES III)
Table 6.11. Variables in the simulation study
Figure 6.7. Histograms of AGE, BMI, HYP and CHL in the population.
(NCHS, 1994). The data were collected by interviews and medical examinations in mobile examination centers. Because many of the sampled persons did not show up for examination, missingness rates for key exam variables exceeded 30%. To keep matters simple, this study is restricted to adult males (age 20+) and four variables. Definitions of the variables are given in Table 6.11. An artificial population of 2000 subjects was created by drawing a simple random sample without replacement of all the adult males in the survey who had complete data for all four variables. Histograms for the variables in this population are shown in Figure 6.7. Because the survey used disproportionate sampling in certain racial, ethnic and age categories, and because we have omitted cases with missing data, these 2000 subjects are not representative of any population of substantive interest; the data and results presented here should not be regarded as estimates for any meaningful segment of the U.S. population. This study is meant only to illustrate the properties of modelbased multiple imputation when applied to a population of real data that do not conform to simplistic modeling assumptions.
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Sampling and response mechanism From the population of 2000 subjects, simple random samples of size n = 100 were drawn without replacement. After a sample was drawn, a random pattern of missingness was imposed on BMI,
Table 6.12 Probabilities for response patterns by AGE, with observed and missing variables denoted by × and ?, respectively
HYP and CHL for each sampled person according to his age. The probabilities for the 23 = 8 possible response patterns by age were estimated from all adult males in the MANES III sample, and are shown in Table 6.12. Because the response probabilities depend only on AGE, which is always observed, this mechanism is ignorable. The mechanism creates missingness rates of approximately 20% for each of the three variables BMI, HYP and CHL over repetitions of the sampling procedure. Imputation After imposing a pattern of missingness, the ’missing’ values were then imputed m = 5 times under a multivariate normal model. AGE was entered into the model as two dummy variables: AGE2 = 1 for AGE = 2 and 0 otherwise; and AGE3 = 1 for AGE = 3 and 0 otherwise. BMI, HYP and CHL were entered without recoding or transformation. The imputations were created by running five independent chains of data augmentation under the standard noninformative prior (5.18). Each chain was started at the ML estimate and allowed to run for 20 cycles. The final value of Ymis from each chain was
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taken as an imputation, and the continuous imputes for HYP were rounded off to the nearest category. 6.4.2 Completedata inferences After imputing five times, five sets of completedata point and variance estimates were calculated for a variety of scalar estimands, and the results were combined in the usual way (Section 4.3.2). Eighteen different estimands were examined, including population means, proportions, quantiles, a correlation coefficient and an odds ratio. Methods of completedata inference for means and proportions are well 2 known. If µ is the mean of a population, and y and S are the sample mean and variance, respectively, from a simple random sample of size n, then the standard point and variance 2 estimates are y and S /n. Similarly, if p is a population proportion and pˆ is a sample proportion, the point and variance estimates are pˆ and pˆ (1 − pˆ ) / n . Completedata inferences for quantiles, correlations and odds ratios are described below. Quantiles The following approximate method for quantiles was described by Woodruff (1952). Suppose that Q is the p th quantile of a distribution function F, and Qˆ is an estimate of Q based on a simple random sample of size n. Then
Q1 ≤ Q ≤ Q2 will be true if and only if
F (Q1 ) ≤ p ≤ F (Q2 ) , because F and F − are strictly increasing. Rather than finding an interval estimate for Q directly, we instead construct an interval estimate for the proportion of the population that lies below Q, and then translate the endpoints of this interval into 1
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quantiles. For example, an approximate 95% interval for p ranges from
p1 = p − 2
p(1 − p) p(1 − p) to p2 = p + 2 . n n
If we set Q 1 and Q2 equal to the p 1ith and p2ith sample quantiles, respectively, then the approximate 95% confidence interval for Q ranges from Q 1 to Q 2. This interval is not necessarily symmetric about Qˆ . It is well known, however that under mild smoothness conditions for F the sample quantiles are asymptotically normally distributed (e.g Serfling, 1980), and for large samples we can take (Q2 − Q 1)/4 as an estimated standard deviation for Qˆ (Francisco and Fuller, 1991). Correlation coefficients Suppose that r is a correlation coefficient from a simple random sample of n units, and ρ is the corresponding population value. The familiar transformation due to Fisher (1921),
z (r ) = tanh −1(r ) = 12 log
1+ r , 1− r
makes z(r) approximately normally distributed about z(ρ) with variance 1/(n−3). This result is derived under an assumption of bivariate normality. An interval estimate for ρ can be calculated by first finding an interval for z(ρ) using the normal approximation, and then applying the inverse transformation 1 z− (⋅) = tanh(⋅) to the endpoints. Because z(r) ≈ r for values of r near zero (they agree to two decimal places for r j (Rubin, 1974; Little and Rubin, 1987). A monotone pattern is shown in Figure 6.8. Monotone patterns often arise in repeatedmeasures or longitudinal datasets, because if a subject drops out of the study in a given time period, then his or her data will typically be missing in all
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subsequent time periods. Sometimes a nonmonotone dataset can be made monotone or nearly so by reordering the variables according to their missingness rates. Let n j denote the number of rows of the data matrix for which Y j is observed. If the pattern is monotone, then np ≤ np−1 ≤ ⋅⋅⋅ ≤ n1 = n. We will assume that the rows of the monotone dataset have been sorted as in Figure 6.8, so that Yj (and hence Y1,..., Yj − 1 as well) is observed for rows 1,...,nj and missing for rows nj + 1,...,n. Factoring the observeddata likelihood When the observed data Yobs are monotone, the observeddata likelihood function can be expressed in a very convenient form. Let φ = (φ1,φ2,...,φp), where φ1 denotes the parameters of the marginal distribution of variable Y1, φ2, the parameters of the conditional distribution of Y2 given Y1, φ3 the parameters of the conditional distribution of Y 3 given Y1 and Y 2, and so on. In other words, φj contains the intercept, slopes and residual variance from the normal linear regression of Yj on Y1,...,Yj − 1. It is easy to show that φ = φ (θ) is a onetoone function of the usual parameters θ = (µ ,Σ ). Moreover, if no prior restrictions are imposed upon θ , then the components
φ1,...,φp are distinct in the sense that the parameter space of φ is the crossproduct of the individual parameter spaces for φ1,...,φp. Expressions for φ1,...,φp in terms of θ = (µ,Σ ) can be found by partitioning µ and Σ and applying the formulas given in Section 5.2.4. When Y obs is monotone, the observeddata likelihood function for φ factors neatly into independent likelihoods for φ1,...,φp. To see this, notice that the joint density of the variables Y1,...,Yp can be factored as
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(
)
(
P Y1,..., Y p  φ = P(Y1  φ1 )P Y2  Y1,φ 2
(
)
)
L P Y p  Y1,..., Y p −1, φ p , which allows us to write the completedata likelihood as
L(φ  Y ) =
n
∏ P( yi1,..., yip  φ ) i=1 n
=
p
∏ ∏ P( yij  yi1,..., yi, j −1, φ j )
(6.3)
i=1 j =1 p
=
n
∏ ∏ P( yij  yi1,..., yi, j −1, φ j ). j =1 i=1
The inner product in (6.3), n
∏ P( yij  yi1,..., yi, j −1, φ j ), i=1
can also be written nj
n
∏ (
P yij  yi1,..., yi, j −1, φ j
i=1
) ∏ P( yij  yi1,..., yi, j −1, φ j ).
(6.4)
i= n j +1
The observeddata likelihood L(φYobs) is by definition the integral of (6.3) over Y mis. But notice that the first product in (6.4) does not involve Ymis, because variable Yj is observed in rows 1,..., nj, whereas the second product integrates to unity because Yj is missing in rows nj + 1,..., n. It follows that
L(φ  Yobs ) =
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p
∏ L(φ j  Yobs ), j =1
(6.5)
where
(
nj
) ∏ P( yij  yi1,..., yi, j −1, φ j ).
L φ j  Yobs =
(6.6)
i=1
Under the multivariate normal model, (6.6) is simply the likelihood for the normal linear regression of Yj on Y1,..., Yj−1, based on the rows 1,..., nj of the data matrix. Thus the factorization (6.5) effectively reduces the problem of inference about φ to a sequence of completedata regressions over subsets of the rows of the data matrix. 6.5.2 Computing alternative parameterizations When the data are monotone, the observeddata likelihood has a convenient form when expressed in terms of φ = (φ1,...,φ p). The parameters of the multivariate normal, however, are usually expressed in terms of θ = (µ,Σ), a vector of means and a covariance matrix. To make use of the convenient form of the likelihood, we will need to switch back and forth between the two parameterizations. 1 A numerical procedure for computing φ = φ (θ) or θ = φ − (φ) can be formulated in terms of the sweep operator (Section 5.2.4). For convenience, we introduce a slight generalization of sweep which gives a compact notation to the process of sweeping a square submatrix of a larger matrix. Let G be a p × p symmetric matrix with elements gij, and let A be a subset of the p columns (and rows) of G . The generalized sweep operator SWPA performs the usual sweep computations on the rows and columns of G in the set A but leaves the rest of G unchanged. Formally, SWPA[k] for some k ∈ A operates on G by replacing it with another p × p matrix H,
H = SWPA [ k ]G,
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where the elements of H are given by
hkk = −1 / gkk , g jk / gkk j ∈ A, j ≠ k h jk = hkj = j ∉ A, g jk g jl − g jk gkl / gkk j ∈ A, l ∈ A, j ≠ k , l ≠ k , h jl = hlj = j ∉ A or l ∉ A. g jl This operation will be referred to as sweeping submatrix A of G on position k. Similarly, the corresponding reverse sweep operator RSWA applies the usual reversesweep computations to the rows and columns of G in set A, while leaving the rest of G unchanged. Formally, RSWA[k] for some k ∈ A operates on G by replacing it with another p × p matrix H,
H = RSWA [ k ]G, where the elements of H are given by
hkk = −1 / gkk , − g jk / gkk j ∈ A, j ≠ k h jk = hkj = j ∉ A, g jk g jl − g jk gkl / gkk j ∈ A, l ∈ A, j ≠ k , l ≠ k , h jl = hlj = j ∉ A or l ∉ A. g jl When the sweep or reverse sweep operators are written without a subscripting set, as in SWP[k] or RSW[k], it will be understood that the operation is being applied to the entire matrix. We are now ready to give a compact notation to the process of computing φ from θ and viceversa. Let
(
φ j = β Tj , γ
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)
T
j
(6.7)
where β j is the j × 1 vector of coefficients (including the intercept) from the linear regression of Yj on Y1, Y2,...,Yj−1 and
γj is the residual variance, so that
((
)
)
Y j  Y1,..., Y j −1, φ j ~ N 1, Y1,..., Y j −1 β j , γ j ,
(6.8)
Let µj denote the jth element of µ and σjk the (j, k) element of Σ, and define
(
θ j = µ j , σ 1 j , σ 2 j ,..., σ jj
)T ,
(6.9)
so that θ = (θ 1,θ2,...,θp). As in Section 5.2.4, let us express θ as a symmetric (p + 1) × (p + 1) matrix,
where the lower portion of the matrix is not shown to avoid redundancy. Finally, let the row and column labels for this matrix run from 0 to p, so that θj appear in column j. To convert θ to
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note that sweeping θ on positions 1,2,..., j−1 produces a new matrix whose jth column is φj. Therefore, if we sweep the full
θ matrix on positions 1, 2,..., p−1, then φ(p) appears in the pth column. If we then reversesweep all but the last row and column on position p−1, then φ p −1 appears in column p−1. Reversesweeping all but the last two rows and columns on position p−2 makes φp−2 appears in column p−2, and so on. This procedure can be expressed very concisely in pseudocode. Let A j = {0,1,..., j} for j = 1, 2,..., p. The following two lines will overwrite a θ matrix, replacing it with φ = φ(θ). for j:=1 to p−1 do θ:=SWP[j]θ for j:=p−1 down to 1 do θ:=RSWAj[j]θ
The transformation from φ back to θ is simply a reversal of these steps. The following two lines will overwrite a φ matrix, replacing it with θ = φ− (φ). 1
for j:=1 to p−1 do φ=SWPAj [j] φ for j:=p−1 down to 1 do φ:=RSW[j] φ
6.5.3 Noniterative inference for monotone data Maximumlikelihood estimation When Yobs has a monotone pattern, the factorization of the likelihood in terms of φ=(φ1,...,φp),
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L(φ  Yobs ) =
p
∏ L(φ j  Yobs ), j =1
enables us to calculate ML estimates without iteration (Little and Rubin, 1987, Chapter 6). Because the parameters φ1,...,φ p are distinct, maximizing L(φYobs) is equivalent to maximizing each factor L(φjYobs) separately for j=1,...,p. The ML estimate of φ is φˆ , where φˆ j is the maximizer of L(φjYobs). The maximization of each factor L(φjYobs) is accomplished by ordinary leastsquares regression of Yj on Y1,..., Y j−1, using rows 1,..., nj of the data matrix. Let zj denote the observed data in column j,
(
z j = y1 j , y2 j ,..., yn j , j
)
T
,
(6.10)
and X j the upperleft n j x (j−1) submatrix augmented by a column of ones,
1 y11 1 y 21 Xj= M M 1 yn j ,1
y12 L y1, j −1 y22 L y2, j −1 . M M yn j , 2 L yn j , j −1
(6.11)
By (6.8), the conditional distribution of zj given Xj and φj is
(
)
z j  X j,φ j ~ N X j, β j,γ jI , so the likelihood for φj is
(
)
L φ j  Yobs ∝ γ
−n j 2 exp − j
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(
)(
T 1 z j − X jβ j z j − X jβ j 2γ j
).
Using wellknown properties of the normal linear regression model, the ML estimate of φj is given by
(
βˆ j = X Tj X j
)
−1
X Tj z j ,
ˆ Tj ∈ ˆ j, γˆ j = n −1 j ∈
(6.12)
(6.13)
ˆ j = z j − X j βˆ j (e.g. Draper and Smith, 1981). Notice where ∈ that γˆ j , the ML estimate of the residual variance, is biased because its denominator is nj rather than nj−j. Calculating (6.12)(6.13) for j = 1, 2,..., p yields φˆ , the ML estimate of φ. Because ML estimates are invariant under transformations of the parameter, the ML estimate for θ can be calculated as θˆ = φ −1 φˆ .
()
Bayesian inference Similarly, when Yobs has a monotone pattern, we can also conduct Bayesian inferences without iteration provided that the prior distribution has a certain form. If we apply a prior density to φ that factors into independent densities,
( )
π (φ ) = π1(φ1 )π 2 (φ2 ) Lπ p φ p ,
(6.14)
then it is obvious that the posterior distribution P(θYobs) will also factor into independent posteriors for φ1,...,φp, a structure that Rubin (1987) calls monotone distinct. Bayesian inferences for φ can then be carried out as a sequence of independent inferences based on the posteriors
(
) (
) ( )
P φ j  Yobs ∝ L φ j  Yobs π j φ j
for j = 1,..., p. For example, we can simulate a value of φ from P(θYobs) by drawing φj from P(θjYobs) independently for j =
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1,...,J. A simulated value of θ from P(θYobs) can then be obtained by applying the backtransformation θ = φ− (φ) to the 1
simulated value of φ. The noninformative prior most commonly used for multivariate normal data,
π (θ ) ∝ Σ −(
p +1 2
),
(6.15)
can be factored as in (6.14). To avoid confusion, let us refer to the density (6.15) as πθ(θ), and the corresponding density for φ induced by (6.15) as πθ(φ). The relationship between πθ and πφ is
(
)
π φ (φ ) = π θ φ −1(φ ) J
−1
,
(6.16)
where θ=φ−1(φ ) is the inverse of the transformation φ=φ(θ ), J is the Jacobian or firstderivative matrix of the transformation
φ=φ(θ ) and J is the absolute value of the determinant of J. By a well known property of determinants, Σ can be written as
Σ11 Σ12 −1 = Σ11 Σ 22 − Σ 21Σ11 Σ12 Σ 21 Σ 22 −1 for square submatrices Σ 11 and Σ 22 . But Σ 22 − Σ 21Σ11 Σ12 is the residual covariance matrix from the regression of the variables corresponding to Σ22 on the variables corresponding
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to Σ 11 (Section 5.2.4). Taking Σ22=σpp, the determinant of Σ becomes
Σ = Σ11 γ p ,
(6.17)
where Σ 11 is Σ without the last row and column. Applying (6.17) recursively to Σ11 leads to p
Σ =
∏γ j.
(6.18)
j =1
To find π φ,(φ ) we also need to evaluate J . In Section 5.4.2, we derived the determinant of the Jacobian that arises when we condition on a subset of the variables Y1,..., Yp. Suppose that we first transform θ to the intermediate parameter (ξp−1,φp), where ξp−1 represents the portions of µ and Σ pertaining to the marginal distribution of Y1,..., Yp—1, andφp pertains to the regression of Yp on Yp−1 (5.28), the determinant of the Jacobian for going from θ to (ξp−1,φp) is Σ11− , where 1
Σ 11 is the covariance matrix for Y 1,...,Yp−1. But Σ11=
γ1,γ2⋅⋅⋅γp−1, so the determinant of the Jacobian of this intermediate transformation is (γ 1,γ2⋅⋅⋅γp−1)− . If we then 1
transform ξp−1 to (ξp−2,φp−1), where ξp−2 contains the portions of µ and Σ pertaining to Y 1,...,Yp−2 and φp−1 pertains to the regression of Y p−1 on Y1,...,Yp−2. the determinant of the Jacobian is γ1γ2⋅⋅⋅γp−2. We can repeat this procedure until we have reached the final parameterization φ=(φ1,...,φp), and the determinant of the Jacobian for φ=φ(θ ) will be the product of the determinants for each of the intermediate transformations. The result is
J = γ1 (
− p −1) −( p − 2) γ2 Lγ −p1−1.
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(6.19)
Substituting (6.19) and (6.18) into (6.16) gives
π φ (φ ) ∝
p
∏
−
γj
(
p +1 2
p+ j
)
(6.20)
j =1
as the prior density for φ=φ(θ) induced by (6.15). Now we show the posterior that results when this prior is combined with the observeddata likelihood from a monotone dataset. Consider the likelihood factor for φj,
(
)
−n j 2 j
L φ j  Yobs ∝ γ
1 T exp − z j − X jβ j z j − X jβ j γ 2 j
(
)(
).
With some algebraic manipulation, it can be shown that
( z j − X j β j )T ( z j − X j β j ) =∈ˆ Tj ∈ˆ j +(β j − βˆ j )
T
(
βˆ j = X T X j
)
−1
(
)
X T X β j − βˆ j ,
ˆ j = X j βˆ j . When ∈ L(φjYobs) is combined with the factor in (6.20) involving φj,
where
( )
X Tj z j
π j φ j ∝γ
− j
(
and
p +1 2 −
p+ j
),
the resulting posterior can be written as
1 P φ j  Yobs ∝ γ −j j 2 exp − β j − βˆ j 2γ j
(
(
)
((
) )
− n j − p + j −1 / 2 −1
×γ j
)
T
X T X β j − βˆ j
(
)
1 T ˆj∈ ˆ j , ∈ exp − 2γ j
which is the product of a multivariate normal and a scaled invertedchisquare density,
(
β j  γ j , Yobs ~ N βˆ j , γ j X T X
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)
−1
,
(6.21)
ˆ Tj ∈ ˆ j χ n−2− p + j −1. γ j  Yobs ~∈ j
(6.22)
6.5.4 Monotone data augmentation Thus far we have discussed methods of inference that are appropriate when the observed data Yobs are monotone. It often happens in practice that a dataset is not precisely monotone, but would become monotone if a relatively small portion of the missing data were filled in. This situation often arises with double sampling, where investigators attempt to measure certain variables for all units in a sample, and then measure additional variables for only a subsample. If there were no missing values except for those missing by design, then the data would be perfectly monotone; in practice, however, there is usually some additional unplanned missingness which makes the overall pattern deviate slightly from monotonicity. Nearmonotonicity also results in many longitudinal or panel studies, in which variables are measured for individuals on multiple occasions. Subjects who drop out of the study at a particular occasion or wave usually do not reappear in subsequent waves, so that if the variables are ordered by wave the overall pattern is nearly monotone. When this situation arises, we can exploit the nearmonotone pattern to devise simulation algorithms that are computationally more efficient than the data augmentation procedures described in Chapter 5. These new procedures, which we call monotone data augmentation, differ from ordinary data augmentation in that they fill in only enough of the missing data at each Istep to complete a monotone pattern. Suppose that we partition the missing data as Ymis = (Ymis*, Ymis**) where Ymis* is some subset of the missing values which, if filled in, would result in (Yobs, Ymis*) having a monotone pattern. Monotone data augmentation proceeds in the following two steps. 1. Istep: Given the current simulated value θ(t) of the parameter, draw a value from the conditional predictive distribution of Ymis*,
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(
)
( t +1) Ymis ~ P Ymis *  Yobs , θ ( t ) .
(6.23)
( ) , draw a new value of θ from 2. Pstep: Conditioning on Ymis * its posterior given the nowcompleted monotone pattern, t +1
(
)
( t +1) θ ( t +1) ~ P θ  Yobs , Ymis * .
(6.24)
In practice, the Pstep will have to be carried out using the parameterization φ =(φ1,...,φp) that corresponds to the monotone pattern of (Yobs, Ymis*). That is, we will have to draw
φ ( t +1) = (φ1( t +1) ,..., φ (pt +1) ) by drawing
φ (j
t +1)
(
( t +1) ~ P φ j  Yobs , Ymis *
)
independently for j = 1, 2,..., p, and then calculate
(
θ ( t +1) = φ −1 φ ( t +1)
)
using the procedures for numerical transformation described earlier in this section. Monotone data augmentation has two computational advantages over ordinary data augmentation. First, it requires fewer random number draws per iteration, i.e. it is typically faster to fill in Ymis* than the full Ymis. Second, it will achieve approximate stationarity in fewer iterations. Liu, Wong and Kong (1994) show that `collapsing’ the data augmentation by drawing only a subset of the unknown quantities at each iteration leads to faster convergence
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Figure 6.9. Possible missingness patterns for a threevariable dataset, with observed and missing variables denoted by 1 and 0, respectively.
and smaller autocorrelations between successive iterates. With ordinary data augmentation, convergence is governed by the amount of information contained in Y mis relative to Yobs (Section 3.5.3). With monotone data augmentation, however, convergence is governed by the amount of information in Ymis* relative to Y obs. When Yobs is not far from monotone, Ymis* is relatively small; the distribution P(θYobs,Ymis*) is then nearly independent of Ymis*, and only a few steps of monotone data augmentation will be needed to achieve approximate stationarity. In the extreme case where Y obs is precisely monotone, Y mis* is empty and the algorithm reaches stationarity in one step. Monotone data augmentation was first proposed by Li (1988) who demonstrated its use in simple bivariate examples. The algorithm presented here, which assumes multivariate normal data and the customary noninformative prior
π (θ ) ∝ Σ −(
p +1 2
),
has also been described by Liu (1993). Choosing the monotone pattern to be completed To identify a Ymis* it helps to group the rows of the data matrix by their patterns of missingness. For example, the possible patterns of missingness for a threevariable dataset are shown in Figure 6.9. The missing values in the unshaded region constitute Ymis* and need to be filled in at every Istep; missing
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data in the shaded region constitute Ymis** and do not need to be filled in. In most cases, of course, there is no unique set of missing data Ymis* that will complete a monotone pattern. By simply reordering the columns Y1, Y2,..., Yp of the data matrix, we can identify alternative sets of missing values that are candidates for Y mis*. For computational efficiency, it is advantageous to choose Ymis* to be ’small’ in two senses. First, the actual number of missing values contained in Ymis*. should be small, to reduce the number of random variates that need to be drawn at each Istep. Second, Ymis* should contain as little information as possible about the unknown parameters, to reduce the number of steps required to achieve approximate stationarity. These two objectives may sometimes conflict. In a normal dataset, for example, there may be a tradeoff between filling in a large number of relatively noninfluential observations and filling in a smaller number with high leverage. Finding a set Ymis* to maximize the efficiency of the algorithm is a difficult problem, as it involves questions about the convergence of Markov chain Monte Carlo algorithms that are not easy to answer at present. Moreover, finding such an optimal set may itself require substantial computation, offsetting the potential gains of a more efficient algorithm. To choose Ymis*, we suggest the naive approach of simply ordering the columns of Y according to their fractions of missing observations. That is, choose Y1 to be the variable with the fewest missing values, Y2 the variable with the second fewest, and so on. This approach is attractive because it is computationally trivial. Moreover, it has the feature that if Yobs is already monotone, it will find the monotone pattern and identify Ymis* to be empty. 6.5.5 Implementation of the algorithm In discussing how to implement monotone data augmentation for the multivariate normal model, we will need to build on the bookkeeping notation of Chapter 5. Suppose that the rows of the data matrix have been grouped together according to their patterns of missingness as shown in Figure 6.10. Index the missingness patterns by s = 1, 2,..., S. Let sj denote the last
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pattern for which variable Yj may need to be filled in to complete the overall monotone pattern, so that
S = s1 ≥ s2 ≥ L ≥ s p . Following Section 5.3.1, let
1 if Y j is observed in pattern s, rsj = 0 if Y j is missing in pattern s.
Figure 6.10. Arrangement of missingness patterns for monotone data augmentation, with 0 denoting a variable that is missing and × denoting a variable that is either observed or missing.
Let O(s) and M(s) denote the column labels corresponding to variables that are observed and missing, respectively, in pattern s, O(s) = {j : rsj = 1} M(s) = {j : rsj = o} Also, let M*(s) denote the subset of M(s) that must be filled in to complete the monotone pattern, and let M **(s) be the remainder of M(s), M*(s) = {j : rsj = 0 and sj ≥ s}
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M**(s) = {j : rsj = 0 and sj < s} For any s, M*(s) lists the columns with missing values in the unshaded region of Figure 6.10, and M**(s) lists the columns in the shaded region. Finally, let I(s) denote the subset of {1, 2,..., n} corresponding to the rows of the data matrix Y in pattern s. The I and Psteps The Istep for monotone data augmentation is nearly identical to the Istep for ordinary data augmentation; the only difference is that rather than imputing all the missing values Ymis, we need only impute the portion Y mis* to complete the monotone pattern. Consequently, the pseudocode for the Istep shown in Figure 5.6 can be used for monotone data augmentation with only one modification: replace every occurrence of M(s) with the potentially smaller set M*(s). The four lines of code in Figure 5.6 preceded by the character ’C’ are not needed and may be removed. The Pstep, however, is computationally more complicated than the Pstep for ordinary data augmentation, because the posterior distributions of φ1,φ2,...,φp depend on different sets of sufficient statistics. The posterior of φj, given by (6.21)
(
)
−1 ˆ Tj ∈ ˆ j , which are (6.22), depends on βˆ j X Tj X j , and ∈
obtained from the regression of Yj on Y1,...,Yj−1 over the rows of the data matrix in missingness patterns s = 1,..., s j. To perform this regression, we need to accumulate sums of squares and crossproducts for variables Y1,...,Yj and patterns s = 1,..., sj. As in Section 5.3.3, define T(s) to be the (p + 1) × (p + 1) matrix of completedata sufficient statistics from missingness pattern s,
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ns T (s ) =
Σyi1 Σyi21
Σyi2
Σyip Σyi1 yi2 L Σyi1 yip Σy22 L Σyi2 yip , O M 2 Σyip L
where all sums are taken over i∈I(s), and ns =
∑ i∈I (s) is the
sample size in pattern s. For simplicity, we will number the rows and columns of T(s) from 0 to p rather than from 1 to p + 1. Let Tmis (s) and Tobs (s) be matrices of the same size as T(s) with elements defined as follows: the (j, k)th element of Tmis (s) is equal to the (j, k)th element of T(s) if j∈M(s) or k∈M(s), and zero otherwise; and T obs(s) = T(s) − T mis(s). Notice that Tmis(s) contains the sufficient statistics that depend on Ymis, whereas Tobs(s) contains the sufficient statistics that are functions only of Yobs. Finally, let Tmis*(s) be a matrix identical to T mis(s), but with the following exception: the rows and columns corresponding to variables that are not needed to complete the monotone pattern are set to zero. That is, set the (j, k)th element of Tmis*(s) equal to zero if j∈M**(s) or k∈M**(s), otherwise set it equal to the (j, k)th element of Tmis(s). Thus Tmis* contains the sufficient statistics that depend on Ymis* but not on Ymis**. Suppose that the unknown values in Ymis* have been filled in by an Istep so that Tmis*(s) can be calculated. If we let
Tj =
sj
sj
s =1
s =1
∑ Tobs (s ) + ∑ Tmis* (s ),
then
X Tj X j T j = z Tj X j 0
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X Tj z j z Tj
zj 0
0 0, 0
where zj and Xj, given by (6.10) and (6.11), are the response vector and covariate matrix needed for the regression of Yj on Y1,..., Yj−1. If this matrix is swept on positions 0, 1,..., j−1, the result is
(
)
( X Tj X j ) X Tj z j −1 ) z Tj z j − z Tj X j ( X Tj X j ) X Tj z j
−1 T − Xj Xj −1 T T z j X j X j X j 0
(
−1
0
0 0. 0
But notice that
( X Tj X j )
−1
X Tj z j = βˆ j
is the vector of estimated coefficients from ordinary leastsquares regression of zj on Xj. Moreover, it is straightforward to show that
(
z Tj z j − z Tj X j X Tj X j
)
−1
ˆ Tj ∈ ˆ j, X Tj z j =∈
ˆ j = z j − X j βˆ j is the vector of estimated residuals. where ∈ The quantities needed to describe the posterior distribution of φj, given the observed data Yobs and imputed data in Ymis* can thus be obtained by sweeping the matrix Tj. Note that all of the elements of Tj in rows and columns j + 1,..., p are zero before and after sweeping. Superfluous arithmetic can be avoided by applying the generalized sweep operator (Section 6.5.2) to sweep only the nonzero portions of T. The regression computations become
(
T − X j X j βˆ Tj SWPA j [0,..., j − 1]T j = 0
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)
−1
βˆ j ˆ Tj ∈ ˆj ∈ 0
0 0, 0
where Aj = {0, 1,..., j}. An implementation of the Pstep is shown in Figure 6.11. The components of φ are simulated in the reverse order
φ1,φp−1,...,φ1 and placed in a (p + 1) × (p + 1) matrix as shown in Section 6.5.2. This implementation requires two matrix workspaces
Figure 6.11. Pstep for monotone data augmentation.
of the same size as φ : T, in which the sufficient statistics Tj are accumulated and swept; and C, which holds the Cholesky factors required for simulating the vectors of regression coefficients βj. In addition, a vector workspace v = (v0, v1,..., vp−1) is needed for temporary storage of normal random variates. The quantity Nj, which appears in the degrees of freedom of the chisquare random variate, is sj
Nj =
∑ ns , s =1
the total number of rows of the data matrix Y for which variable Yj is either observed or imputed. The algorithm of Figure 6.11 operates as follows. After the elements of T are initialized to zero, the sufficient statistics for
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Y1,..., Yp are accumulated in T over missingness patterns 1,..., sp. The matrix T is swept on positions 0,..., p − 1, producing statistics from the regression of Yp on Y1,..., Yp−1. A random value of φ p = (γp,βp) is then drawn from its posterior, distribution. If additional rows of Y will enter into the next regression, i.e. if sp−1 > sp then T is reverseswept on positions 0,..., p−1 to prepare for the accumulation of sufficient statistics over these additional rows. Otherwise, T is reverseswept only on position p−1, yielding the results from the regression of Yp−1 on Y1,..., Yp−2. Continuing in this fashion, the algorithm draws
φp−1,φp−2,...,φ1. Upon completion, the resulting value of φ should be transformed to the θscale (Section 6.5.2) to prepare for the next Istep. The accumulation of sufficient statistics in line 4 of this algorithm may be rewritten as sj
T := T +
∑
Tobs ( s ) +
s = s j +1 +1
sj
∑
Tmis * ( s ). s = s j +1 +1
(6.25)
The first sum on the righthand side of (6.25) depends only on the observed data Yobs and does not need to be recalculated at each Pstep. Calculating sj
Bj =
∑
Tobs ( s ) s = s j +1 +1
once at the outset of the program and storing it for future iterations can substantially reduce the amount of computation required at each Pstep. Notice that we do not need to calculate and store Bj for every j = 1, 2,..., p, but only for those values of j for which sj+1 < sj. The second sum on the righthand side of (6.25) depends on Ymis* , the missing values imputed at the Istep, so these terms will need to be recalculated at each Pstep.
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6.5.6 Uses and extensions Like ordinary data augmentation, monotone data augmentation enables us to (a) simulate values of θ from the observeddata posterior P(θYobs), and (b) create proper multiple imputations of Ymis. The output stream is a sequence
(Y ( ) ,θ ( ) ), (Y ( ) ,θ ( ) ),..., (Y ( ) ,θ ( ) ),... 1 mis *
2 mis *
1
2
t mis *
t
with P(Ymis*, θYobs) as its stationary distribution. After a sufficient burnin period, successive values of θ,
θ ( t ) , θ ( t +1) , θ ( t + 2) ,..., constitute a dependent sample from P( θYobs) and may be summarized using any of the methods described in Chapter 4. Iterates of Y mis* that are sufficiently far apart in the output stream, say ( t + k ) ( t + 2k ) ( t) Ymis * , Ymis * , Ymis * ,...
for some large value of k, can be taken as proper multiple imputations of Ymis*. In many applications, we will want proper multiple imputations of all the missing data in Ymis, not just the missing data Ymis*. needed to complete a monotone pattern. To obtain m proper multiple imputations of Ymis, we should first generate m values of θ that are approximately independent, say
q (t) ,q (t+k) ,...,q (t+mk) and then draw a value of Ymis given each one,
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( Y ( ) ~ P(Y
)
() Ymis ~ P Ymis  Yobs , θ ( t ) , 1
2 mis
mis
M ( m) Ymis
)
 Yobs , θ ( t + k ) ,
(
)
~ P Ymis  Yobs , θ ( t + mk ) ,
using the Istep for ordinary data augmentation described in Chapter 5. Of course, to obtain independent values of θ we do not necessarily need to subsample every kth value from a single chain of monotone data augmentation; we can also run m independent chains of length k from a common starting value, or better still, from m independent starting values drawn from an overdispersed starting distribution (Section 4.4.2). Alternative priors The monotone data augmentation algorithm described above uses the customary noninformative prior distribution
π (θ ) ∝ Σ −(
p +1 2
).
It is occasionally helpful to use other priors. For example, in sparsedata situations where some aspects of the covariance structure are poorly estimated, we may want to apply the ridge prior described in Section 5.2.3. A strategy for monotone data augmentation under an arbitrary invertedWishart prior distribution for Σ is outlined by Liu (1993). Liu’s algorithm uses a clever factorization of the posterior distribution under monotone data, derived using an extension of the Bartlett decomposition (Section 5.4.2). 6.5.7 Example Section 6.4 presented a small simulation study designed to mimic the types of data and missingness found in a national health examination survey. The response mechanism shown in Table 6.12, which was estimated from an actual survey, tends
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to produce samples that are nearly monotone. The most common missingness pattern, which occurs for about 70% of sampled individuals, has all four survey variables (AGE, BMI, HYP, CHL) observed. The next most common pattern, which occurs about 15% of the time, has AGE observed and the other three variables missing. If AGE is placed in the first column of the data matrix, then at least 85% of the sampled individuals will tend to conform to a monotone pattern. This is precisely the type of situation in which monotone data augmentation should outperform ordinary data augmentation. To illustrate, a simple random sample of n = 25 individuals was drawn from the study population, and a random pattern of missingness was imposed on the sample according to the estimated mechanism. The simulated data and missingness patterns are shown in Table 6.14. Overall there are 27 missing values, but only three of them (one value of HYP and two of BMI) are needed to complete a monotone pattern. As in the simulation study, we replaced AGE by two dummy indicators (AGE2 = 1 for AGE = 2 and 0 otherwise; AGE3 = 1 for AGE = 3 and 0 otherwise) and modeled the resulting fivevariable dataset as multivariate normal. An exploratory run of the EM algorithm revealed that the worst fraction of missing information, as estimated from the elementwise rates of convergence, is about 66%. Runs of data augmentation and monotone data augmentation under the customary noninformative prior verified that monotone data augmentation does indeed converge faster. Sample autocorrelations for two functions of the parameter θ , calculated over 5000 iterations of each algorithm, are displayed in Figure 6.12. Figure 6.12 (a) shows ACFs for the correlation between BMI and CHL, and Figure 6.12 (b) shows ACFs for the worst linear function of θ, which was estimated from the trajectory of EM. With respect to these two parameters, data augmentation appears to be approximately stationary by lag k = 4, whereas monotone data augmentation seems nearly stationary at lag k = 1. For a dataset of this size, iterations of either algorithm can be executed so quickly on modern computers that the
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advantage of monotone data augmentation is of little practical importance.
Table 6.14. Sample data from a health examination survey with simulated Pattern of missingness (1=observed, 0=missing)
In a large database, however, a fourfold reduction in the time required to produce a given number of multiple imputations can be a substantial improvement. Moreover, the gains tend to become even more dramatic as the rates of missing information increase. In studies that employ double sampling or matrix sampling, it is not uncommon for the rates of missing information for some parameters to be 90% or more. These high rates of missingness, due primarily to data that are missing by design, can make the convergence of ordinary data augmentation painfully slow. It is easy to envision scenarios where exploiting a nearmonotone pattern that
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Figure 6.12. Sample ACFs of series from ordinary data augmentation (dashed line) and monotone data augmentation (dotted line) for (a) the correlation between BMI and CHL, and (b) the worst linear function of the parameter.
arises by design can reduce the computations by one or more orders of magnitude.
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CHAPTER 7
Methods For Categorical Data
7.1 Introduction The past three decades have seen enormous growth in the theory and application of models for categorical data. Categoricaldata techniques such as logistic regression and loglinear modeling are now commonplace in the social and biomedical sciences and nearly every other major area of statistical application. For the most part, however, principled methods for handling missing values in categorical data analysis have not been readily available. We have already demonstrated that, under certain circumstances, categorical variables can be handled quite reasonably by applying the multivariate normal distribution (Sections 6.3 and 6.4). In other situations, however, it is desirable to use a model specifically designed for categorical data. This chapter develops techniques for parameter simulation and multiple imputation for incomplete categorical data under the saturated multinomial model. The saturated multinomial is more general than the multivariate normal in the sense that it allows for threeway and higher associations among the variables; the multivariate normal captures simple (twoway) associations only. When maintaining higherorder associations among continuous variables is a priority, it may even be worthwhile to categorize them and apply the methods of this chapter rather than normalbased methods, even though the categorization may result in a slight loss of information. The generality of the saturated multinomial model can also be a drawback, however, because in many applications (particularly as the number of variables grows) some of the
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higherorder associations may be poorly estimated. In these situations, it often helps to simplify the model by selectively removing some of these complex associations. Elimination of higherorder associations will be discussed within the framework of loglinear modeling, which will be covered in Chapter 8. Section 7.2 lays the groundwork for our categoricaldata methods by reviewing fundamental properties of two multivariate distributions, the multinomial and the Dirichlet. Basic EM and data augmentation algorithms for the saturated multinomial model are developed in Section 7.3. Section 7.4 introduces a class of algorithms that tends to be more efficient when the missing values fall in a pattern that is nearly monotone.
7.2 The multinomial model and Dirichlet prior 7.2.1 The multinomial distribution Let Y 1, Y2,..., Y p denote a set of categorical variables. For notational convenience, we will suppose that the levels of each variable are coded as positive integers, so that
Y j takes possible values 1, 2, ..., d j for j = 1, 2,..., p. Throughout this chapter, we will regard the levels 1, 2,..., dj as nominal or unordered categories; we do not consider models that explicitly account for ordering, e.g. the models for ordinal variables discussed by Agresti (1984) and Clogg and Shihadeh (1994). Incomplete ordinal data can sometimes be handled, at least approximately, by pretending that they are normally distributed and applying the methods of Chapters 56. Alternatively, one can disregard the order of the levels and apply the methods described here. Disregarding the order results in some loss of information and may lead to models that are more complex (i.e. having more parameters) than necessary to describe the essential relationships among the variables. For developing models that are parsimonious and scientifically meaningful, it is usually desirable to retain
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the ordering of the levels, if possible. On the other hand, if the immediate goal is to create plausible multiple imputations of missing data for future analyses, then disregarding the order and applying the methods of this chapter may be a perfectly reasonable approach. If values of Y 1, Y2,..., Yp are recorded for a sample of n units, then the complete data can be expressed as an n × p data matrix Y. If the sample units are independent and identically distributed (iid), then without loss of information we can reduce Y to a contingency table with D cells, where p D= j =1 d j is the number of distinct combinations of the
∏
levels of Y 1, Y2,..., Y p. In practice, logical constraints among the variables may render some of these combinations impossible. For example, if Y 1 represents age (1=09 years, 2=1019 years,...) and Y 2 represents marital status (1=never married, 2=currently married,...) then under most circumstances (Yi = 1, Y 2 = 2) should be regarded as an impossible event. Cells of the contingency table that are necessarily empty due to logical constraints are called structural zeroes (e.g. Agresti, 1990). Structural zeroes present only minor complications, most of which are notational. For now, we will proceed as if there are no structural zeroes. Let us index the cells of the contingency table by the single subscript d = 1, 2,..., D. Let xd be the number of sample units that fall into cell d, and let
x = ( x1, x2 ,..., xD ) denote the entire set of cell frequencies or counts. If the D sample units are iid and the sample size n = d =1 xd is
∑
regarded as fixed, then x has a multinomial distribution. We will write
x  θ ~ M ( n, θ ) to indicate that x is multinomial with index n and parameter
θ = (θ1, θ2 ,..., θ D ),
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where θd is the probability that a unit falls into cell d. The probability distribution for x is given by
P( x  θ ) = for
n! x x x θ 1 θ 2 Lθ DD x1! x2!L xD ! 1 2
(7.1)
∑ dD=1 xd = n and 0 otherwise. Because the total sample
size n is fixed, one of the elements of x is redundant; we can D −1 replace X D by d =1 xd and regard (7.1) as the probability distribution for (x1,..., xD − 1)
∑
Notice that the cell probabilities must satisfy
∑ dD=1θ d = 1,
so the multinomial model has only D − 1 free parameters; θD can be replaced by 1 −
∑ dD=−11θ d . Alternatively, we can regard
the full vector θ = (θ 1, ...,θD) as the unknown parameter, with the understanding that it must lie within the simplex
{
Θ = θ : θ d ≥ 0 for all d and
∑ dD=1θ d = 1},
(7.2)
a (D − 1)dimensional subset of Ddimensional space. When D = 3, for example, Θ is the region encompassed by the triangle with vertices (1, 0, 0), (0, 1, 0) and (0, 0, 1). The simplex Θ is the natural parameter space for the multinomial, i.e. the set of all possible values of θ for which (7.1) is a valid probability model. Throughout this chapter, we allow θ to lie anywhere in Θ . Such a model is said to be saturated, because it includes the maximum number of free parameters (D − 1). The saturated model is very general; it allows for any kind of relationships to exist among the variables Y1, Y2,..., Yp. In many applications, however, such generality is undesirable because the information contained in the observed data may not be sufficient to estimate so many parameters adequately. Moreover, when the goal is to develop a model that is scientifically meaningful, models that are more
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parsimonious (i.e. having fewer parameters) than the saturated model may be easier to interpret. In Chapter 8, we will show how to reduce the number of free parameters by imposing loglinear constraints on the elements of θ. When the multinomial vector x has only D = 2 cells, x2 and θ2 can be replaced by n − x 1 , and 1 − θ1, respectively, and (7.1) reduces to a binomial distribution,
P( x  n) =
n! n− x x θ 1 (1 − θ1 ) 1 . x1! ( n − x1 )! 1
In this special case, we will sometimes use the notation
x1  θ1 ~ B( n, θ1 ) as an alternative to
( x1, n − x1 )  θ ~ M ( n, (θ1, n − θ1 )). The first two moments of the multinomial distribution are given by
E( xd  θ ) = nθ d , = nθ d (1 − θ d ), V ( xd  θ ) Cov( xd , xd ′  θ ) = − nθ dθ d ′ , d ′ ≠ d Further properties of the multinomial distribution can be found in texts on discrete data (e.g. Bishop, Fienberg and Holland, 1975; Agresti, 1990). Maximumlikelihood estimation The likelihood function for the multinomial parameter is
L(θ  Y ) ∝
D
∏ θ dx d =1
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d
I Θ (θ ),
(7.3)
where IΘ(θ ) is an indicator function equal to 1 if θ ∈Θ and 0 otherwise. Notice that we have written L(θY) rather than L(θ x). We are allowed to do this because all relevant information about θ in the data matrix Y is captured in the contingency table x; that is, we can reconstruct Y from x except for the order of the sample units, which under the iid assumption is statistically irrelevant. The loglikelihood is
l(θ  Y ) =
D
∑ xd log θ d ,
(7.4)
d =1
defined over the simplex Θ. The multinomial is a regular exponential family distribution whose sufficient statistics are simply the cell counts x = (x1,..., XD). Therefore, completedata ML estimates can be obtained simply by equating each observed cell count xd to its expectation E(xdθ ) = nθd, leading to the wellknown result that the ML estimates for the cell probabilities are the observed proportions
x θˆd = d , d = 1,..., D. n
(7.5)
7.2.2 Collapsing and partitioning the multinomial The multinomial distribution has two convenient properties that enable us to factor the probability distribution P(xθ) and the likelihood L(θY). Suppose that we collapse two cells of the contingency table, say x1 and x2, adding the frequencies together to produce a new table x* = (z,x3, ...,xD) where z = x1 + x2. Then (a) the distribution of x* is multinomial,
x*  θ ~ M ( n, θ *),
(7.6)
where θ* = (ξ,θ3, . . . , θD) and ξ = θ1+θ2; and (b) the conditional distribution of (x1, x2) given z is also multinomial,
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( x1, x2 )  z, θ ~ M ( z, (θ1 / ξ, θ2 / ξ )).
(7.7)
Property (a) is derived by summing the multinomial probabilities for all xvectors consistent with x1 + x2 = z,
P( x*  θ ) =
z
∑ P( x1 = j, x2 = z − j, x3 ,..., xD ) j =0 z
=
∑ j!( z − j )! x3!L xD! θ1jθ2z − jθ3x n!
j =0
=
n! x x θ3 3 Lθ DD z! x3!L xD !
z
3
x
Lθ DD
∑ j!( z − j )!θ1jθ2z − j , z!
j =0
and noting that z
∑ j!( z − j )! θ1jθ2z − j = (θ1 + θ2 ) z z!
j =0
by the Binomial Theorem. Property (b) can be deduced as follows. Notice that if we repeatedly apply Property (a) to collapse the table down to x1 + x2 = z and x3 +⋅⋅⋅+ xD = n − z, we obtain
( z, n − z )  θ ~ M ( n, (ξ, 1 − ξ )). Moreover, if we collapse x3 +⋅⋅⋅+ xD = n − z but leave x1 and x2 intact, then
( x1, x2 , n − z )  θ ~ M ( n, (θ1, θ2 , 1 − ξ )). The conditional distribution of (x1, x2) given z is by definition
P( x1, x2  z, θ ) =
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P( x1, x2 , n − z  θ ) P( z , n − z  θ )
(7.8)
for x 1 + x2 = z and 0 otherwise. Substituting expressions for the numerator and denominator, the righthand side of (7.8) becomes
n! n! n− z n− z x x θ1 1 θ2 2 (1 − ξ ) ξ z (1 − ξ ) x1! x2! ( n − z )! z! ( n − z )!
−1
which reduces to x
x
z! θ1 1 θ2 2 P( x1, x2  z, θ ) = , x1! x2! ξ ξ the desired result. We have stated these results in terms of collapsing just two cells (x 1 + x2 = z), but they extend to arbitrary types of collapsing. Suppose that we partition the cell numbers {1, 2,..., D} into subsets A1, A2,...,A K that are mutually exclusive and collectively exhaustive. Denote the part of x corresponding to Ak by
x( k ) = { xd : d ∈ Ak }. The collection {x(1), x(2),..., x(K)} of these parts will be called the partitioned table, and x(k) will be called the kth part of x. Denote the total frequency for the kth part by
zk =
∑ xd .
d ∈Ak
The collection z = (z 1, z2,...,zK) of these total frequencies will be called the collapsed table. Denote the probability that a sample unit falls into the kth part by
ξk =
∑θd ,
(7.9)
d ∈Ak
and the conditional probability that a sample unit falls into cell d given that it falls into the kth part by
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φ kd = θ d / ξ k for all d ∈ Ak .
(7.10)
Denote the collection of all such conditional probabilities for the kth part by
φ k = {φ kd : d ∈ Ak }. Notice that φk is simply the kth part of θ , rescaled so that its elements sum to one. Under these conditions, it can be shown that (a) the marginal distribution of the collapsed table is multinomial,
z  θ ~ M ( n, ξ ),
(7.11)
where ξ = (ξ 1,ξ2,...,ξK); and (b) the conditional distribution of the partitioned table given the collapsed table is a set of independent multinomials,
x(1)  z, θ ~ M ( z1, φ1 ), x(2)  z, θ ~ M ( z2 , φ2 ), M
(7.12)
x( K )  z, θ ~ M ( z K , φ K ). A set of independent multinomial distributions over a partitioned contingency table is often called a product multinomial. For any collapsing scheme, we can thus factor the multinomial distribution into a multinomial for the frequencies in the collapsed table, whose parameters are obtained by summing or collapsing θ in the same manner that x was collapsed, and a product multinomial for the conditional distribution of the partitioned table given the collapsed table, whose parameters are obtained by partitioning θ and rescaling each part to sum to one.
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Factoring the likelihood It is easy to see that the parameters for the collapsed table and the partitioned table, which we denote collectively by
ψ = (ξ, φ1,..., φ K ), are a onetoone function of θ = (θ1,...,θd); the forward transformation ψ = ψ(θ ) is defined by (7.9)(7.10), and the back transformation θ=ψ − (ψ) is 1
θ d = ξ k φ kd for all d ∈ Ak ,
(7.13)
k = 1, 2,..., K. Moreover, the parameters for the collapsed table and each part of the partitioned table are mutually distinct; any values of ξ,φ1,...,φK in their respective simplexes will produce a value of θ in its simplex Θ. It follows that the likelihood function for ψ can be factored into a sequence of independent multinomial likelihoods,
(
) (
)
L(ψ  x) = L(ξ  z )L φ1  x(1) L L φ K  x( K ) . Likelihoodbased inferences about each part of ψ can be carried out independently, and the results can then be combined to produce a valid overall inference. For example, ML estimates for each part ξ, φ 1 ,...,φ K can be calculated independently; they are
z ξˆk = k n
x and φˆkd = d for all d∈Ak. for all d ∈Ak. zk
Applying the back transformation θ = ψ (ψ ) to these values gives θˆd = xd/n, the ML estimates for θ. Bayesian inferences for each part can also proceed independently, provided that the 1
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prior distribution applied to ψ factors into independent priors for ξ,φ1, . . . , φK. Nonmultinomial sampling This factorization of the multinomial likelihood has important implications for statistical inference. In many datasets, the distribution of one or more categorical variables is not random but fixed by design. Common examples of this include (a) treatment indicators in randomized experiments and (b) variables used to define strata in sample surveys. When the distribution of one or more variables is fixed by design, the cell frequencies x = (x 1, x2,..., XD) are not multinomial; rather, they follow a productmultinomial distribution. If we erroneously apply a multinomial model, however, we can still obtain valid likelihoodbased or Bayesian inferences about the parameters of the nonfixed portion of the model. This result holds for incomplete data, provided that the missing values are confined to variables that are not fixed (Section 2.6.2). In addition, the multinomial likelihood may lead to valid conditional inferences in situations where the total sample size n is random (e.g. Poisson sampling) (Bishop, Fienberg and Holland, 1975; Agresti, 1990). Although we will speak almost exclusively of the multinomial model throughout this chapter and the next, the reader should be aware that the methods presented here can be reasonably applied in many nonmultinomial situations. 7.2.3 The Dirichlet distribution The simplest way to conduct Bayesian inference with a multinomial model is to choose a parametric family of prior distributions whose density has the same functional form as the likelihood (7.3). Suppose that θ = (θ1,...,θD) is a vector of random variables with the property that θd ≥ 0 for d = 1,...,D and
∑ dD=1θ d = 1.
Then θ is said to have a Dirichlet
distribution with parameter α = (α1, . . . , αD) if its density is
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P(θ  α ) =
Γ (α 0 ) α −1 α −1 α −1 θ 1 θ2 2 Lθ D D Γ (α1 )Γ (α 2 ) L Γ (α D ) 1
over the simplex Θ, where α0 =
(7.14)
∑ dD=1α d and Γ (⋅) denotes
the gamma function. As a shorthand for (7.14), we will write
θ  α ~ D(α ). The righthand side of (7.14) is a valid probability density provided that αd > for 1, . . . , D. When the Dirichlet is used as a prior distribution for the parameters of the multinomial, we will typically omit the normalizing constant and write the prior density as α −1 α −1
π (θ ) ∝ θ1 1 θ2 2
α −1
Lθ D D
,
(7.15)
where α1,...,αD are understood to be userspecified hyperparameters. Although this appears to be a joint density for D random variables, we must remember that one of the elements of θ is redundant. In taking expectations, for example, we would replace θ D by 1 −
∑ dD=−11θ d and integrate
with respect to θ1,...,θD−1 In the special case of D = 2, θ2 =
1−θ1 and the Dirichlet reduces to a beta distribution for θ1. In this special case we may write
θ1  α ~ Beta(α1, α 2 ) as an alternative to
(θ1, θ2 )  α ~ D(α ).
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Figure 7.1. Dirichlet densities for (a) α = (5,3,4) and (b) α = (3,1,2), plotted as functions of θ1 and θ2.
Properties of the Dirichlet distribution Here we state without proof some basic properties of the Dirichlet distribution. For a more detailed treatment, see Wilks (1962). The first two moments are given by
E(θ d ) = V (θ d ) =
αd , α0
α d (α 0 − α d ) α 02 (α 0 + 1)
Cov(α d , α d ′ ) = −
α dα d ′
α 02 (α 0 + 1)
, , d ′ ≠ d.
If the means αd/α0 are held constant but α 0 is allowed to increase, then the variances and covariances are of order O α 0−1 . For this reason, α0 may be regarded as a precision
( )
parameter; as it increases, the distribution becomes more tightly concentrated about the mean. The mode of the Dirichlet can be found by noting that its density is equivalent to the likelihood function from a multinomial contingency table x = (x1,...,XD) with xd = α d−1, d = 1,...,D. This function is maximized at θd = xd /
∑ dD′=1 xd ′
provided that every xd is nonnegative. Therefore, the mode of the Dirichlet density occurs at
θd =
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αd − 1 d = 1,..., D, α0 − D
(7.16)
provided that every αd ≥ 1. Two examples of the Dirichlet density for D = 3 are shown in Figure 7.1. Because one of the elements of θ = (θ1,θ2,θ3) is redundant, the densities are plotted as functions of θ 1 and θ2 over the triangular region θ1 ≥ 0, θ 2 ≥ 0, θ 1 + θ2 ≤ 1. Figure 7.1 (a) shows the density for α = (5,3,4), and Figure 7.1 (b) shows the density for α = (3,1,2). Notice that in (a) the mode lies in the interior of the parameter space Θ, whereas in (b) the mode lies on the boundary. It is true in general that if every αd > 1, then the density has a unique mode in the interior of Θ. If every αd = 1, then the Dirichlet density is uniform over Θ. If one or more of the parameters ad is equal to one but none are less than one, then the density is bounded and the mode occurs on the boundary. Finally, if αd < 1 for any d then the density function becomes infinite on the boundary. These properties suggest that if θ ~ D(α) represents the current state of knowledge about θ, and if one or more elements of a are less than or equal to one, then the mode may not be a sensible point estimate for θ; a better estimate would be the mean. Relationship to the gamma distribution An important relationship exists between the Dirichlet distribution and the gamma distribution. A random variable v is said to have a standard gamma distribution with parameter a > 0 if its density is
P ( v  a) =
1 a −1 − v v e Γ ( a)
for v > 0, and we write
v  a ~ G( a). The gamma distribution is usually presented as a twoparameter family, with one parameter determining the shape
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and the other determining the scale. The standard gamma distribution is obtained by setting the usual scale parameter to one. The mean and variance of the standard gamma are both equal to a. The standard gamma also has the following reproductive property: if v 1 ~ G (a1) and v 2 ~ G(a2) are independent, then v1+v2 ~ G(a1+a2). The Dirichlet distribution can be obtained from the standard gamma as follows. Suppose that v1, v2,..., vD are independent standard gamma variates with parameters α1,α2, . . . , αD, respectively. If we take
θd =
∑
vd D d ′=1 vd ′
, d = 1, 2,..., D,
then θ = (θ 1,θ2, ...,θ D ) will have a Dirichlet distribution with parameter α = (α 1,α2, ...,αD ). This property enables us to simulate a Dirichlet random vector using a standard gamma variate generator. Methods for efficient generation of gamma random variates are reviewed by Kennedy and Gentle (1980). Limitations of the Dirichlet prior From a purely conceptual standpoint, the Dirichlet distribution is not the most attractive prior for crossclassified contingency tables. One of its drawbacks is that it treats the cells of the table in an unordered fashion, ignoring its crossclassified structure. We have adopted the Dirichlet prior mainly for computational convenience, because with complete data it leads to posterior distributions that are easily summarized. If the parameters of the data model are not well estimated by the data, and it becomes apparent that the choice of prior has a substantial impact on the results, then one should be wary of drawing firm conclusions from an analysis under a Dirichlet prior or, for that matter, any other type of prior. 7.2.4 Bayesian inference It is easy to see what happens when a Dirichlet prior is applied to the parameters of the multinomial. Suppose that a
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contingency table x = (x 1,...,XD) has a multinomial distribution with parameter θ = (θ1,...,θD), and the prior distribution for θ is Dirichlet with hyperparameter α = (α1, ...,αD),
x  θ ~ M ( n, θ ),
(7.17)
θ ~ D(α ).
(7.18)
Multiplying the Dirichlet density (7.15) by the multinomial likelihood (7.3) produces α + x1 −1 α 2 + x 2 −1 α + x −1 θ2 Lθ D D D ,
P(θ  Y ) ∝ θ1 1
(7.19)
which is a Dirichlet density with parameters
α ′ = (α1′ , α 2′ ,..., α D ′ )
= (α1 + x1, α 2 + x2 ,..., α D + xD ) = α + x.
(7.20)
The posterior distribution of θ under (7.17)(7.18) is thus
θ  Y ~ D(α ′). The posterior mean is
α′ α′ α′ E(θ  Y ) = 1 , 2 ,..., D α 0′ α 0′ α 0′ where α 0′ = mode is
∑ dD=1(α d + xd ) = α0 + n ,
and the posterior
α ′ − 1 α 2′ − 1 α ′ − 1 mode(θ  Y ) = 1 , ,..., D α 0′ − D α 0′ − D α 0′ − D provided that every α d′ ≥ 1.
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The Dirichlet prior (7.17) is a proper probability distribution if α1,α2,...,αD are all positive. Notice, however, that for (7.19) to be proper, we only need the updated hyperparameters α′d = αd+xd to be positive. This means that we can adopt an improper prior density function such as
π (θ ) ∝ θ1−1θ2−1 Lθ D−1,
(7.21)
which is the limiting form of the D(α) density as a approaches (0, 0,..., 0), and still obtain a proper posterior if αd+xd > 0 for every d. In a slight abuse of terminology, we will call (721) the Dirichlet density with α = (0, 0,..., 0); it should be understood that this is not a density per se, but it leads to a proper Dirichlet posterior if there are no empty cells (i.e. if every xd ≥ 1). 7.2.5 Choosing the prior hyperparameters Because of the rule (7.20) for updating the hyperparameters α = (α1,..., αD), it is tempting to think of these as imaginary prior counts in the cells of the contingency table. This notion is certainly correct in a relative sense; increasing αd by one has the same inferential effect as observing one additional sample unit in cell d. In an absolute sense, however, we hesitate to interpret αd as the number of prior observations in cell d, because it is not necessarily true that αd = 0 represents no prior observations in cell d. Noninformative priors When little prior information is available about 0, it may be sensible to take α1,α2, . . . , αD equal to a common value that is, to set α = (c,c,...,c) for some constant c. However, there is no unique choice for c that clearly represents a state of ignorance about θ . Most statisticians would agree that without strong prior information, the ML estimate
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x x x θˆ = 1 , 2 ,..., D n n n
(7.22)
is a reasonable point estimate for θ. This is particularly true if θˆ lies in the interior of the parameter space, i.e. if there are no empty cells. Notice that (7.22) is the posterior mean of θ under the improper prior with c = 0, assuming that there are no empty cells. But it is also the posterior mode under the uniform prior with c = 1. From the standpoint of estimating θ, one could thus argue that either (or neither!) of these priors is noninformative. Moreover, the Jeffreys invariance principle for choosing a noninformative prior (e.g. Box and Tiao, 1992) leads to the choice c = 1/2. Therefore, it seems reasonable to regard the whole range of values of c between zero and one as potentially noninformative. With certain techniques or algorithms, there may be a natural choice for a noninformative prior. For example, with a modefinding algorithm such as EM, the uniform prior (c = 1) will cause the procedure to converge to an ML estimate. In a generalpurpose implementation of EM, therefore, it would be natural to adopt c = 1 as a default noninformative prior. In other situations, however, the choice is less clear. In data augmentation, for example, the parameters of interest are typically estimated by their simulatedposterior means. Under the prior with c = 0, the posterior mean coincides with the ML estimate (at least in the completedata case) for parameters that are linear functions of θ1,...,θ D , but not for nonlinear parameters (e.g. odds ratios). Unlike ML estimates, posterior means are not invariant under nonlinear transformations. Therefore, we cannot really claim that c = 0 is a good default prior for a generalpurpose data augmentation routine. The c = 0 prior is also unattractive because it is improper; the existence of a proper posterior under this prior is not guaranteed. If the sample size is large relative to the number of parameters being estimated, the choice of prior will tend to have little impact on the final inferences. For the examples in this book, we will adopt the Jeffreys prior (c = 1/2) as a
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default noninformative prior for simulations where the sample size is large. This choice is admittedly somewhat arbitrary. If there is any doubt that the influence of the prior is minimal, one should always conduct a sensitivity analysis, applying a variety of alternative priors to see how the resulting inferences change. If the results vary dramatically over a range of plausible priors, then the only scientifically justifiable conclusion may be that no firm conclusions are possible. Sparse tables and flattening priors When the sample size n is not much larger than the number of cells D , a substantial number of cells may contain no observations. A table x = (x1,...,xD) in which a high proportion of the frequencies xd are zero is said to be sparse. It is well known that when common models for discrete data (e.g. loglinear or logistic models) are fit to sparse tables, the empty cells can lead to inestimable parameters and/or ML estimates on the boundary. For this reason, it has often been suggested that a small positive number such as 1/2 should be added to every cell of a sparse table prior to model fitting. The use of such a number, called a flattening constant, is reviewed by Clogg et al. (1991). The effect of a flattening constant is to smooth the estimate of θ toward a uniform table in which all cell probabilities are equal. When x = (x1,..., xD) represents a crossclassification by discrete variables Y 1,Y2,...,Yp, a uniform table has no relationships whatsoever among the variables. Adding a constant ∈ > 0 to every cell thus tends to be conservative, in the sense that it makes us less likely to conclude that relationships among the variables exist when in fact they do not. A prior distribution that smooths parameter estimates toward a uniform table will be called a flattening prior. Flattening priors can be helpful for ensuring that the mode of θ is unique and lies in the interior of the parameter space. For modefinding algorithms, the prior with α = (c,c,...,c) for some c > 1 is flattening; it adds the equivalent of ∈ = c−1
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prior observations to every cell. Values of c less than one are not recommended for modefinding algorithms because they are ’antiflattening,’ pushing the estimate of θ away from a uniform table. For simulations in which the results are summarized by posterior means, any prior with c > 0 has a flattening effect on the elements of θ, adding the equivalent of
∈ = c observations to every cell relative to the ML estimate. For odds ratios and other nonlinear parameters, however, the effect of these priors when c is near zero may hardly be flattening. For such parameters, priors with c close to zero may place too much mass near the boundary, causing inferences about nonlinear parameters to be unstable when the table is sparse. In sparsedata situations, it is always advisable to apply a variety of reasonable alternative priors and see how the results change. When using a flattening prior, care should be taken not to oversmooth the data. Adding ∈ imaginary counts to every cell introduces information equivalent to D∈ prior observations. In a very sparse table, adding, say, 1/2 to every cell may result in an effective prior sample size comparable to or greater than the actual sample size. In the absence of strong prior beliefs about θ , it is probably unwise to add prior information that amounts to more than about 1020% of the actual sample size, so that the integrity of the observed data is not seriously compromised. If inferences about the parameters of interest cannot be stabilized by these modest amounts of prior information, then the model is probably too complex to be supported by the observed data. In such situations, it would be wise to simplify the model by eliminating unnecessary variables or by imposing loglinear constraints (Chapter 8). Datadependent priors One obvious potential drawback of flattening priors is that when they are applied to crossclassified contingency tables, they smooth the data toward a model in which each variable Yj has a uniform distribution over its levels 1,2,...,d j. In many contexts, it is more desirable to smooth toward a model of
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mutual independence among the variables but to leave the marginal distributions of the variables unaffected. This can be achieved by making the prior datadependent. Suppose that one of the variables (say Y1) represents the response of greatest interest, and the other variables are potential predictors of Y1. Clogg et al. (1991) advocate a strategy in which prior observations are divided among cells of the contingency table in such a way that the marginal distribution of Y1 in the observed data is preserved. For example, suppose that Y1 is dichotomous, with Y1 = 1 and Y1 = 2 observed for 30% and 70% of the sample units, respectively. After an appropriate total number of prior observations n0 has been chosen, 30% of this total can be allocated to cells of the table corresponding to Y1 = 1, with the remaining 70% going to cells corresponding to Y1 = 2. This strategy, which has an empirical Bayes flavor, smooths the estimates of θ toward a null model in which none of the predictors has any effect on Y1, but it does not affect the overall distribution of Y1 itself. This strategy can be extended to formulate a prior that simultaneously preserves the marginal distributions of all the variables in the dataset (Fienberg and Holland, 1970, 1973). Suppose that cell d of a frequency table corresponds to the event Y1 = y1, Y 2 = y2,..., Yp = yp. If Y 1, Y 2,..., Yp are mutually independent, then the probability associated with this cell is
(
)
θ d = P(Y1 = y1 )P(Y2 = y2 ) L P Y p = y p .
(7.23)
The probabilities on the righthand side of (7.23) can be estimated by the observed proportions in the sample. Substituting these estimates into (7.23), and multiplying the resulting estimate of θ d by the desired total number of prior observations n0, gives the number of prior observations to be allocated to cell d . For modefinding algorithms, the hyperparameter associated with cell d would be p
α d = 1 + n0
∏ Pˆ(Y j = y j ) j =1
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(7.24)
(
)
where Pˆ Y j = y j is the observed proportion of sample units for which Yj = yj. For simulations, p
α d = n0
∏ Pˆ(Y j = y j )
(7.25)
j =1
is a more natural choice, at least when we are concerned with linear functions of the elements of θ . These datadependent priors can be thought of as discretedata versions of the ridge prior for the parameters of the multivariate normal (Section 5.2.3), which also smooths toward a model of mutual independence among variables. If the marginal distribution of each Yj is not far from uniform (i.e. if the levels 1, 2,..., dj occur with roughly the same frequency), then these datadependent priors will have nearly the same effect as flattening priors. If some levels are relatively much rarer than others, however, then flattening priors may exert undue influence on these rarer categories, inflating their probabilities and distorting the inferences about certain functions of θ . When this is the case, datadependent priors can be an attractive alternative to flattening priors, particularly when the data are sparse. 7.2.6 Collapsing and partitioning the Dirichlet A Dirichlet random vector can be collapsed and partitioned in a manner analogous to that already described for the multinomial (Section 7.2.2), and the resulting vectors will have Dirichlet distributions. Let us first consider what happens when we collapse two elements. Suppose that θ = (θ 1,...,θD) has a Dirichlet distribution with parameter α = (α 1,...,αD ). If we form a new vector θ* = (ξ,θ3,...,θD), where ξ = θ1+θ2, then (a) the distribution of θ* is Dirichlet with parameter α * = (β,α3, ...,αD ), where β = α1+α2; and (b) the conditional distribution of (θ1/ξ,θ2/ ξ ) given ξ is Dirichlet with parameter (α1,α2). Proofs of these properties are given by Wilks (1962);
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they can also be justified by appealing to the relationship between the Dirichlet and the standard gamma distribution (Section 7.2.3). More generally, suppose that θ = (θ 1,...,θD) represents the cell probabilities for a multinomial vector x = (x 1 ,...,xD), and we apply the transformation described in Section 7.2.2 to θ , transforming it into the cell probabilities for the collapsed and partitioned versions of x. That is, suppose that A1, A2,..., AK are mutually exclusive and collectively exhaustive subsets of {1, 2,..., D} let
x( k ) = { xd ; d ∈ Ak } be the kth part of x; and let
zk =
∑ xd
d ∈Ak
be the total frequency for the kth part. The cell probabilities for the collapsed table z = (z 1, z 2,..., zk) are ξ = (ξ 1,ξ2,...,ξK), where
ξk =
∑θd ,
d ∈Ak
and the conditional probability of falling into cell d given that we are already in the kth part of the table is
φ kd = θ d / ξ k for all d ∈ Ak . If θ has a Dirichlet distribution with parameter α = (α1,...,αD), then it can be shown that the distribution of ξ is Dirichlet,
ξ  α ~ D( β ),
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where the parameters β = (β1,...,βK) are obtained by summing the elements of α in the same way the elements of θ were summed to obtain ξ,
βk =
∑αd ,
d ∈Ak
Moreover, if φ k = {φ k d : d∈Ak} is the set of conditional probabilities for the k th part of x, then the conditional distribution of φ = (φ1,φ2, ...,φ K ) given ξ is a set of K independent Dirichlet distributions,
( )  ξ, α ~ D(α ( ) ),
φ1  ξ, α ~ D α (1) , φ2
2
M
(7.26)
( )
φ K  ξ, α ~ D α ( K ) , where α(k) = {αd : d∈Ak} denotes the kth part of α. These properties imply that if a Dirichlet prior is applied to the parameter θ of a multinomial contingency table x, then the prior distribution of ψ = (ξ,φ) which is a onetoone function of θ can be factored into independent Dirichlet distributions for ξ,φ1,...,φ K . This ability of the Dirichlet distribution to be collapsed and partitioned makes it a very attractive prior for use in simulation algorithms, and provides the basis for a monotone data augmentation routine to be described in Section 7.4.
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7.3 Basic algorithms for the saturated model 7.3.1 Characterizing an incomplete categorical dataset This section presents EM and data augmentation algorithms for incomplete categorical datasets under the saturated multinomial model, which imposes no restrictions on the types of relationships that may exist among the variables Y1, Y2,...,Yp. These algorithms are conceptually simple, but the notation needed to describe them in a general setting is somewhat unwieldy. To characterize the information contained in an incomplete multivariate categorical dataset, we must extend our notation for contingency tables in several ways. First, we must account for the fact that the completedata contingency table x = (x 1 , x2,...,X D ) is actually a crossclassification by the levels of Y1, Y2,..., Yp, and as such can be regarded as a pdimensional array. Suppose that variable Y j takes possible values 1, 2,...,dj. Let xy, where y = (y1, y2,..., yp), be the total number of units in the sample for which the event Y1 = y1, Y2 = y2,..., Yp = yp occurs, and let θy be the probability of this event for any unit. Here we are using y to represent a generic realization of (Y1, Y2,..., Yp) for a single unit, i.e. a possible row of the n × p data matrix Y. We will denote the set of all possible values of y by Y. Assuming for the moment that there are no structural zeroes, Y is the Cartesian crossproduct of the sets for {1, 2,...,dj} for j = 1, 2,..., p. When a cell count or probability appears with the vector subscript y = (y1, y2,..., yp) it should be interpreted as an element of an array with dimensions d1 × d2 × ⋅⋅⋅ × d p, but when it appears with the scalar subscript d it should be interpreted as the dth element of p a vector of length D = j =1 d j . Depending on the context,
∏
we will sometimes think of the tables x and θ as vectors,
x = ( x1, x2 ,..., xD ),
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θ = (θ1, θ2 ,..., θ D ),
and at other times as pdimensional arrays,
{
}
x = x y : y ∈Y ,
{
}
θ θ y : y ∈Y .
The distinction between the two forms is simply a matter of notational convenience, because it is always possible to turn an array into a vector by assigning a linear ordering to its cells. Now we must extend the notation to allow for missing data. Let us assume that observations have been grouped according to their missingness patterns. Index the missingness patterns that appear in the dataset by s = 1, 2,..., S and define a set of binary response indicators
1 if Y j is observed in pattern s, rsj = 0 if Y j is missing in pattern s.
x(ys ) denote the number of sample units within missingness pattern s for which (Y1, Y2,..., Yp) = y, and let
Let
{
x( s ) = x(ys ) : y ∈ Y
}
denote the full set of these counts for pattern s. If any variables are missing in pattern s, then x (s) is not observed; rather, we observe the counts for a lower dimensional table in which the sample units have been crossclassified only by the observed variables. Let Os and Ms be functions that extract from y = (y1, y2,..., yp) the elements corresponding to the variables that are observed and missing, respectively, in pattern s,
{ } M s ( y) = { y j : rsj = 0}.
Os ( y) = y j : rsj = 1 ,
Also, let O s and Ms be, respectively, the sets of all possible values of Os(y) and M s (y). For example, suppose that in a dataset with p = 4 variables, missingness pattern s has Y1 and
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Y4 observed but Y2 and Y3 missing; then O s(y) = (y1,y4), Ms(y) = (y2,y3),
{( y1, y4 ) : y1 = 1, 2,..., d1; y4 = 1, 2,..., d4 }, M s = {( y2 , y3 ) : y2 = 1, 2,..., d2 ; y3 = 1, 2,..., d3 }.
Os =
When the units within missingness pattern s are crossclassified only by their observed variables, the result is a table with counts that we shall denote by
zO( )( y ) =
∑ x(ys) for all Os ( y) ∈Os .
s
s
(7.27)
M s ( y )∈M s
The marginal probability that an observation falls within cell Os(y) of this table will be called
βO ( y ) = s
∑θ y .
(7.28)
M s ( y )∈M s
Observeddata likelihood When x = (x 1, x2,..., X D) has a multinomial distribution with parameter θ = ( θ 1 ,θ2,...,θ D ), then the completedata loglikelihood function for θ is
l(θ  Y ) =
D
∑ xd log θ d d =1
over the simplex Θ (Section 7.2.1). Equivalently, viewing x and θ as pdimensional arrays, we can write the loglikelihood as
l(θ  Y ) =
∑ x y log θ y . y ∈Y
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(7.29)
When some data are missing, the observeddata loglikelihood can be calculated as follows. For any missingness pattern s, the observed data are summarized by the table
{
}
z ( s ) = zO( )( y ) : Os ( y) ∈ Os . s s
(7.30)
Notice that z(s) is a collapsed version of the unobserved x(s). By our rules for collapsing multinomial tables (Section 7.2.2), it follows that the contribution of z(s) to the observeddata loglikelihood is equivalent to that of a multinomial distribution with index
ns =
∑ x(ys) y ∈Y
and parameter
{
}
β ( s ) = βO ( y ) : Os ( y) ∈ Os . s
(7.31)
That is, the contribution of z(s) to the observeddata loglikelihood is
∑ zO(s)( y) log βO ( y) .
Os ( y )∈Os
s
s
The observeddata loglikelihood is the sum of these contributions for missingness patterns s = 1, 2,..., S,
l(θ  Yobs ) =
S
∑ ∑ zO(s)( y) log βO ( y) . s =1 Os ( y )∈Os
s
s
(7.32)
Despite the concise appearance of (7.32), it is a rather complicated function of the individual elements of θ . Evaluating l(θYobs) at specific numerical values of θ is not difficult, but calculating analytic expressions for its first two derivatives can be tedious. For this reason, it is inconvenient to maximize l(θYobs) by gradient methods. The EM algorithm
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is straightforward, however, because it involves only the repeated maximization of the completedata loglikelihood (7.29). 7.3.2 The EM algorithm EM for the saturated multinomial model was first described by Chen and Fienberg (1974) in the special case of p = 2 variables, and extended by Fuchs (1982) to arbitrary p. A description also appears in Chapter 9 of Little and Rubin (1987). The algorithm, which was already presented in Section 3.2.2 for two binary variables, is simple and intuitive. For each missingness pattern s = 1,..., S, we allocate the counts in the observed table z(s) to the cells of the full pway table x(s). This allocation is carried out in the proportions implied by the current estimate of θ . When the allocation is complete, the (1)
(2)
( )
proportions in the resulting table x = x +x +⋅⋅⋅x S provide the updated estimate of θ. Before running EM, the observed data for each missingness pattern should first be crossclassified according to the observed variables; that is, the data should be reduced to (1) (S) (1) (S) z ,..., z . Notice that z ,..., z can be regarded as arrays of (s) varying dimensions; the number of dimensions for z is equal to the number of variables observed in pattern s. When (1) (S) implementing EM on a computer, however, storing z ,..., z as multidimensional arrays tends to be cumbersome and inefficient. As the number of variables p grows, the number of arrays S can increase very rapidly. Moreover, these arrays can be very sparse; many of them may contain only a few or perhaps even just one observation each. A generalpurpose computer program should be efficient in its use of memory, and the data structures it creates should have predictable size and shape. A more efficient way to store and manipulate the (1) (S) counts in z ,..., z is outlined in Appendix B. The E and Msteps The completedata loglikelihood (7.29) is a linear function of the elements of x = {xy : y∈Y}, the unobserved pdimensional
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table that crossclassifies all sample units by their values of Y1, Y2,..., Yp. To perform the Estep, we must find the expectation of each count xy given the observed data and an assumed value S x( s ) , the for θ . Notice that x can be expressed as x =
∑ s =1
sum of individual tables for missingness patterns 1,..., S. Moreover, the observed data z(s) for pattern s is a collapsed version of x(s), and by our rules for collapsing and partitioning (Section 7.2.2) it follows that the conditional distribution of x(s) given z(s) is productmultinomial. Let
{
xO( )( y ) = x(ys ) : M s ( y) ∈ M s s s
}
(7.33)
denote the portion of x(s) that is obtained by fixing Os(y) at a (s) specific value but varying Ms(y) over M s; that is, xO ( y ) is s simply the set of all cell counts in x(s) that contribute to the (s) (s) observed count zO ( y ) . By the partitioning rules, xO ( y ) has, s s (s) (s) given zO ( y ) , a multinomial distribution with index zO ( y ) s s and parameters
γOs ( y) = θ y / βO ( y ) : M s ( y) ∈ M s ; s
{
}
(7.34)
s s s xO( )( y ) zO( )( y ) , θ ~ M zO( )( y ) , γ O ( y ) . s s s s
(7.35)
that is,
Notice that (7.34) is simply the portion of θ corresponding to s xO( )( y ) , rescaled so that its elements sum to one. It follows that s the conditional expectation of an element of x(s) is s E x(ys ) z ( s ) , θ = zO( )( y )θ y / βO ( y ) . s s
(7.36)
The Estep consists of calculating (7.36) for every s = 1,...,S and summing the results,
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S
(
) ∑ zO(s)( y)θ y / βO ( y) .
E x y  Yobs , θ =
s =1
s
s
(7.37)
Once the Estep has been completed, the Mstep is trivial. The completedata loglikelihood (7.29) is maximized at θy = xy/n, so the Mstep is simply to
(
)
estimate θ y by E x y  Yobs , θ / n
(7.38)
for all y∈Y A pseudocode implementation of the E and Msteps is shown in Figure 7.2. Given the observed counts z(1),..., z(S) and the current value of θ, this code overwrites θ with its updated value. A temporary workspace x of the same size as θ is required for accumulating
Figure 7.2. Single iteration of EM for the saturated multinomial model.
the expected sufficient statistics. The algorithm cycles through the missingness patterns and checks to see whether the current pattern s has any missing variables (i.e. if M s(y) is nonempty). If not, then the observed counts for pattern s are added into the
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elements of x; otherwise, the expectations (7.36) are calculated and added into x. After this is done for s = 1, 2,..., S, the resulting elements of x are divided by n, which yields the updated value of θ. Starting values and posterior modes If the starting value of θ lies on the boundary of the parameter space Θ , i.e. if some of its elements are zero, then an inconsistency could arise in the initial Estep. It could happen that a nonzero count appears in one of the cells of the observeddata tables z(1),..., z(S) for which the probability implied by the starting value of θ is zero. If this occurs, then the algorithm may halt due to attempted division by zero. To prevent such inconsistencies from arising, a starting value should be chosen in the interior of the parameter space. A good default starting value is a uniform table, in which all the elements of θ are equal. The algorithm in Figure 7.2 calculates an ML estimate, but with a slight modification it can also be used to find a posterior mode under a Dirichlet prior. The Estep remains the same, but the Mstep must be altered to maximize the completedata posterior density rather than the completedata likelihood. If the prior distribution of θ is Dirichlet with hyperparameter α = {αy : y∈Y}, then the last line in Figure 7.2 should be changed to
(
)
for y ∈ Y do θ y := x y + α y − 1 / ( n + α 0 − D),
(7.39)
where α0 and D is the total number of cells in θ. Taking αy = 1 for all y∈ Y results in a uniform prior, under which the posterior mode and the ML estimate coincide. Notice that if any α y < 1 and the corresponding cell count xy is zero, then (7.39) will produce a negative estimate for θy. For computing posterior modes, priors with αy < 1 are not recommended.
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Random zeroes and structural zeroes (1)
When cells of the observeddata tables z ,..., z(S) are empty not because the events corresponding to those cells are impossible but merely as an artifact of chance, the cells are (1) said to contain random zeroes. Random zeroes in z ,..., z(S) may have two undesirable effects. First, they may produce an ML estimate on the boundary of Θ . Such an estimate is conceptually unattractive, because it implies that some events in the discrete sample space have zero probability even though they have not been deemed impossible on a priori grounds. Second, random zeroes may render certain functions of θ inestimable, in which case the ML estimate will not be unique; the observeddata likelihood will be maximized along a ridge, and EM will converge to different stationary values depending on the starting value (Fuchs, 1982). When random zeroes result in inestimable parameters or ML estimates on the boundary, the algorithm in Figure 7.2 does not experience any numerical difficulty; it still converges reliably from any starting value in the interior of Θ. The value to which it converges, however, may be a poor estimate for certain functions of θ. When this happens, it is often helpful to apply a Dirichlet prior distribution in which all the hyperparameters are greater than one, e.g. a flattening prior with α = (c, c,..., c) for some c > 1, which adds the equivalent of c − 1 prior observations to each cell. Another good choice is a datadependent prior that smooths the estimate toward a null model of independence (Section 7.2.5). A cell that is empty because the corresponding event is logically impossible is said to contain a structural zero. Structural zeroes are qualitatively different from random zeroes and should not be handled in the same way. Because the probabilities associated with structural zeroes are known to be zero a priori, those cells should be omitted from the estimation procedure. In the algorithm of Figure 7.2, structural zeroes can be handled by providing a starting value for θ in which the elements corresponding to structural zeroes have
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been set to zero. If the initial value of θy is zero, then the first Estep will not allocate any portion of the observed counts in (1) z ,...,z(S) to cell y, and the resulting expectation E(x yYobs,θ) will be zero. To ensure that the estimate of θy remains zero for all subsequent iterations, the last line of the algorithm should be revised to
)(
(
)
for y ∈ Y * do θ y := x y + α y − 1 / n + α 0* − D * ,
(7.40)
where Y* is the set of all possible values of y excluding the structural zeroes, α 0* = y ∈Y *α y is the sum of the prior
∑
hyperparameters and D* is the number of elements in Y* (i.e. the total number of cells excluding structural zeroes). Observeddata loglikelihood The observeddata loglikelihood function l(θYobs), given by (7.32), and the observeddata logposterior density
log P(θ  Yobs ) = l(θ  Yobs ) + log π (θ ), are not difficult to calculate for specific values of θ . Evaluating the loglikelihood or logposterior density can be helpful for monitoring the progress of EM and data augmentation (Sections 3.3.4 and 4 . 4 . 3 ). Pseudocode for evaluating l(θYobs) is shown in Figure 7.4. The loglikelihood at the current value of θ is calculated and stored in l. Notice that this code is very similar to the Estep and could easily be woven into EM. 7.3.3 Data augmentation Data augmentation for the saturated multinomial model is quite similar to the EM algorithm described above. Recall that in data augmentation, we alternately draw from the predictive distribution of the missing data given the observed data and the parameters (the Istep) and from the completedata
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posterior distribution of the parameters (the Pstep). The observed data consist of the tables z(s) for missingness patterns s = 1,..., S, and the missing data consist of the information needed to expand each z(s) into a full pdimensional table x(s). The predictive distribution of x(s)
Figure 7.3. Evaluation of the observeddata loglikelihood function.
given z(s) and θ is the product multinomial given by (7.33)(7.35). Therefore, the Istep consists of drawing each x(s) from its product multinomial distribution and summing them to (1) (2) obtain a simulated completedata table x = x + x +⋅⋅⋅+ x(S). Under the Dirichlet prior θ ~ D(α), the Pstep is then just a simulation of θ from its completedata posterior D(α + x). In the pseudocode of Figure 7.2, the line s for M s ( y) ∈ M s do x y := x y + zO( )( y )θ y / sum s
(7.41)
(s) allocates an observed count zO ( y ) to the cells of the s completedata table in fixed proportions determined by the current value of θ . To convert this Estep into an Istep, the proportional allocation must be replaced by a random
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allocation; that is, we must replace (7.41) by a routine that will draw s s xO( )( y ) ~ M zO( )( y ) , γ O ( y ) s s s
and add the result into x. One method for simulating the multinomial counts, called table sampling, is to compare standard uniform U (0, 1) random variates to cumulative sums of the probabilities in γ O ( y ) . Pseudocode for table sampling s
is shown in Figure 7.4. Substituting this code for (7.41) will change the Estep into an Istep. Table sampling can be slow if the counts in the observeddata tables z(s) are large. A more efficient method for simulating multinomial
Figure 7.4. Table sampling for the data augmentation Istep.
draws in that situation, which relies on a Poisson variate generator, is described by Brown and Bromberg (1984). To complete the conversion of the EM algorithm to data augmentation, the Mstep (the final line of Figure 7.2) must be changed to a Pstep; that is, the estimation of θ from the completedata table x must be replaced by a random draw of θ from the Dirichlet posterior D(α+x). The Dirichlet is easily simulated using standard gamma variates (Section 7.2.3). If any structural zeroes are present, those cells should be omitted
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from the Pstep and their probabilities should be set to zero. If (1) random zeroes occur in z ,..., z(S) and the improper Dirichlet prior with α = (0,0, ,0) is being used, then depending on the pattern of the zeroes the Pstep could be undefined, because some elements of a + x could be zero. For this reason, the prior α = (0, 0,..., 0) should be avoided whenever random zeroes are present. A proper prior, e.g. a flattening prior with α = (c, c,..., c) for some positive value of c, should be used instead. Imputation of unitlevel missing data The I and Psteps of the data augmentation algorithm described above operate on the sufficient statistics stored in the workspace x. After enough steps have been taken to achieve approximate stationarity, x will contain a simulated draw from the posterior predictive distribution of the completedata contingency table P(xYobs). If the algorithm is being used for multiple imputation, however, it may be necessary at the end of the simulation run to impute the missing values at the unit level, i.e. to fill in the missing elements Ymis of the n × p data matrix Y. Figure 7.5 shows pseudocode for a modified Istep that imputes the missing elements of Y. Executing this code once at the end of a sufficiently long data augmentation run will result in a proper imputation of Y mis i.e. a simulated draw from P(θYobs,Ymis). In Figure 7.5, y i (obs) and yi(mis) denote the observed and missing portions, respectively, of the ith row of the data matrix Y, and I(s) denotes the rows of Y that exhibit missingness pattern s. The vector workspace y = (y 1, y2,..., yp) serves as a counter, indexing the cells of the pdimensional contingency table. For any row i in missingness pattern s, the subvector Os(y) of y is first set equal to the observed data in yi(obs), so that the remaining portion Ms(y) indexes all the cells of the contingency table into which observation i might fall. The missing values in yi(mis) are then drawn simultaneously by table sampling, comparing a single uniform variate u to the set of probabilities derived from θ that describe the conditional distribution of yi(mis) given yi(obs).
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7.3.4 Example: victimization status from the National Crime Survey Recall the data of Table 3.3 from the National Crime Survey, in which households were classified according to whether they had been victimized by crime in two sixmonth periods. In the sample of 756 households, 38 had victimization status missing for the first period, 42 had status missing for the second period and 115 had status missing for both periods. Using the EM algorithm and likelihoodratio tests, we found very strong evidence that victimization status on the two occasions was related; the pvalue for testing the hypothesis of independence was essentially zero. Moreover, we found fairly strong evidence that the rates of victimization in the two periods were not equal; the pvalue for testing the hypothesis of marginal homogeneity/symmetry was 0.06 (Section 3.2.4). Analysis by parameter simulation Tests of independence and marginal homogeneity/symmetry can also be readily carried out by parameter simulation. To test a hypothesis by parameter simulation, we first select a function of the
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Figure 7.5. Istep for imputing missing values at the unit level.
cell probabilities θ that measures the degree to which θ departs from the null hypothesis, and simulate the posterior distribution of this quantity given the observed data. For independence, a natural quantity to examine is the odds ratio
ω=
θ11θ22 , θ12θ21
(7.42)
where θij denotes the probability of (Y1 = i, Y2 = j) for i, j = 1, 2. The proportion of simulated values of ω that are less than or equal to one can be taken as an approximate onesided pvalue for testing the hypothesis of independence (ω = 1) against the alternative that households victimized in the first period were more likely to be victimized in the second period (ω > 1). For marginal homogeneity/symmetry, we can examine the difference in victimization rates between the second period (θ+2 = θ12 + θ22) and the first period (θ+2 = θ12 + θ22).
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δ = θ +2 − θ2 + = θ12 − θ21.
(7.43)
Figure 7.6. Histograms of (a) ω = (θ11θ22)/(θ12θ21) and (b) δ = θ12−θ21 over 5000 iterations of data augmentation, and of (c) the likelihood ratio statistic 2 dL with the χ 3 density superimposed.
The proportion of simulated values of δ that fall above zero is an approximate onesided pvalue for testing the hypothesis of no change (δ = 0) against the alternative that the victimization rate has dropped (δ < 0). One interesting question is whether the 115 households for which both variables are missing should be included in the simulations. From an inferential standpoint it does not matter; under the ignorability assumption these sample units contribute nothing to the likelihood function for θ , so likelihoodbased or Bayesian inferences for θ will be the same whether these units are included or not. From a computational standpoint, however, it is slightly better to omit them, because their presence needlessly increases the fractions of missing information and slows the convergence of data augmentation. In this particular example, the difference is barely noticeable. Without these 115 cases, the worst fraction of missing information as estimated from the iterations of EM (Section 3.3.4) is about 13%. Including these cases, it rises to 26%. Either way, data augmentation converges very quickly; in preliminary runs under the Jeffreys prior (all α = 1/2), the autocorrelations in scalar functions of θ essentially died out after lag 2 or 3 even when the 115 cases were included.
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Starting from the ML estimate of θ , we simulated 5000 steps of data augmentation under the Jeffreys prior following a burnin period of 100 steps. Histograms of the simulated values of ω and δ are shown in Figure 7.6 (a) and (b), respectively. All 5000 values of w were greater than one, so the simulated pvalue for the test of independence is zero. Of the 5000 values of δ, 164 fell above zero, so the simulated pvalue for testing δ = 0 against the onesided alternative δ < 0 is 164/5000 = 0.033; the pvalue against the twosided alternative is 2 × 0.033 = 0.066. Notice that these simulated pvalues agree closely with those from the likelihoodratio tests performed in Chapter 3. Because of the large sample size and the small number of parameters in this example, Bayesian and likelihoodbased inferences are essentially identical. Further evidence that the largesample properties are working well is provided by the posterior distribution of the likelihoodratio statistic. The quantity
[(
)
]
dL = 2 l θˆ  Yobs − l(θ  Yobs ) , where θˆ is the ML estimate, has (when regarded as a function of θ ) a posterior distribution that is asymptotically chisquare with three degrees of freedom, because the multinomial model for this example has three free parameters. A histogram of the 5000 simulated values of dL is shown in Figure 7.6 (c) with the χ 32 density function superimposed; the two are nearly indistinguishable. By averaging the 5000 iterates of ω and δ , we obtain simulated posterior means
Eˆ (ω  Yobs ) = 3.67 and Eˆ (δ  Yobs ) = −0.036. Comparing these to the ML estimates obtained in Section 3.2.2,
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ωˆ = 3.57 and δˆ = −0.037, we find that the agreement is close. Simulated 95% posterior intervals for ω and δ based on sample quantiles of the 5000 iterates are (2.20, 5.77) and (0.076, 0.001), respectively. Analysis by multiple imputation In this example, it is also straightforward to conduct inferences by multiple imputation. We generated a set of m = 10 imputations by running ten independent chains of data augmentation for 100 steps, starting each chain from the ML estimate. To speed convergence, the 115 households for which both variables were missing were omitted from the sample. At the final Istep of each chain, however, these households were restored to the sample so that their missing data could be imputed. Because these households contribute nothing to the observeddata likelihood, inferences will be essentially the same whether they are included or not. We decided to include them in the final Isteps so that the variation among the imputed datasets would more accurately reflect the real levels of missingdata uncertainty. The observed data and ten imputations of the completedata table are shown in Table 7.1. Table 7.1. Victimization status for households in the National Crime Survey, with m = 10 multiple imputations
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As before, let us make inferences about the odds ratio ω = (θ11θ22)/(θ12θ21) and the difference δ = θ 12 − θ21. The standard method for obtaining a point estimate and confidence interval for an odds ratio with complete data is given in Section 6.4.2. The obvious completedata estimate of the difference δ i s δˆ = θˆ − θˆ , where θˆ = x / n and θˆ = x / n . In large 12
21
12
12
21
21
samples δˆ will be approximately normally distributed, and a consistent estimate of its variance is
() [ (
)
(
)
1 Vˆ δˆ = θˆ12 1 − θˆ12 + θˆ21 1 − θˆ21 + 2θˆ12θˆ21 n
]
by elementary properties of the multinomial distribution (Section 7.2.1). Given these completedata methods and the ten imputations in Table 7.1 (b), multipleimputation point and interval estimates were obtained by Rubin’s method for scalar estimands (Section 4.3.2). The resulting point estimates for ω and δ are 3.60 and −0.039, respectively, which agree closely with the ML estimates and the simulated posterior means. The resulting 95% interval estimates are (2.15, 6.04) and (0.079, 0.001), which also agree well with the intervals obtained through parameter simulation. Estimated fractions of missing information for ω and δ are 35% and 26%, respectively. 7.3.5 Example: Protective Services Project for Older Persons Fuchs (1982) analyzed data from the Protective Services Project for Older Persons, a longitudinal study designed to measure the impact of enriched social casework services on the wellbeing of elderly clients (Blenkner et al., 1971). For 101 clients in the study, six dichotomous variables were recorded:
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Variable Group membership Age Sex Survival status Physical status Mental status
Levels 1 = experimental, 2 = control 1 = under 75, 2 = 75+ 1 = male, 2 = female 1 = deceased, 2 = survived 1 = poor, 2 = good 1 = poor, 2 = good
Code G A S D P M
For an additional 63 clients, values of physical and/or mental status were missing. The observed dataset, including complete and incomplete cases, is shown in Table 7.2. Results from this project generated considerable controversy in the social work literature. Some (Fischer, 1973) argued that the enriched services seemed to be detrimental to the clients, because the mortality rate for the experimental group was actually higher than for the control group. Classifying the subjects by only G and D, both of which are observed for the entire sample, we obtain the marginal frequencies displayed in Table 7.3. The test for independence 2 in this table, based on the wellknown Pearson X statistic, 2 yields X = 5.03 with one degree of freedom; the approximate pvalue is 0.025, which provides fairly strong evidence that G and P are related. The estimated odds ratio is 2.04, suggesting that subjects in the experimental group were about twice as likely (on the odds scale) to die than subjects in the control group. If subjects had been assigned to treatments in a random fashion, then Table 7.3 would indeed provide evidence that the services given to the experimental group were detrimental. If we examine the relationships between G and the other variables, however, we find that the treatment assignments were not random. Subjects in the experimental group tended to be older, and also tended to have poorer physical and mental status, than subjects in the
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Table 7.2. Data from the Protective Services Project for Older Persons
Table 7.3 Classification of subjects by G and D
control group. It appears that the investigators tended to give the enriched services to clients who appeared to have the greatest need for them. The marginal association between G and D could thus be due, at least in part, to the fact that the subjects in the experimental group were simply more prone to die than the subjects in the control group, regardless of any services they received. Rather than examining the marginal association between G and D, we ought to focus on their
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conditional associations given the covariates A, S, P and M, to see whether G and D are still related after the possibly confounding effects of these covariates have been removed. That is, we should examine the odds ratios for G and D within the sixteen 2 × 2 tables that correspond to the unique combinations of the levels of A, S, P and M. The completedata contingency table has 26 = 64 cells; with a sample size of n = 164, this results in an average of only 2.6 observations per cell. As noted by Fuchs (1982), the ML estimate of θ under the saturated model is not unique due to the pattern of random zeroes in the observeddata tables. Moreover, the suprema of the likelihood function lie on the boundary of the parameter space. To make EM converge to a unique mode in the interior, a Dirichlet prior was applied with α=(c,c, ...,c) for c = 1.1 which adds the equivalent of 6.4 prior observations and spreads them uniformly across the 64 cells. Then, taking this mode as a starting value, single chains of data augmentation were simulated under two alternative priors: c = 0.1 and c = 1.5. Each chain was run for 1000 steps following a burnin period of 200 steps. Boxplots of the simulated GD odds ratios for each of the sixteen ASPM combinations are shown in Figure 7.7. The odds ratios are plotted on the natural log scale, with positive values indicating a positive association between enriched services (G = 1) and death (D = 1). Under the c = 0.1 prior, the simulated odds ratios show enormous variability; this prior assigns high probability to regions of the parameter space near the boundary, where odds ratios can approach 0 or +∞. Under the stronger prior c = 1.5 the situation has improved, but the range of the simulated odds ratios is still implausibly wide. Notice that under either prior, all of the boxplots straddle the null value of zero, and there is no overwhelming tendency for the boxplots to be centered either to the left or to the right of zero. Thus there seems to be no strong evidence against the null hypothesis that G and D are unrelated. To further sharpen the posterior distributions, we could increase the value of c even more. But this does not seem appropriate,
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Figure 7.7. Boxplots of simulated logodds ratios from 1000 iterations of data augmentation under two flattening priors.
because with c = 1.5 we have already added the equivalent of 1.5 × 64 = 96 prior observations with respect to estimation of the elements of θ . It appears that modest amounts of prior information are not sufficient to stabilize the inference; the observed data are simply too sparse to support the estimation of separate odds ratios within each cell of the ASPM classification. We will deal with this problem of sparseness in Chapter 8 by fitting a simpler model that assumes a common odds ratio for all sixteen levels of ASPM.
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7.4 Fast algorithms for nearmonotone patterns 7.4.1 Factoring the likelihood and prior density In Chapter 6 we introduced a class of algorithms called monotone data augmentation. Monotone data augmentation is similar to ordinary data augmentation except that in each Istep we impute only enough of the missing values to complete a monotone pattern. The advantage of monotone data augmentation is that it tends to converge very quickly when the observed data are nearly monotone. In this section we present monotone data augmentation for the saturated multinomial model. Monotone data augmentation is feasible when the prior and likelihood for the complete data factor neatly into independent pieces corresponding to the marginal distribution of Y1, the conditional distribution for Y 2 given Y1, the conditional distribution for Y3 given Y1 and Y 2, and so on. Let us first consider the likelihood. Until now we have been describing the data by a single multinomial distribution for the completedata contingency table x, but we can equivalently characterize this model as a sequence of productmultinomials. Suppose we write
(
)
P Y1,..., Y p  θ = P(Y1  φ1 )P(Y2  Y1, φ2 )
(
)
L P Y p  Y1,..., Y p −1, φ p ,
(7.44)
where φj denotes the parameters governing the conditional distribution of Yj given (Y1,..., Yj−1). Each of the factors of the righthand side of (7.44) corresponds to a productmultinomial distribution on a collapsed version of x. To be more precise we need some additional notation. Suppose that y = (y1, y2,..., yp) is a generic realization of (Y1, Y2,..., Yp) for a single unit. Let Fj be a function that extracts from y the first j elements,
(
)
F j ( y) = y1,..., y j ,
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and let Lj extract the last p − j elements,
(
)
L j ( y) = y j +1,..., y p . Let F j and L j, respectively, be the sets over which Fj(y) and Lj(y) are allowed to vary; that is, Fj will be the Cartesian crossproduct of the sets {1, 2,..., dk} for k = 1,..., j, and Lj the crossproduct for k = j + 1,..., p. We will write the probability of the event Y1, y1, Y2 = y2,..., Yj = yj as
ξF ( y ) = j
∑
θ y, L j ( y )∈L j
and the full set of parameters governing the marginal distribution of (Y1, Y2,..., Yj) as
{
}
ξ j = ξ F ( y ) : F j ( y) ∈ F j . j The conditional probability of the event Yj = yj given that Y1 = y1, Y2 = y2,..., Yj − 1 = yj − 1 will be
φ F ( y ) = ξF ( y ) / ξF ( y ) , j j j −1
(7.45)
and the full set of parameters governing the conditional distribution of Yj given (Y1, Y2,..., Yj − 1) is
{
}
φ j = φ F ( y ) : F j ( y) ∈ F j . j Suppose we collapse the pdimensional contingency table x on its last p − j dimensions, producing a table that crossclassifies the units by (Y1, Y2,..., Yj). Denote a frequency in this table by
zF ( y) = j
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∑
xy, L j ( y )∈L j
and the entire jdimensional table by
{
}
z j = z F ( y ) : F j ( y) ∈ F j . j By the rules for collapsing and partitioning (Section 7.2.2), z j has a multinomial distribution with index n and parameter ξj. Moreover, the conditional distribution of zj given zj−1 is a productmultinomial whose parameters are contained in φ j. More specifically, suppose we partition zj into a set of d1 × d2 × ⋅⋅⋅ dj−1 vectors, each of length dj. Denote one of these vectors by
{
}
z j; F ( y ) = z F ( y ) : y j = 1, 2,..., d j , j −1 j which is simply the portion of zj obtained by fixing (y1,..., yj−1) at a specific value but letting y j vary over {1, 2,..., dj}. The table zj is then the collection of these vectors,
{
}
z j = z j; F ( y ) : F j −1( y) ∈ F j −1 . j −1 If we partition φj in the same fashion, as
{
}
φ j = φ j , F ( y ) : F j −1( y) ∈ F j −1 j −1 where
{
}
φ j; F ( y ) = φ F ( y ) : y j = 1, 2,..., d j , j −1 j then the conditional distribution of zj given zj − 1 is
z j; F ( y )  z j −1, φ j ~ M z F ( y )φ j; F ( y ) j −1 j −1 j −1 independently for all Fj−1(y)∈Fj−1.
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(7.46)
By these properties, it follows that the multinomial likelihood function for any ξj can be factored as
(
) (
)(
)
L ξ j  z j = L ξ j −1  z j −1 L φ j  z j , the product of a multinomial likelihood for ξj−1 whose sufficient statistics are contained in zj−1 a n d
a
productmultinomial likelihood for φj whose sufficient statistics are contained in z j. Applying this factorization recursively, first to ξp = θ, then to ξp−1, and so on down to ξ2, we obtain
L(φ  Y ) =
p
∏ L(φ j  z j ), j =1
where each factor L(φjzj) is a productmultinomial likelihood. The full set of parameters φ = (φ1,φ2, ...,φp) forms a onetoone transformation of θ , and it follows from (7.45) that the backtransformation is
θ y = φ F ( y )φ F ( y ) Lφ F ( y ) . p 1 2 Factoring the prior Just as the likelihood function factors into independent pieces for φ 1 ,φ2, ...,φp, the density function for φ induced by the ordinary Dirichlet prior on θ also factors into a product of independent densities. Suppose that a priori θ has a Dirichlet distribution,
θ ~ D(α ),
(7.48)
where the hyperparameters are regarded as an array with the same dimensions as θ,
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{
}
α = α y : y ∈Y . By the collapsing rules for the Dirichlet discussed in Section 7.2.6, the distribution for ξj implied by (7.48) is also Dirichlet. The parameters of this distribution, which we shall call
{
}
β j = β F ( y ) : F j ( y) ∈ F j , j are obtained by summing the elements of a in the same way the elements of θ were summed to produce ξj,
βF ( y) = j
∑α y.
L j ( y )∈L j
Moreover, by the results of Section 7.2.6, the conditional distribution of φ j given ξj−1 for any j is a product of independent Dirichlet distributions. That is, if we partition the jdimensional table β j in precisely the same manner as we partitioned φj, as
{
}
β j = β j; F ( y ) : F j −1( y) ∈ F j −1 j −1 where
{
}
β j; F ( y ) = β F ( y ) : y j = 1, 2,..., d j , j −1 j the conditional distribution of φj given ξj−1 is
φ j; F ( y )  ξ j −1 ~ D β j; F ( y ) j −1 j −1
(7.49)
independently for all Fj−1(y)∈Fj−1. Now from (7.45) it is clear that ξ j is a onetoone function of (φ1, ...,φj) for any j. The prior density for φ = (φ1, ...,φj) can thus be written
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p
∏ π j (φ j  φ1,..., φ j −1)
π (φ ) = π1(φ1 )
j =2
(7.50)
p
∏ π j (φ j  ξ j −1).
= π1(φ1 )
j =2
But notice that ξj−1 does not appear on the righthand side of (7.49); thus φj is independent of ξj−1, and (7.50) becomes
π (φ ) =
p
∏ π j (φ j ),
(7.51)
j =1
where each of the terms π j(φj) is a product of independent Dirichlet densities whose parameters are contained in βj. 7.4.2 Monotone data augmentation By the factorizations described above, it immediately follows that completedata Bayesian inferences under the saturated multinomial model and Dirichlet prior,
x  θ ~ M ( n, θ ), θ ~ D(α ), can be carried out as a sequence of independent Bayesian inferences for φ1,φ2, ...,φp,
P(φ  Y ) =
p
∏ P(φ j  z j ). j =1
By combining (7.46) with (7.49), we see that the completedata posterior distribution for any term φj is
φ j; F ( y )  z j ~ D β j; F ( y ) + z j; F ( y ) j −1 j −1 j −1
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(7.52)
independently for all Fj−1(y)∈Fj−1. This factorization of the posterior applies not only when the data are complete; more generally, it holds whenever the observed data form a monotone pattern as described in Section 6.5. Suppose that the observed data are monotone in the sense that if Yj is missing for a unit, then Y j+1,..., Yp are missing as well (Figure 6.8). By essentially the same argument as was given in Section 6.5.1, the observeddata likelihood for φ given Yobs can be factored as
L(φ  Yobs ) =
p
∏ L(φ j  z *j ), j =1
where z *j is the contingency table that cross classifies all the units for which Yj is observed by their values of Y1,..., Yj. If we denote a cell of this table by z F* ( y ) and let j
{
z *j; F ( y ) = z *j; F ( y ) : y j = 1, 2,..., d j j −1 j
(
}
)
be a subvector within this table, L φ j  z *j will be the likelihood that arises from the productmultinomial distribution z *j: F ( y )  z *j −1, φ j ~ M z F ( y ) , φ j; F ( y ) j −1 j −1 j −1 for all F j −1 ( y ) ∈ F j −1. Combining this new likelihood with the prior (7.49) leads to the observeddata posterior
P(φ  Yobs ) =
(
p
∏ P(φ j  z *j ),
(7.53)
j =1
)
where P φ j  z *j is given by
φ j; F ( y )  z *j ~ D β j; F ( y ) + z *j; F ( y ) j −1 j −1 j −1
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(7.54)
for all F j −1 ( y ) ∈ F j −1 . Monotone data augmentation capitalizes on (7.53) to create an efficient simulation algorithm for situations where Yobs is nonmonotone. Suppose that Yobs is no longer monotone, but we have identified a subset Ymis* of Y mis, such that (Yobs, Ymis*) is monotone. The monotone data augmentation algorithm alternates between the following two steps. 1. Istep: Simulate a value of Ymis* from its predictive distribution given the current value of θ,
(
)
( ) ( t) . Ymis * ~ P Ymis *  Yobs , θ t +1
2. Pstep: Draw a new value of θ from its posterior distribution given Yobs and the new value of Ymis*,
(
)
( t +1) θ ( t +1) ~ P θ  Yobs , Ymis * .
In practice, the Istep is identical to that of ordinary data augmentation (Section 7.3.3) except that we need only draw the elements of Ymis* rather than the full Ymis. The Pstep is carried out by drawing φ 1 , ...,φp from the factored posterior (7.53), and then numerically transforming the resulting value of φ = (φ1, ...,φp) back to the θscale using (7.47). Interleaving the I and Psteps Notice that the simulation of φ j within a Pstep does not require knowledge of the most recent simulated value of the entire Ymis* rather, it requires only the most recent value of the jdimensional table z *j . This allows us to interleave portions of the I and Psteps in the following manner. Suppose that the data are grouped by missingness pattern and sorted as shown in Figure 6.10. Let sj denote the last pattern for which variable Yj may need to be filled in to complete the overall monotone
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pattern, so that sp ≤ s p−1 ≤⋅⋅⋅≤ s1, and for convenience define sp+1 = 0. Let T 1, T2 and T3 be three workspace arrays, each of dimension d1×d2× d p. Initialize T1 and T 2 to be equal to the current parameter value θ(t) and α , respectively, and initialize all the elements of T3 to one. Then, for j: =p, p − 1,...,1, perform the following steps: 1. If sj > sj+1, impute the missing data for variables Y1,..., Yj within patterns sj+1 + 1 up to sj. These data should be drawn from their predictive distribution given the observed data and the parameters stored in T1. 2. Crossclassify the units in patterns sj + 1 + 1 up to sj by their observed or imputed values for Y1,..., Yj, and add the resulting counts into the corresponding cells of the workspace T2. Upon completion of this step, T2 will contain βj plus the simulated value of z *j . 3. Draw a value of φj from its productmultinomial posterior distribution (7.54) given the value of β j + z *j in T 2. Multiply the elements of the array T3 by the corresponding elements of this simulated φj. 4. If j > 1, collapse T1 by summing along its jth dimension, thereby reducing its size to d 1 ×⋅⋅⋅× d j−1. Now T 1 contains the current value of ξ j−1 (the parameters of the joint distribution of Y1,..., Yj  1) which will be necessary for the next Step 1. Perform this same collapsing operation for T2, preparing it for the next Step 2. After all pcycles of Steps 14 have been completed, the +1 workspace T3 will contain the updated parameter θ(t ). Running this algorithm from a starting value θ
(0)
generates
a sequence of parameter values {θ (t):t = 1,2, ... } which
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converges in distribution to the correct observeddata posterior,
(
)
P θ ( t )  Yobs , θ (0) → P(θ  Yobs ) as t → ∞ Convergence tends to be faster than for the ordinary data augmentation algorithm described in Section 7.3, because Ymis* contains less information about the parameter than does Ymis. The most dramatic improvements are seen when Yobs is nearly monotone, because then Y mis* is only a small subset of Ymis. When the observed data happen to be monotone, Ymis* is empty and the algorithm converges from any starting value in a single step. This algorithm can be used to generate proper multiple imputations of the missing data Ymis as follows. First, simulate a small number of independent draws of θ from P( θYobs), either by running multiple chains or subsampling a single chain. Then, under each of these θ values, impute the full set of missing data Y mis using the ordinary data augmentation Istep (Figure 7.5). 7.4.3 Example: driver injury and seatbelt use The data in Tables 7.4 and 7.5, previously analyzed by Hochberg (1977) and Chen (1989), concern the effectiveness of seatbelts in reducing the risk of driver injury in automobile accidents. Table 7.4 classifies 80 084 automobile accidents according to four variables obtained from police reports: driver’s sex, car damage (low, high), belt use (no, yes) and injury (no, yes). At first glance, these data suggest that the use of seatbelts substantially reduces the risk of injury. The estimated odds of injury are
199 + 117 + 583 + 297 = 0.167 3006 + 1262 + 2155 + 728 for belted drivers and
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1687 + 1422 + 6746 + 3707 = 0.233 22536 + 11199 + 17476 + 6964
Table 7.4. Classification of accidents by police reports of driver’s sex, car damage, injury and belt use
for unbelted drivers, giving an odds ratio of 0.717; an approximate 95% confidence interval for this ratio is (0.673,0.765). This simple analysis is unconvincing, however, for a number of reasons. First, the belted and unbelted groups tend to differ with respect to a variety of characteristics (e.g. sex), and to the extent that these characteristics may be related to the risk of injury, our estimate of the effectiveness of seatbelts may be biased upward or downward. Another difficulty with this analysis is that the data provided by the police reports are not always accurate, especially with respect to belt use and injury. Experience has shown that the police were prone to overestimate the proportion of drivers who were not injured and unbelted, and that the biases toward not injured were especially severe for lowdamage accidents. Even small rates of misclassification with respect to belt use and injury can have a large impact on the estimated effect of wearing a seatbelt. To examine the effect of misclassification errors, followup data were collected for an additional sample of 1796 accidents. Subsequent to the police reports, investigators obtained more reliable data on belt use and injury from hospital records and personal interviews. We will assume that
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the information obtained in this followup effort is correct. Data from the followup study are shown in Table 7.5, with the policereported and followup values of belt use and injury indicated by (p) and (f), respectively. The followup data in Table 7.5 may be used in a variety of ways. For example, we may ignore the police reports entirely and estimate the seatbelt effect from the followup data alone. Presumably, such estimates would be less biased than those we obtained from Table 7.5. Classification of accidents by driver’s sex, car damage, injury and belt use obtained from police reports (p), and injury and belt use obtained from followup (f)
Table 7.4, because they would be less prone to misclassification error. On the other hand, they would have greater variability because they would be based on a much smaller sample. A more effective approach would be to combine the data from Tables 7.4 and 7.5 and analyze them as a sixvariable dataset with two of the variables partially missing. Combining the two sources would allow us to make use of the policereport data for all 81880 accidents, but would calibrate them to correct for occasional misclassification errors in keeping with the error rates seen in the followup study. In other words, a combined analysis would allow the policereport data to serve as a proxy for the followup data among the
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initial 80084 cases, taking into account the fact that the correlation between the two data sources is less than perfect. The sixvariable combined dataset has a monotone pattern, with followup belt use and injury missing for 97.8% of the cases. Because of the high rate of missingness for these two variables, the EM and ordinary data augmentation algorithms described in Section 7.3 converge very slowly. To illustrate, we ran a single chain of ordinary data augmentation for 5000 steps beginning from the
Figure 7.8. Sample ACFs for the worst linear function of θ, estimated from 5000 iterations of ordinary data augmentation, with dashed lines indicating approximate critical values for testing ρk=ρk+1=⋅⋅⋅=0.
ML estimate using the Jeffreys prior (all hyperparameters equal to 1/2), and monitored the worst linear function of θ as estimated from the trajectory of EM (Section 4.4.3). The sample autocorrelation function for this parameter, plotted in Figure 7.8, reveals extreme longrange dependence. Monotone data augmentation, however, converges in a single step because the observed data are precisely monotone. A sequence of 0 values generated by monotone data augmentation will be
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an actual independent sample from the observeddata posterior P(θYobs). Using monotone data augmentation, we simulated 1000 independent draws of θ from the observeddata posterior under the Jeffreys prior, and calculated the odds ratios relating seatbelt use to driver injury (both from the police reports and from the followup reports) within each of the four sexbydamage cells. Boxplots of these simulated odds ratios are shown in Figure 7.9. The odds ratios based on the police reports are highly concentrated to the left of one; the beneficial effects of seatbelts thus appear to be ’statistically significant’ if we ignore the problem of misclassification. The odds ratios based on followup data, however, are much more dispersed, with all of the distributions straddling one; when misclassification errors are taken into account, the evidence that seatbelts reduce the risk of injury is no longer overwhelming. Simulated posterior means, 95% interval estimates and pvalues for these odds ratios are shown in Table 7.6. The pvalues are simply the proportions of simulated odds ratios exceeding one; they are appropriate for testing whether a given odds ratio is one, versus the onesided alternative that it is less than one. Because the policereport versions of belt use and injury are
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Figure 7.9. Boxplots of 1000 simulated odds ratios showing the relationship of seatbelt use and injury within classes of damage and sex, both from police reports and from followup data.
highly correlated with the followup versions, one might think that the rates of missing information for the followup variables should be much smaller than their actual missingness rates (98%). The fact that the followupbased intervals are so much wider than the policebased intervals, however, indicates that rates of missing information for these variables are still quite high. The main reason for this is the complexity of the saturated multinomial model. The saturated model allows for a full sixway association among the variables. The misclassification mechanism is described by the fourway table that relates the followup versions of beltuse and injury to the police versions. The saturated model estimates a full fourway association in this table; moreover, it allows the fourway association to vary freely across the four sexbydamage cells. It is apparent that some of these highorder associations are poorly estimated, because the data in some parts of Table 7.5 are sparse. We will address this issue in Chapter 8 by applying models that are more parsimonious.
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Table 7.6. Simulated posterior means, 95% intervals and pvalues for odds ratios from 1000 iterations of monotone data augmentation
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CHAPTER 8
Loglinear Models
8.1 Introduction In Chapter 7 we examined methods based on the saturated multinomial model. That model was quite general, allowing the associations among the categorical variables to be arbitrarily complex. In many realistic examples, however, unless the number of variables is very small, the observed data cannot support such complexity. This chapter presents methods for a flexible class of models which allows the associations among variables to be simplified. Loglinear models have been used extensively, particularly in the social sciences, for almost two decades. In loglinear models, the cell probabilities for the crossclassified contingency table are decomposed into multiplicative effects for each variable and for the associations among them. Eliminating certain terms from this decomposition imposes equality constraints on odds ratios in the crossclassified table. A large part of this chapter is devoted to loglinear modeling with complete data, in particular, to the classical estimation technique of iterative proportional fitting (IPF) and a new simulation algorithm known as Bayesian IPF. These methods, which will be unfamiliar to many readers, are easily extended to calculate ML estimates and simulate posterior draws of parameters and missing data. Sections 8.3 and 8.4 concentrate on IPF and Bayesian IPF, respectively, and extensions to incompletedata problems are presented in Section 8.5.
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8.2 Overview of loglinear models 8.2.1 Definition Suppose x = (x 1, x2,..., x D ) is a contingency table having a multinomial distribution, χ  θ ~ M (n, θ),
(8.1)
where the cell probabilities θ = (θ1, θ2 ,..., θ D ) lie within the simplex
{
}
Θ = θ : θd ≥ for all d and ∑ dD=1 θ d = 1 . A loglinear model does not alter the distributional assumption (8.1), but imposes further constraints on the elements of θ. Let
ηd = logθd, d=1, 2,..., D and
η = (η1, η2, η ., D)T . In the most general sense, a loglinear model is any constraint of the form
η = Mλ
(8.2)
where λ is an r × 1 parameter vector and M is a fixed and known D × r design matrix. Thus, in addition to requiring that the elements of θ sum to one, we also require η=log θ to lie in the linear subspace spanned by the columns of M . The meaning of the elements of λ will depend on the coding method used in M . In typical applications of loglinear modeling, x represents a crossclassification of sample units by categorical variables Y1, Y 2,..., Yp, and M is a design matrix of the type used in the analysis of variance (ANOVA) for
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factorial experiments; each variable Yj represents a ’factor,’ and the elements of λ represent the ’main effects’ and ’interactions’ associated with the factors. Models for three categorical variables For expository purposes, let us temporarily assume that there are only three categorical variables (p = 3). This assumption is purely a matter of convenience, and all results will immediately generalize to any number of variables. Also, we will temporarily switch to a notation more consistent with that of standard texts on categorical data (e.g. Agresti, 1990); in later sections, we will return to the notation developed in Chapter 7. Suppose we have three categorical variables: A with levels i = 1, 2,..., I; B with levels j = 1, 2,..., J; C with levels k = 1, 2,..., K. Let xijk denote the number of sample units for which we observe A = i, B = j, C = k. Let θ ijk = P( A = i, B = j , C = k ) and ηijk = log θ ijk . Finally, let ’+’ in place of a subscript denote summation over that subscript, as in J
I
x+ jk =
∑
xijk and θ i+ + =
i=1
K
∑ ∑ θ ijk . j =1 k =1
The total sample size is n=x+++. As in a factorial ANOVA model, we can decompose nijk into additive terms corresponding to the ’main effects’ and ’interactions’ ofA, B, and C, AC ABC ηijk = λ0 + λ iA + λBj + λCk + λ ijAB + λ ik + λBC jk + λ ijk ,
(8.3)
where for identifiably the λ terms are constrained to sum to zero over any subscript,
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I
I
∑
λ iA
= 0,
∑
J
λ ijAB
i=1
i=1
=
∑ λijAB = 0 ,
(8.4)
j =1
and so on. To see how this relates to the general specification (8.2), consider the special case where I = J = K =2; taking
η111 1 η211 1 η121 1 η 1 221 η = η , M = 1 112 1 η212 1 η122 1 η 222
1 −1 1 −1 1 −1 1 −1
1 1 −1 −1 1 1 −1 −1
1 1 1 1 −1 −1 −1 −1
1 −1 −1 1 1 −1 −1 1
1 −1 1 −1 −1 1 −1 1
1 −1 −1 1 −1 1 1 −1
1 1 −1 −1 −1 −1 1 1
yields
[
AB AC BC ABC λ = λ0 , λ1A , λ1B , λ1C , λ11 , λ11 , λ11 , λ111
]
T
,
and the other λ terms follow from the identifiably constraints, AB AB AB AB , λ1A = − λ2A , λ11 = − λ12 = − λ21 = λ22
and so on. In many respects the loglinear model (8.3) is like the classical linear model for a factorial experiment. There are two important differences, however, that distinguish the loglinear model from its linear counterpart. First, the term λ0, which appears to be like the ’grand mean,’ is not a free parameter but a normalizing constant chosen to make the cell probabilities sum to one,
{
(
)}
ABC λ0 = − log Σ ijk exp λ iA + λBj + L + λ ijk .
Second, the linear equation (8.3) does riot represent the mean of a response variable given A = i, B = j, C = k; rather, it represents the logprobability of the event A = i, B = j, C = k itself. The loglinear model is not a regression model
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describing the effects of A, B and C on an additional response variable, but a true multivariate model describing the relationships among the variables A, B and C . Thus the meaning of the λ terms is quite different from the usual interpretation of main effects and interactions in a linear model. For example, the set of terms λ AB = λ ijAB describes
{ }
the association between A and B, not their interaction with respect to a third variable. The terms in λAB are essentially the logodds ratios describing the association between A and B, ABC and the terms λ ABC = λ ijk are the differences in logodds
{
}
ratios describing how the association between any two variables varies across levels of the third. For details on the exact correspondence between the λ terms and logodds ratios, see Bishop, Fienberg and Holland (1975) or Agresti (1990). 8.2.2 Eliminating associations The number of free parameters in the loglinear model (8.3) can be counted in the same manner as the number of degrees of freedom in a factorial ANOVA.
Notice that the total number of free parameters in (8.3), IJK1, is the same as in the saturated multinomial; hence (8.3) is nothing more than a reparameterization of the saturated model, with the cell probabilities θ reexpressed in terms of the loglinear coefficients λ. The loglinear representation has a great advantage, however, in that it allows us to selectively eliminate associations among variables by setting groups of λ terms to zero. Suppose we set all the terms in AC λ AB = λ ijAB , λ AC = λ ik , λBC = λBC a n d jk ,
{ }
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{ }
{ }
{
ABC λ ABC = λ ijk written as
} to zero. The loglinear model can then be (
)
θ ijk ∝ exp λ iA + λBj + λCk , which implies that A, B and C are mutually independent,
θ ijk = P( A = i)P( B = j )P(C = k ) Setting λBC = λ AC = λ ABC = 0 but allowing λAB to vary leads to
θ ijk = P( A = i, B = j )P(C = k ) , which means that A and B may be related but requires them to be jointly independent of C. Setting λ AB = λ ABC = 0 gives θ ijk = P( A = i  C = k )P( B = j  C = k )P(C = k ) , which means that A and B are conditionally independent given C. Finally, setting λABC=0 results in a model of homogeneous association. This model does not imply any form of independence or conditional independence, but has the property that the association between any two variables (in terms of odds ratios) is constant across levels of the third. Hierarchical models In most applications of loglinear modeling, it would not make sense to specify a model that contains an association but omits a main effect. A model that includes λAB but omits λA allows A to be related to B, but requires the average logprobability across levels of B to be the same within every level of A. Under ordinary circumstances one would not expect this to happen except by chance. Similarly, it would rarely make sense to fit a model that contains the threeway association λABC but omits one or more of the twoway associations λAB,
λAC or λBC.
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A loglinear model is said to be hierarchical if omitting a λ term implies that all higherorder associations containing that term are omitted as well; for example, if setting λAB=0 requires that we also set λABC = 0. Putting it another way, a model is hierarchical if no association is present unless all lowerorder terms within that association are also present. Thus a hierarchical model containing λABC must also contain λ A , λB,
λC, λAB, λAC and λBC. The class of hierarchical models includes models of independence and conditional independence, as well as some other models that may be of interest, e.g. the model of homogeneous association in a threeway table (λABC=0). Nonhierarchical models, however, rarely correspond to sensible hypotheses about the underlying categorical variables. For the remainder of this book, we will restrict our attention to hierarchical models only. 8.2.3 Sufficient statistics The loglikelihood for the saturated model in terms of the cell probabilities θ is
l(θ  x) = Σ ijk xijk log θ ijk . When expressed in terms of the loglinear coefficients, the loglikelihood becomes
l(λ  x) = Σ ijk = nλ0
(
ABC xijk λ0 + λ iA + L + λ ijk
)
+ Σ i xi+ + λ iA + Σ j x + j + λBj AC + Σ k x+ + k λCk + Σ ij xij + λ ijAB + Σ ik xi+ k λ ik ABC + Σ jk x+ jk λBC jk + Σ ijk xijk λ ijk
{ }
{ } and so
We will use x A = { xi+ + }, x AB = xij + , x ABC = xijk
on to denote the marginal frequencies that result when units are crossclassified by subsets of variables. Following Bishop, Fienberg and Holland (1975), we will call these marginal tables configurations. Because the configurations x A , x B , xC,
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xAB, xAC, and xBC can be obtained by summing the elements of xABC the loglikelihood for the saturated model is a linear function of the configuration xABC. If we simplify the model by eliminating some of the λ terms, the corresponding configurations drop out of the loglikelihood. For example, if we set λBC = λ ABC = 0 the loglikelihood becomes
l(λ  x) = nλ0 + Σ i xi+ + λ iA + Σ j x+ j + λBj AC + Σ k x+ + k λCk + Σ ij xij + λ ijAB + Σ ik xi+ k λ ik
Because x A , x B and xC follow from xAB and xAC, the latter two configurations constitute a minimal set of sufficient statistics for this model. If a model is hierarchical, then the configuration for any set of variables present in the model can be derived from the highestorder configuration containing that set. Consequently, the configurations for these highestorder terms form a minimal set of sufficient statistics. We will call these the sufficient configurations. It has become standard practice to identify loglinear models by their sufficient configurations. For example, the model Table 8.1. Hierarchical loglinear models for three categorical variables
λABC=0 can be denoted by (xAB, xAC, xBC) or, more simply, (AB, AC, BC).
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8.2.4 Model interpretation The hierarchical models that can be fitted to a threevariable dataset are listed in Table 8.1, along with their sufficient configurations. This list does not include any model that omits one or more of the main effects λA , λ B , λC . Setting a main effect to zero is equivalent to saying that the marginal distribution of the corresponding variable is uniform across its levels, a hypothesis which is rarely of interest. Models in four or more dimensions are interpreted in a similar fashion. For example: (AB, CD) means that A and B are jointly independent of C and D; (ABC, BCD) means that A and D are conditionally independent given (B, C); (AB, AC, AD, BC, BD, CD) means that the odds ratios for any two variables are constant across levels of the other two; and (ABC, ABD, ACD, BCD) means that the associations among any three variables are constant across levels of the fourth. Correspondence to logit models If one of the variables is regarded as a response and the others are regarded as potential predictors, then certain loglinear models are equivalent to standard logistic regression or logit models (Goodman, 1970). Consider the saturated model for A, B and C, where C is a binary variable considered to be a response, and let π ij = P(C = 1  A = i, B = j ) . The logit model for predicting the probability of C = 1 from A and B can be written as
π ij log 1 − π ij
θ ij1  θ ij + = log θ ij2  θ ij + , = ηij1 − ηij2 ,
where nijk = log θ ijk . But notice that
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ηij1 − ηij2
=
(λ1C − λC2 ) + (λiAC1 − λiAC2 ) BC ABC ABC +(λBC j1 − λ j2 ) + (λ ij1 − λ ij2 )
BC ABC = 2λ1C + 2λ iAC 1 + 2λ j1 + 2λ ij1
so this logit model is of the form
π ij A B AB log = β0 + β i + β j + β ij , 1 − π ij
(8.5)
where the coefficients satisfy
Σ iβ iA = Σ j β Bj = Σ iβ ijAB = Σ j β ijAB = 0 . Thus, the saturated model (ABC) implies a standard logit model for C that includes main effects for A and B as well as the AB interaction. Notice that if we set λ ABC = 0 in the loglinear model, (8.5) becomes
π ij B A log = β0 + β i + β j 1 − π ij a logit model with main effects only. Setting λ ABC = λ AC = 0 and λ ABC = λBC = 0 removes the effects of A and B, respectively, and λ ABC = λ AC = λBC = 0 produces the null model
π ij log = β0 1 − π ij Omitting λ AB, however, would require A and B to be conditionally independent, an assumption not found in the standard logit model. The standard logit model, like other regression models, makes no assumptions about the predictors; it allows their joint distribution to be arbitrary.
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The relationships between loglinear models and logit models for three categorical variables are summarized in Table 8.2. In general, Table 8.2. Loglinear and corresponding logit models for three categorical variables
a loglinear model is equivalent to a standard logit model provided that it includes all potential associations among the variables considered to be predictors. A twoway association between a response and a predictor in the loglinear model introduces a main effect for that predictor in the logit model; a threeway association between a response and two predictors introduces an interaction between the two predictors; and so on. The response variable need not be binary; if it has more than two categories, the loglinear model implies a generalized logit model for an unordered multinomial response (e.g. Agresti, 1990).
8.3 Likelihoodbased inference with complete data 8.3.1 Maximumlikelihood estimation To derive the ML estimates for a loglinear model, we could try to differentiate the loglikelihood with respect to some set of free parameters and set the resulting expressions to zero. However, as demonstrated by Birch (1963), we may also apply the method that leads to ML estimates for any regular exponential family model: solve the system of equations that results when the minimal sufficient statistics are set equal to their expectations (Section 3.2.1). In many cases, this system
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can be solved immediately to yield the ML estimates for the cell probabilities θ. For example, consider the saturated model (ABC). Setting the elements of the sufficient configuration xABC equal to their expectations E xijk  θ = nθ ijk produces
(
)
θˆijk = xijk / n for all i, j and k. For the model (AB, C), the moment equations are
xij + = nθˆij + ,
x+ + k = nθˆ+ + k . But because this model implies θ ijk = θ ij +θ + + k , we obtain
(
)
θˆijk = xij + x+ + k / n2 . The model (AB, BC) gives
xij + = nθˆij + , x+ jk = nθˆ+ jk , and because this model implies
θ ij + θ + jk θ ijk = θ + j + , θ+ j+ θ+ j+ The model (AB, AC, AC), however, produces
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xij + xi+ k x+ jk
= nθˆij + , = nθˆi+ k , = nθˆ , + jk
a system for which there is no closedform solution. It turns out that (AB, AC, BC) is the only hierarchical model in three dimensions for which the ML estimates cannot be written in closed form. In four or more dimensions, however, there are many more models for which this is so. When the moment equations do not yield explicit ML estimates for θ, they may be solved numerically by the method of iterative proportional fitting. 8.3.2 Iterative proportional fitting Iterative proportional fitting (IPF) is a simple and intuitive method for solving the moment equations. First, start with an arbitrary value of θ that satisfies the loglinear constraints, typically a uniform table (all cell probabilities equal). Then proportionately adjust the elements of θ to satisfy the moment equations for a single configuration. Do this for each sufficient configuration in turn, and repeat the entire process until the elements of θ stabilize. For example, consider the model (AB, AC, BC). Given the current estimate θ(τ), IPF updates it as follows: ( θ ijk
( t + 0 / 3) xij + / n = θ ijk t + 0 / 3 for all i, j, k; ) θ( ij +
(8.6)
( θ ijk
( t +1 / 3) xi+ k / n for all i, j, k; = θ ijk ( t +1 / 3) θ i+ k
(8.7)
t +1 / 3)
t + 2 / 3)
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( θ ijk
t + 3 / 3)
( = θ ijk
t + 2 / 3)
x + jk / n for all i, j, k. θ ( t + 2 / 3) + jk
(8.8)
Notice that θ(τ+1/3) satisfies the required conditions for xAB but not necessarily those for xAC or xBC. Similarly, θ(τ+2/3) satisfies those for x A C but not necessarily for xAB or x B C, and θ(τ+3/3) satisfies those for xBC but not necessarily xAB or xBC. Repeating θ which the cycle (8.6)(8.8) produces a sequence θ (1) (2) ∞) ( = θˆ satisfying all three sets of converges to a value θ moment equations simultaneously; this is the unique ML estimate of θ. This IPF algorithm immediately generalizes to loglinear models of any dimension. When the moment equations can be solved in closed form, IPF may still be used, and it typically converges to the correct ML estimates in a single cycle. In other cases, IPF exhibits linear convergence near the mode. Proofs of the convergence of IPF are given by Bishop, Fienberg and Holland (1975) and their references. Because IPF operates on the cell probabilities θ , it does not automatically yield explicit estimates of the loglinear coefficients λ. Estimates of λ may be obtained in a variety of ways. One simple way is to define a fullrank design matrix M such that log logθ=Μλ, and calculate the ordinary leastsquares estimates
(
λˆ = M T M
)
−1
M T log θˆ
using standard regression software. Except for rounding errors, the regression model will have perfect fit because log θˆ is required to lie in the space spanned by the columns of M. Another way to obtain the elements of λ is to express them as linear contrasts of the elements of logθ ; see Bishop, Fienberg and Holland (1975) for details.
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Comparison with other methods IPF has been in existence for more than half a century. Deming and Stephan (1940) discussed raking, a method of proportionately adjusting survey data to make the observed distributions of certain variables agree with census totals; their algorithm is essentially equivalent to IPF. Early work on loglinear modeling (e.g. Bishop, Fienberg and Holland, 1975) relied almost exclusively on IPF, but more recent books emphasize other methods. Software packages capable of fitting loglinear models typically use NewtonRaphson or Fisher scoring (e.g. Agresti, 1990). These methods tend to be quicker than IPF because their convergence behavior is quadratic; moreover, they provide asymptotic standard errors as an automatic byproduct, because they make use of the loglikelihood function’s second derivatives. Standard errors can be obtained with IPF, but computing them requires additional formulas that are not an integral part of the fitting algorithm. Yet IPF maintains some advantages because of its simplicity and computational stability. In this text, we focus on IPF because of its intimate relationships to ECM and the simulation algorithms described later in this chapter. Random and structural zeroes If the contingency table contains random zeroes, θˆ may lie on the boundary of its parameter space with estimated probabilities of zero in one or more cells. When this occurs, some of the IPF equations may be undefined at θˆ because they may involve division by zero. This difficulty is easily overcome by the following modification: if any probability falls below a small positive constant c, set it to zero and omit that cell from further iterations. Structural zeroes, whose cell probabilities are taken to be zero a priori, can also be handled quite easily. The usual way of handling them is to omit them from the model and assume that the loglinear specification (8.2) holds for the remaining cells. With IPF, we simply choose a starting value of θˆ that has zeroes in the structuralzero cells and uniform values
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elsewhere. Because 0 < ∈, the estimated probabilities for these cells will remain at zero for all iterations. An implementation in pseudocode Returning now to the general notation of Chapter 7, suppose that Y1, Y2,..., Yp are categorical variables recorded for a sample of n units, where Yj takes values 1, 2,..., d j. Let y =
y
be (y1,..., yp) denote a generic realization of (Y1,..., Y p); let the set of all possible values of y; and let x y and θy be the frequency and cell probability, respectively, associated with cell y of the pdimensional
Figure 8.1. Single cycle of iterative proportional fitting.
crossclassified table. Let C be a subset of {1, 2,..., p} identifying a generic configuration; for example, C = {1, 3} indicates the configuration Y1Y3. For brevity, we will also use C to denote the function that extracts from y = (y 1,..., yp) the elements corresponding to the configuration C, for example, if C = {2, 4} then C = (y2, y4). Finally, a generic marginal count
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within C and its corresponding marginal probability will be denoted by
xC ( y ) =
∑ x y and θC( y) = ∑ θ y ,
C ′( y )
C ′( y )
respectively, where C is the complement of C. Pseudocode for a general implementation of IPF is shown in Figure 8.1. Given the observed counts in the workspace x and the current parameter θ (t) in θ , this code performs one cycle of IPF, overwriting θ (t) with the updated cell means nθ(t+1). Prior to execution, the values stored in θ need not sum to one; we will obtain the same result if this workspace contains c θ (t) for any constant c. Therefore, the updated cell means nθ(t+1) from one cycle may be used directly as input to the next cycle; there is no need to rescale them to sum to one at each cycle. 8.3.3 Hypothesis testing and goodness of fit A loglinear model’s quality of fit may be assessed by its deviance. The deviance, typically denoted by G 2, is the likelihoodratio statistic for testing the current model against the alternative of a saturated model,
G2 = 2
∑
y ∈y
{
where θˆ = θ y : y ∈
x y log
xy nθˆ
(8.9)
y
y *} is the maximizer of the likelihood model, and y * is the set of all cells
under the current excluding structural zeroes. The deviance is asymptotically equivalent to the wellknown goodnessoffit statistic due to Pearson,
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( x y − nθˆ y )
2
X = 2
∑
y ∈y
(8.10)
nθˆ y
If the sample size is sufficiently large, G2 and X 2 are 2 under the null hypothesis distributed approximately as xdf that the current model is true, where df is equal to the difference in the number of free parameters in the saturated and current models. An approximate pvalue for testing the current model against a general alternative is thus 2 2 P xdf ≥ G2 or P xdf ≥ X 2 . The chisquare approximation
(
)
(
)
for these goodnessoffit tests is traditionally regarded as accurate if nθˆ y ≥ 5 for all y ∈ * ; in addition, some empirical studies have shown that it may be reasonably accurate if a small proportion of the cells have nθˆ y as small as 2 or even 1; see Agresti (1990, pp. 246247) for further details. The G 2 and X2 statistics also provide a basis for general comparisons between models that are nested. Suppose we want to test the null hypothesis that Model A is true against the alternative that Model B is true, where Model A is a special case of Model B. Then
y
∆G2 = G2 for Model A — G 2 for Model B and ∆X2 = X2 for Model A — X 2 for Model B are distributed approximately as chisquare with degrees of freedom equal to ∆df = df for Model A — dffor Model B Large values of ∆G2 or ∆X2 indicate that Model B fits the data substantially better than Model A. Asymptotic arguments suggest that the chisquare approximations for ∆G2 and ∆X 2
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may be quite accurate even when the approximations for the individual goodnessoffit statistics for Model A and Model B are poor. Consequently, for ∆G 2 and ∆X2 can be useful even with sparse tables, provided that (a) the number of observations is large relative to ∆df, and (b) the observed frequencies are of approximately the same order of magnitude (Haberman, 1977). Effects of random zeroes and boundary estimates Notice that the presence of a random zero (xy = 0) will cause G2 to be undefined. This problem can be overcome by taking 0 log 0 to be 0. When a pattern of random zeroes causes the ML estimate to lie on the boundary (θy = 0 for some y), however, neither G 2 nor X 2 can be calculated directly. In these situations, it is customary to omit the cells with zero estimates from consideration and adjust the degrees of freedom to reflect the fact that some parameters may not be estimable. Rules for adjusting the degrees of freedom (e.g. Bishop, Fienberg and Holland, 1975) are quite complicated and are difficult to implement in generalpurpose computer code. Users of generalpurpose software for loglinear modeling should be wary of these adjustments, because situations exist for which nearly every popular software package gives misleading results (Clogg et al., 1991). When sparseness in the data table x leads to boundary estimates, a simpler and more reliable procedure is to introduce a small amount of prior information to smooth the data and move θˆ away from the boundary; see Clogg et al. (1991) and Section 8.4.2 below. 8.3.4 Example: misclassification of seatbelt use and injury Recall the data of Table 7.5 from a followup study on misclassification error in seatbelt use and injury in automobile accidents. Hochberg (1977) and Chen (1989) investigated loglinear models for the six dichotomous variables: Code
D S
B1
Variable car damage (1=low, 2=high) driver's sex (1=male, 2=female) belt use, police report (1=no, 2=yes)
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I1 B2 I2
injury, police report (1=no, 2=yes) belt use, followup study (1=no, 2=yes) injury, followup study (1=no, 2=yes)
Table 8.3. Goodnessoffit statistics for nine loglinear models fitted to the automobile accident followup data
Instead of working with these six variables directly, let us consider models for D, S, B2, I2 and the two error indicators EB = 1 if B1 = B2, 0 otherwise; E1 = 1 if I1 = I2, 0 otherwise.
Working with D, S, B2, I2, E B and EI rather than the six original variables may result in a model for the misclassification mechanism that is easier to interpret, because the associations between the underlying ’true’ state of an accident DSB 2 I2 and the error indicators EBEI may be somewhat simpler than the associations between DSB2I2 and the police report B 1 I1. Regarding E B and EI as response variables and D, S, B2, and I 2 as potential predictors, perhaps the simplest loglinear model worth considering is (DSB2I2, EBEI), which states that the bivariate response is unrelated to the predictors. Goodnessoffit statistics for this model and eight other loglinear models are shown in Table 8.3. None of the nine models shown in Table 8.3 produced ML estimates on the boundary of the parameter space, but all of them had estimated expected counts falling below 1.0 for some cells; pvalues based on the chisquare approximations for G 2 and X2 are not shown because they are not trustworthy. Chisquare approximations for comparisons among these models are probably more accurate, however, and results from hypothesis tests for various pairs of nested models are shown in Table 8.4. Model 1, the null model of no relationships between the response and predictors, appears to fit the data
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very poorly. The fit improves dramatically when the actual belt use/injury status B2I2 is allowed to influ Table 8.4. Hypothesis tests for various pairs of nested models
ence the response (Model 2). In addition, the data provide fairly strong evidence for the associations D EB, DEI, and perhaps SEI. Among these nine, the simplest models that capture the essential relationships between the response and the predictors appear to be Models 5 and 8.
8.4 Bayesian inference with complete data 8.4.1 Prior distributions for loglinear models In our work with the saturated multinomial model in the last chapter, we adopted the simple Dirichlet prior distribution θ ~ D(α ) , with the elements of a typically chosen to be equal. This prior is ’naive’ in the sense that it treats the cell probabilities in an unordered fashion, i.e. it does not describe the special structure that exists in a crossclassified contingency table. This was not regarded as a serious drawback, because the saturated multinomial model does not make use of this crossclassified structure either. The fundamental quality of loglinear models, however, is that they take this structure into account. Many alternative types of prior distributions have been proposed to sensibly incorporate prior information about the structure of a loglinear model. Bishop, Fienberg and Holland
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(1975) decomposed the Dirichlet hyperparameters a into ’main effects’ and ’associations’ in a loglinear fashion. Good (1967) proposed a secondstage prior distribution on a, resulting in a mixture of Dirichlet priors that can potentially reflect a crossclassified structure. Several authors have applied normal prior distributions to the loglinear coefficients λ; variations of this approach are discussed by Good (1956), Leonard (1975), Laird (1978), and Knuiman and Speed (1988). The normal priors, although conceptually attractive, lead to nonnormal posteriors that are computationally more difficult to handle than the Dirichlet. The constrained Dirichlet prior For our purposes, it will be convenient to adopt a prior distribution that has the same functional form as the Dirichlet, but which requires the parameters to satisfy the constraints imposed by a loglinear model. Let M denote the design matrix for a loglinear model
log θ = Mλ
(8.11)
{
y}
and let Θ M denote the set of all parameters θ = θ y : y ∈ that lie in the simplex and satisfy (8.11) for some λ. Let us take the prior density for θ to be
π (θ ) ∝
∏θ y
α y −1
y ∈y
for θ ∈Θ M and zero elsewhere. We will call this the constrained Dirichlet prior with hyperparameter α = αy : y∈ . This is not a Dirichlet distribution per se,
{
y}
but the conditional distribution of θ ~ D(α ) given that the event θ ∈Θ M has occurred. The normalizing constant
θy ∫Θ ∏ y ∈y
α y −1
M
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dθ
is not generally tractable, but this will not be a problem because in the algorithms to follow this integral will not be explicitly evaluated. The advantage of the constrained Dirichlet prior is that it retains the same functional form as the multinomial likelihood and thus forms a conjugate class; the posterior distribution of θ given the contingency table x is another constrained Dirichlet with updated hyperparameters α ′ = α + x . A potential disadvantage is that this prior makes the strong assumption that the given loglinear model is true; it assigns zero probability to values of θ not satisfying (8.11). Just as in likelihoodbased inference, however, it will be possible to examine the adequacy of a model by performing goodnessoffit tests against alternative models that are more general. 8.4.2 Inference using posterior modes Under the constrained Dirichlet prior, the completedata posterior density for θ is
P(θ  x) ∝
∏θ y ∈y
xy + α y − 1 y
(8.12)
for θ ∈Θ M and zero elsewhere. Notice that this is equivalent to the likelihood function for θ given a modified contingency table with cell counts x ′y = x y + α y − 1 . Any algorithm that computes ML estimates for loglinear models can thus be trivially modified to find posterior modes for θ as well; all we need to do is to augment each cell count x y by the amount α y − 1 . In particular, the IPF algorithm of Section 8.3.2 will find posterior modes if we simply replace each x y by x ′y = x y + α y − 1 . It might seem natural to call this modified IPF algorithm ’Bayesian IPF.’ However, we will reserve that name for another algorithm, to be discussed shortly, for simulating random draws from a constrained Dirichlet distribution.
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Notice that the posterior mode for θ is identical to the ML estimate under the uniform prior α = (1, 1,..., 1). A prior α = (c, c,..., c) for some c > 1 has a ’flattening’ effect; the posterior mode under this prior will represent a compromise between the ML estimate and a uniform table in which all cell probabilities are equal (Section 7.2.5). Flattening priors can be especially useful for ensuring that the mode lies within the interior of the parameter space Θ M , avoiding complications that arise with sparse data when ML estimates lie on the boundary. Posterior modes and goodness of fit The goodnessoffit statistics G 2 and X2 defined in Section 8.3.3 can be used with posterior modes. One possibility is to use the same expressions (8.9)(8. 10) and simply replace the ML estimate θˆ with a posterior mode. In many situations, however, it will be more natural to work with statistics that are based on the augmented cell counts x ′y = x y + α y − 1 . For a
{
y}
given loglinear model, let θ˜ = θ˜ y : y ∈ * represent the posterior mode under the constrained Dirichlet prior with
y
hyperparameter α where * is the set of all cells excluding structural zeroes. The posterior mode under the saturated model and the unrestricted D(α) prior occurs at
θy =
x ′y n′
, n′ =
∑ x′y .
(8.13)
y ∈ y*
The statistic
G2 = 2
x′
∑ x′y log n′θyˆ y
(8.14)
y ∈ y*
represents twice the increase in the logposterior density as we move from the mode under the current model to the mode under the saturated model. Like its likelihoodbased
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counterpart, the modified statistic (8.14) is approximately 2 over repeated samples, where df is the distributed as xdf difference in the number of free parameters under the two models. This result holds because as the sample size grows, the cell counts x y become appreciably larger than the hyperparameters αy, and the influence of the prior becomes negligible. The analogue of (8.14) corresponding to Pearson’s goodnessoffit statistic is
( x′y − n′θ˜ y )
2
X2 =
∑
y ∈ y*
(8.15)
n′θ˜ y
One attractive feature of (8.14) and (8.15) is that these two statistics are easy to calculate; they will be generated automatically by standard software for loglinear modeling if the cell counts xy are replaced by x ′y . Moreover, if all the hyperparameters αy are greater than one, both θˆ and the unrestricted mode (8.13) are guaranteed to lie in the interior of the parameter space, and there is no need to worry about adjusting df for estimates on the boundary (Clogg et al., 1991). 8.4.3 Inference by Bayesian IPF Here we present a clever but still relatively unknown technique for simulating random draws from a constrained Dirichlet posterior (8.12). This iterative method, first presented by Gelman et al. (1995), bears a striking resemblance to iterative proportional fitting and has thus been named Bayesian IPF. For simplicity, we describe Bayesian IPF for three categorical variables A, B and C under the model of homogeneous association (AB, AC, BC). Following the notation of Section 8.2, let xijk, θijk and αijk denote the observed frequency, probability and prior hyperparameter, respectively, corresponding to the event (A = i, B = j, C = k). Let θ(t) denote
{ }
the simulated value of the parameter θ = θ ijk at cycle t. Bayesian IPF updates the parameter in three steps: first,
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( θ ijk
t +1 / 3)
( t + 0 / 3) gij + / g+ + + = θ ijk for all i, j, k, θ ( t + 0 / 3) ij +
(8.16)
where the gij+ are independent random variates drawn from standard gamma distributions with shape parameters
α ij′ + =
∑ (α ijk + xijk ) k
(Section 7.2.3), and g+ + + = Σ ij gij + is their sum; second, ( θ ijk
t + 2 / 3)
( t +1 / 3) gi+ k / g+ + + for all i, j, k, = θ ijk ( t +1 / 3) θ i+ k
(8.17)
where the g i+k are standard gamma variates with shape parameters
α i′+ k =
∑ (α ijk + xijk ) j
drawn independently of those in (8.16), and g+ + + = Σ ik gi+ k is the new sum; and third, ( θ ijk
t + 3 / 3)
( t + 2 / 3) g+ jk / g+ + + = θ ijk for all i, j, k, θ ( t + 2 / 3) + jk
(8.18)
where the g +jk are standard gamma variates with shape parameters
α +′ jk =
∑ (α ijk + xijk ) i
drawn independently of those in the first two steps, and g+ + + = Σ jk g+ jk is the new sum. Given any starting value that satisfies the constraints of the loglinear model (AB, AC, BC), these three steps (8.16)(8.18) define a Markov chain
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{θ ( ) : t = 1, 2,...} t
which converges in distribution to the
constrained Dirichlet posterior with hyperparameters α ijk ′ = α ijk + xijk ; a heuristic argument for this result will be given below. Thus, for a suitably large value of t, we can regard θ(t) as a random draw from the correct posterior P(θ  x). The subsequent output stream θ ( t +1) , θ ( t + 2) ,... represents a dependent sample from P(θ  x) which can be summarized by any of the methods described in Chapter 4. Relationship to conventional IPF It is easy to see the relationship between this algorithm and conventional IPF. Consider the first step of conventional IPF under the constrained Dirichlet prior, ( θ ijk
t +1 / 3)
θ ( t + 0 / 3) x ′ ij + ijk = t +0 / 3 for all i, j, k, ( ) θ n′ ij +
(8.19)
where xijk ′ = xijk + α ijk − 1 and n′ = Σ ijk xijk ′ . The first term in parentheses on the righthand side of (8.19) is the estimate of the conditional probability of C = k given (A = i, B = j) from the previous step. The second term in parentheses is the posterior mode of the marginal probability of (A = i, B = j). Thus (8.19) represents the marriage between an old estimate of P(C = k  A = i, B = j ) and a new estimate of P(A = i, B = j). Now consider the first step of Bayesian IPF, ( θ ijk
t +1 / 3)
θ ( t + 0 / 3) g ijk ij + = t +0 / 3 for all i, j, k. ) g θ( + + + ij +
(8.20)
The first term in (8.20) is the old simulated value of P(C = k  A = i, B = j ) . The second term, gij+/g+++, simulates new values of the marginal probabilities θ ij + Σ k θ ijk from the
{ }
Dirichlet distribution with parameters α ij′ + , the marginal
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posterior distribution of θ ij + given the data. Thus (8.19) represents the marriage between an old draw of P(C = k  A = i, B = j ) and a new draw of P( A = i, B = J ) . The second and third steps of Bayesian IPF continue in a similar vein, updating the parameters by taking new random draws of the marginal probabilities P( A = i, C = k ) and P( B = j , C = k ) , respectively. An implementation in pseudocode Like its conventional counterpart, Bayesian IPF generalizes immediately to hierarchical loglinear models for any number of variables. Pseudocode for a general implementation of Bayesian IPF is shown in Figure 8.2. This code, which uses the same notation as that of Figure 8.1, performs one cycle of Bayesian IPF and overwrites θ with its updated value. The starting value of θ must lie in the interior of the parameter space; in particular, it must satisfy the following requirements. (a) If any structural zeroes are present, the starting value must have zeroes in those positions. (b) All other elements of the starting value must be nonzero. (c) The starting
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Figure 8.2. Single cycle of Bayesian iterative proportional fitting
. value must satisfy the constraints of the loglinear model. One way to create a starting value with these properties is to fill θ with zeroes corresponding to the structural zeroes and uniform values elsewhere. Another good choice is a posterior mode under a constrained Dirichlet prior with all hyperparameters α(y)> 1, which can be obtained from conventional IPF. One unusual feature of Bayesian IPF is that if a cell probability θ (y) ever becomes zero, it remains at zero for all subsequent iterations. In theory, this should never happen for a nonstructuralzero cell, because true gamma random variates are always positive. In practice, however, if the prior hyperparameters αy are close to zero and the observed contingency table contains random zeroes, the pseudorandom number generator used in draw g ~ G(sum2)
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(8.21)
may occasionally produce a value with a floatingpoint representation of zero. The resulting value of θ will then fall on the boundary and remain trapped there for all future iterations, and the Markov chain will fail to converge to the correct posterior distribution. The problem is that boundary values are absorbing states, whose presence violates the regularity conditions necessary for an iterative simulation algorithm to converge (Section 3.5.2). In principle the probability of ever reaching the boundary should be zero, but due to the limitations of computer arithmetic, the chance of falling within machine precision of the boundary might be nonnegligible. This difficulty is easily overcome by adding a 20 very small positive constant (say 10 ) to the value of g in (8.21), the effect of which will be imperceptible in the statistical inference. 8.4.4 Why Bayesian IPF works Here we present heuristic arguments to establish that Bayesian IPF does indeed converge to the constrained Dirichlet posterior for the given loglinear model. For simplicity, let us consider the homogeneousassociation model for three variables, (AB, AC, BC), where each of the variables A, B and C is binary; extensions to other loglinear models will be immediate. The relationships among the logcell probabilities η = logθ can be expressed as η = Mλ, where
η111 1 η211 1 η121 1 η 1 221 η = η , M = 1 112 1 η212 1 η 122 1 η222
1 −1 1 −1 1 −1 1 −1
1 1 −1 −1 1 1 −1 −1
1 1 1 1 −1 −1 −1 −1
1 −1 −1 1 1 −1 −1 1
1 −1 1 −1 −1 1 −1 1
and
[
AB AC BC λ = λ0 , λ1A , λ1B , λ1C , λ11 , λ11 , λ11
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]
T
.
1 1 −1 −1 −1 , −1 1 −1
The remaining loglinear coefficients follow from the identifiability constraints AB AB AB AB , λ1A = − λ2A , λ11 = − λ12 = − λ21 = λ22
and so on. Until now we have been assuming that the cell frequencies x follow a multinomial distribution,
x  θ ~ M ( n, θ ) , where the sample size n = x+++ is considered fixed, and the cell probabilities θ = θ ijk follow a Dirichlet distribution
{ }
{ }
with hyperparameters α = α ijk . With the restrictions imposed by the loglinear model, η logθ must lie in R(M), the sevendimensional linear space spanned by the columns of M; in addition, θ must satisfy θ+++ = 1. This combination of linear and loglinear constraints on the elements of θ makes the parameter space somewhat difficult to visualize and understand. The geometric features can be simplified, however, if we expand the model by allowing the total sample size n to vary. The Poisson/gamma representation Consider an expanded model in which the cell counts are Poisson,
( )
xijk  µ ~ Poisson µ ijk
(8.22)
{ } are
independently for all i, j, k, and the cell means µ = µ ijk a priori distributed as independent gamma variates
( )
µ ijk ~ cG α ijk
(8.23)
with a common scaling factor c. By wellknown properties of the Poisson model, (8.22) implies that
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η  µ ~ Poisson( µ + + + ) and
x  n, µ ~ M (η, θ ) , where µ + + + = Σ ijk µ ijk and θ ijk = µ ijk / µ + + + (e.g. Agresti, 1990). Moreover, it can be shown that the productgamma prior (8.23) implies that µ+++ and θ are independently distributed as
µ + + + ~ cG(α + + + ) and
θ ~ D(α ) , respectively; the proof is a standard exercise in transformation and will be left to the reader. Thus the expanded model (8.22)(8.23) for x and µ implies our usual multinomialDirichlet model for x and µ; the only difference is that the expanded model allows estimation of an overall intensity parameter µ+++ which is independent of θ in both the prior and the posterior distributions. By standard Bayesian arguments, the posterior distributions of the cell means µ are
( )
µ ijk  x ~ c ′G α ijk ′
(8.24)
where α ijk ′ = α ijk + xijk and c ′ = c / ( c + 1) , and the posterior distribution of the intensity parameter is µ + + +  x ~ c′G(α +′ + + ) where α +′ + + = Σ ijk α ijk + n . As before, the posterior
distribution of θ is D(α ′), α ′ = α + x . The Poisson/gamma representation is geometrically convenient because, unlike θ, the cell means µ are not required to sum to any particular value. The loglinear model for the cell probabilities, logθ=Mλ implies a similar model for the cell means, logµ=Μλ∗, (8.25)
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where
[
AB AC BC λ * = λ*0 , λ1A , λ1B , λ1C , λ11 , λ11 , λ11
]
T
and λ*0 = λ0 + log µ + + + . Unlike λ0 the new intercept λ*0 is a free parameter that can take any value on the real line. Thus the parameter space for p is simpler than the space for θ, because logµ is allowed to lie anywhere in R(M).
The cellmeans version of Bayesian IPF Under the expanded model, we can define a version of Bayesian IPF that operates on the cell means µ. It is similar to the version for θ except that we do not rescale the parameters to sum to one at every step. The cellmeans version is ( t +1 / 3) ( t + 0 / 3) c ′gij + µ ijk = µ ijk (8.26) t + 0 / 3 for all i, j, k, ) µ( ij + ( µ ijk
t + 2 / 3)
( t +1 / 3) c ′gi+ k for all i, j, k, = µ ijk ( t +1 / 3) µ i+ k
(8.27)
( t + 2 / 3) c ′g jk = µ ijk for all i, j, k, (8.28) µ ( t + 2 / 3) ijk where the gij+, gi+k and g +jk are independent gamma variates as before. To see the relationship to the previous version, notice that we can rewrite the first step as θ ( t + 0 / 3) g ijk ij + ( t +1 / 3) µ ijk = t +0 / 3 c ′g+ + + ) g θ( + + + ij + 1 / 3 + t ) c ′g , = θ( ( µ ijk
t + 3 / 3)
ijk
+++
and similarly for the second and third steps. The value of θ at every step is the same under the new version; the only difference is that the intensity parameter
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µ+ + + =
∑ θ ijk c′g+ + + = c′g+ + +
i, j, k
is updated at every step to be a random draw from c ′G(α +′ + + ) .
Without constraints, the productgamma posterior (8.24) for µ implies that the posterior distribution of θ
is D(α ′) .
Constraining µ and θ to lie in R(M ) does not change the functional form of the densities for µ or θ , but only their normalizing constants; so if we are able to show that the cellmeans version of Bayesian IPF converges to the constrained productgamma posterior over R(M ), then we have successfully shown that the original version converges to the constrained Dirichlet posterior over R(M). Heuristic argument for convergence The three steps (8.26)(8.28) define the transition rule of a Markov chain for µ. To show that a particular distribution F is the stationary distribution for this chain, one must establish two facts. First, one must show that the chain ’maps F onto itself,’ in other words, that µ ( t ) ~ F implies µ ( t +1) ~ F . Second, one must show that the chain is ergodic, containing no periodicities or absorbing states: it must be possible for µ ( t ) to reach any value in the support of F for a sufficiently large t, and for every step thereafter, from any starting value µ (0) within the support. To establish the first condition, notice that the first step (8.26) combines conditional probabilities P(C = k  A = i, B = j ) from the previous cycle with updated values for the ( A = i, B = j ) marginal rates,
( µ ijk
t +1 / 3)
where
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t +0 / 3 t +1 / 3 = θ ((ij ) k ) µ ij( + ) ,
( t + 0 / 3) ( t + 0 / 3) θ ijk µ ijk t +0 / 3 θ ((ij )k ) = t + 0 / 3 = t + 0 / 3 θ ij( + ) µ ij( + )
and
µ ij( +
t +1 / 3)
(
~ c ′G α ij′ +
)
(8.29)
independently for all i , j. Because (8.29) is the marginal distribution of µ ij + implied by (8.24), this first step
{ }
represents a draw from the conditional posterior of µ with
{θ( ) } fixed at its previous value, ) ~ P µ  x, θ µ( { ( ) } = {θ(( ) )} . ij k
t +0 / 3 ij k
t +1 / 3
ij k
Now if the old value of µ is drawn from the actual posterior distribution,
µ ( t + 0 / 3) ~ P ( µ  x) , then the old values of θ ( ij ) k are drawn from their actual posterior as well,
{θ(( ) )} ~ P {θ( ) }  x t +0 / 3 ij k
ij k
which implies
µ ( t +1 / 3) ~ P( µ  x) Thus we have established that the first step (8.26) maps P(µx) onto itself. By similar arguments this result holds for the second and third steps (8.27)(8.28) as well. To establish ergodicity, we must examine the structure of the design matrix M in the loglinear model (8.25) for µ. Let
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1 1 1 1 M1 = 1 1 1 1
1 −1 1 −1 1 −1 1 −1
1 1 −1 −1 1 1 −1 −1
1 1 −1 1 −1 1 1 1 1, M 2 = 1 1 −1 1 −1 1 1
1 −1 1 −1 1 −1 1 −1
1 1 1 1 −1 −1 −1 −1
1 −1 1 −1 −1 1 −1 1
and
1 1 1 1 1 1 1 1 1 −1 1 −1 1 −1 1 −1 M 3 = 1 1 −1 −1 1 1 −1 −1 1 −1 −1 1 1 −1 −1 1 denote the portions of M corresponding to the AB, AC and BC effects, respectively. The space spanned by the columns of M1 is the same as that of 1 0 0 0 0 1 0 0 0 0 1 0 * 0 0 0 1 M 1 = 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 Similarly, the range spaces of M2 and M3 are the same as those of 1 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0 1 0 0 0 0 1 0 0 0 1 0 0 M 2* = 0 0 1 0, M 3* = 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0 1 0 0 0 0 1 0 0 0 1 respectively. Notice that the first step of Bayesian IPF represents a proportionate adjustment for each group of cells that contributes to a mean µ ij + for the AB marginal table.
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This first step can thus be written in terms of the logcell means as
log µ ( t +1 / 3) = log µ ( t + 0 / 3) + M1* log γ 1 where γ1 is a vector of four gamma variates. Similarly, the second and third steps can be written
log µ ( t + 2 / 3) = log µ ( t +1 / 3) + M 2* log γ 2 log µ ( t + 2 / 3) = log µ ( t + 2 / 3) + M 3* log γ 3 The complete cycle consisting of all three steps is thus
log µ ( t + 3 / 3) = log µ ( t + 2 / 3) + M 3* log γ 3
(
)
(
(8.30)
)
T
where M * = M1* , M 2* , M 3* and γ = γ 1T , γ 2T , γ 3T . Because the columns of M* span the same space as those of M, and the elements of γ are random gamma variates whose logarithms can lie anywhere on the real line, (8.30) ensures that µ(t+1) has a nonzero chance of falling anywhere in R(M) provided that µ ( t ) ∈ R( M ) . Thus it follows that this Markov chain can reach any state in a single cycle from any state in the parameter space of µ, and ergodicity is established. Further notes Bayesian IPF was first presented by Gelman et al. (1995) who gave brief arguments for convergence under a Poisson/gamma loglinear model. Our version (8.26)(8.28) differs from theirs in that we have included a factor c ′ arising from the scaling parameter in the prior distribution (8.23). Without this factor, the resulting posterior could give misleading inferences about the overall intensity µ+++. Inferences about the cell probabilities θ ijk = µ ijk / µ + + + , however, will be the same under both versions.
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Bayesian IPF bears an interesting relationship to Gibbs sampling (Section 3.4.1). In Gibbs sampling, we partition a random vector Z into nonoverlapping subvectors (Z1, Z2,..., ZJ) and draw from the full conditionals P Z j  {Z k : k ≠ j}
(
)
for j = 1,..., J in turn. In Bayesian IPF, however, the vector is partitioned differently at each step of the cycle; in our example we partition µ as
({µij }, {µijk / µij }) at Step 1, ({µi k }, {µijk / µi k }) at Step 2, ({µ jk }, {µijk / µ jk }) at Step 3. +
+
+
+
+
As noted by several authors (e.g. Gelfand and Smith, 1990), any partitioning scheme will work provided that the complete cycle is ergodic, allowing the random vector to eventually reach any state from any other state. 8.4.5 Example: misclassification of seatbelt use and injury In Section 8.3.4 we examined loglinear models pertaining to errors in police reporting of seatbelt use and injury in automobile accidents. Regarding the error indicators E B and EI, as response variables and D, S, B2 and I2 as predictors, we found convincing evidence for the associations B2I2EBEI, DEB, DEI and mild evidence for SEI. This evidence was based on pvalues from chisquare approximations for the test statistics ∆G2 and ∆X2. Using Bayesian IPF, we can make Bayesian inferences about these associations directly without largesample approximations. To illustrate, we ran Bayesian IPF under Model 8,
(DSB2 I 2 , B2 I 2 EB EI , DEB , DEI , SEI )
(8.31)
Recall that the ML estimate for this model lies in the interior of the parameter space. Taking the ML estimate from IPF as a starting value, we ran 5100 cycles of Bayesian IPF under the Jeffreys
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Figure 8.3. Histograms of 5000 simulated values of (a) SE and (c) λ11 I , respectively, from Bayesian IPF.
DE B
λ11
(b)
DE I
λ11
prior with hyperparameters 0.5; ignoring the first 100 cycles, we stored the results of the remaining 5000. Bayesian IPF appears to converge very quickly in this example; autocorrelation plots of a variety of parameters revealed no significant correlations beyond lag 5. This behavior is consistent with that of ordinary IPF, which converged in only 11 cycles. Let us consider how to draw inferences about the coefficients λ of the loglinear model. The output of each cycle of Bayesian IPF is a table of simulated cell probabilities θ, and each coefficient in λ is a linear contrast among the elements of log θ. Consider the terms in λDE B , which pertain to the effect of damage on errors in the police report of belt use. If we average the elements of the 26 array log θ over the dimensions corresponding to S, B2, I2 and EI, all the coefficients pertaining to these variables drop out due to the linear constraints imposed on them. The result of this averaging is a 2 × 2 table with elements DE EB DE B γ ij B = λ0 + λD i + λ j + λ ij for i, j = 1, 2. The loglinear coefficients can then be obtained as DE λ0 = 14 Σ ijγ ij B , DE B
1 λD i = 2 Σ jγ ij
− λ0 , DE = 12 Σ iγ ij B − λ0 , DE EB = γ ij B − λD i − λ j − λ0 .
E λ jB DE B
λ ij
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By similar manipulations of the elements of log θ we can derive any coefficient in the loglinear model. Histograms of DE DE SE the 5000 simulated values of λ11 B , λ11 I and λ11 I are shown in Figure 8.3 (a), (b) and (c), respectively. Notice that the distribution in Figure 8.3 (a) is located primarily to the left of zero, providing evidence that errors in reporting of belt use (EB = 2) tend to occur more frequently for accidents with low damage (D = 1) than those with high DE damage (D = 2). Among the 5000 simulated values of λ11 B fell above zero, so a twotailed Bayesian pvalue for testing the current model (831) against the reduced model that sets λDE B = 0 is 2 × 182/5000. Similarly, Figures 8.3 (b) and (c) show that low damage (D = 1) and female drivers (S = 2) are associated with higher rates of reporting errors for injury (EI = DE SE 2); twotailed Bayesian pvalues for testing λ I and λ I are 0.03 and 0.13, respectively.
8.5 Loglinear modeling with incomplete data 8.5. 1 ML estimates and posterior modes The two algorithms we have discussed thus far, IPF and Bayesian IPF, can be extended quite easily to handle missing values in the original data matrix. With conventional IPF, the extension is an example of the ECM algorithm, a generalization of EM discussed briefly in Chapter 3. EM for loglinear models Let us now return to our general notation, with Y1, Y2,..., Yp representing categorical variables, and x = x y : y ∈ and
{
y}
{
y}
θ = θy : y∈ the cell counts and probabilities, respectively, in the pdimensional crossclassified contingency table. As described in Section 7.3, the actual cell counts x are not observed when the data are incomplete; rather, we observe
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a table z(s) of potentially smaller dimension for each missingness pattern s = 1, 2,..., S where z(s) contains marginal frequencies for the variables observed in pattern s. The basic EM algorithm (Section 7.3.2) updates the estimate of θ in two steps: the Estep, in which we calculate the predicted mean of x given z(1),..., z(S) under the current estimate of θ; and the Mstep, in which we reestimate θ from the predicted mean of x. Under the saturated multinomial model, the Mstep has a particularly simple form because the completedata ML estimates are θˆ = x y / n for all y ∈ . Now consider what happens to EM when we move from the saturated model to a loglinear model, which requires the parameter θ to lie in a restricted space Θ M. The Estep, which
y
is performed under an assumed value of θ, does not change at all, because the conditional expectation of x given z(1),..., z(S) and θ has the same form whether θ ∈Θ M or θ ∉Θ M . The Mstep, however, becomes a constrained maximization of the (expected) completedata loglikelihood or logposterior over Θ M. For loglinear models, this maximization cannot in general be carried out in closed form, but requires an iterative technique such as IPF. A general EM algorithm for loglinear models would thus be doubly iterative, requiring iteration to convergence within each Mstep. The ECM algorithm In many situations, EM for loglinear models is not unduly cumbersome, especially in modern computing environments. Several authors speculated, however, that it might not be necessary to iterate until full convergence at each Mstep; rather, running only a single cycle of IPF might be enough (e.g. Fuchs, 1982). As shown by Meng and Rubin (1993), this modification produces an example of an algorithm called ExpectationConditional Maximization or ECM. ECM possesses the same reliable convergence properties as EM, increasing the observeddata loglikelihood at each step. The key idea of ECM is that the full Mstep is replaced by a
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quicker CMstep, a single cycle of constrained maximizations which, if repeated over and over, would eventually result in a maximization over the full parameter space Θ M . Each step in a cycle of IPF is a constrained maximization. Consider the three steps of IPF for the model (AB, AC, BC). The first step (8.6) is based on a factorization of the completedata likelihood for the cell probabilities θ ijk = P(A= i, B = j, C = k) into independent factors corresponding to the conditional probabilities for C given A and B,
P(C = k  A = i, B = j ) = θ ijk / θ ij + , and the marginal probabilities for A and B,
P( A = i, B = j ) = θ ij + . The first step fixes
{ }
{θ ijk / θ i+ k }
at its previous value but
replaces θ ij + by an ML estimate or posterior mode; thus it represents a constrained maximization of the likelihood or posterior density for θ . Similarly, the second and third steps (8.7)(8.8) represent maximizations subject to fixed values of θ ijk / θ i+ k and θ ijk / θ + jk , respectively.
{
}
{
}
For a cycle of constrained maximizations to form a valid CM step, it must satisfy a set of technical requirements known as the spacefilling conditions (Meng and Rubin, 1993). These conditions, which have been demonstrated to hold for a single cycle of IPF (Meng and Rubin, 1991b), are similar to those needed to establish ergodicity in a Markov chain (Section 8.4.4). The fact that IPF can reach any point in Θ M from any other point in a single cycle ensures that the ECM algorithm will eventually converge to an unconstrained maximum rather than a constrained one. To obtain a general ECM algorithm for loglinear models, we only need to replace the Mstep of the basic EM (the last line of pseudocode in Figure 7.2) by a single call to the IPF algorithm presented in Figure 8.1. As in ordinary IPF, the starting value for θ should lie in the interior of the parameter
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space; one choice that always works is to assign zeroes to the structuralzero cells and uniform values elsewhere. 8.5.2 Goodnessoffit statistics The goodnessoffit statistics G 2 and X 2 can be readily extended to handle incomplete data. Using the notation of Section 7.3, let y = (y1, y2,..., yp) denote a generic realization of the variables (Y 1, Y2,..., Yp), and let O s(y ) and M s (y), respectively, denote the subvectors of y corresponding to the variables that are observed and missing in pattern s. Let Os and Ms denote the sets over which O s(y ) and Ms(y) can vary, excluding structural zeroes. The marginal table z(s) that we observe for pattern s has counts
zO( )( y ) =
∑ x(ys) for all Os ( y) ∈Os
s
s
(8.32)
M s ( y )∈M s
and the marginal probabilities corresponding to these counts are
βo ( y) = s
∑
θy . M s ( y )∈M s
(8.33)
The most obvious way to extend G2 is to take s zO( )( y ) (s) s , =2 zO ( y ) log s ns βO ( y ) s s =1 Os ( y )∈Os S
2 Graw
∑ ∑
(8.34)
where ns represents the total sample size in pattern s. This statistic, when evaluated at an ML estimate θˆ , increases as the observed frequencies deviate from their estimated expected values. The corresponding extension of Pearson’s X2 is S
2 X raw =
∑ ∑
s =1 Os ( y )∈Os
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2 z (s) − n β s Os ( y ) Os ( y ) . ns βO ( y ) s
(8.35)
As noted by Fuchs (1982) and Little and Rubin (1987), these statistics differ from their completedata counterparts in that they are typically nonzero even when evaluated at the ML estimate for the saturated model. The reason, which is somewhat technical, is buried in the definition of the observeddata likelihood in Chapter 2. The marginal probabilities (8.33) are not really the expected proportions for the observed counts (8.32) within each missingness pattern, unless the missingness happens to be MCAR. Under the less restrictive assumption of MAR, the true expected proportions may differ from (8.33), because MAR does not require the distribution of observed data to be identical across patterns. Using the same cell probabilities θ in the calculation of (8.33) for all patterns is merely a matter of convenience, because, as argued in Section 2.3, likelihoodbased inferences for parameters of the completedata model are identical under any ignorable mechanism. The practical effect of using a common θ for all patterns is 2 2 that Graw and X raw as defined above may be drastically different from zero even when evaluated at the ML estimate for the saturated model. In fact, if the sample is large enough, these statistics can be used to test the null hypothesis that the missingness data are MCAR against the alternative of MAR. In most situations, such a test will not be of great interest, because we are concerned primarily with the parameters of the completedata model; the parameters of the missingness mechanism are a nuisance. Moreover, in all but the most trivial real data examples, the expected cell counts within missingness patterns are rarely large enough for the chisquare approximation to work well. For these reasons, we will not 2 2 attempt to interpret a value of Graw or X raw from the saturated model, except as a baseline for assessing the fit of a smaller model.
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Adjusted goodnessoffit statistics Let G02 denote the value of (8.34) evaluated at the ML estimate for the saturated model. Consider the adjusted goodnessoffit measure s zO( )( y ) (s) s G =2 zO ( y ) log − G02 s ns βO ( y ) s s =1 Os ( y )∈Os S
2
∑ ∑
(8.36)
regarded as a function of θ . This represents twice the difference in the observeddata loglikelihood evaluated at the current value of θ and at the global maximum over the entire simplex Θ . When evaluated at the ML estimate for the saturated model, (8.36) is zero. When evaluated at the ML estimate for a nonsaturated loglinear model, it becomes the likelihoodratio statistic for testing the fit of the model against the saturated alternative. Because this statistic behaves in much the same manner as the deviance (8.9) for complete data, we will adopt (8.36) as our definition for the deviance with incomplete data. The Pearson counterpart to (8.36) is S
X2 =
∑ ∑
s =1 Os ( y )∈Os
2 z (s) − n β s Os ( y ) Os ( y ) − X 02 , ns βO ( y )
(8.37)
s
where X 02 represents the raw version (8.35) evaluated at the MLE for the saturated model. Just as in the completedata case, the chisquare approximation for the distributions of G2 and X2 may be poor when the data are sparse. Even with sparse data, however, chisquare approximations for the differences ∆G2 and ∆X2 can be quite reliable for nested model comparisons, particularly when ∆df is small. Finally, just as with complete data, G2 and X2 become problematic when ML estimates lie on the boundary. The easiest way to handle such situations is to add a small amount of prior information, e.g. in the form of a
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Dirichlet prior with all hyperparameters greater than one, to smooth the posterior mode away from the boundary. 8.5.3 Data augmentation and Bayesian IPF We have seen that IPF can be extended in a straightforward way to handle missing data, resulting in an ECM algorithm. In a similar fashion, we can extend Bayesian IPF to create an algorithm for parameter simulation and multiple imputation. Recall the basic data augmentation procedure for the saturated multinomial model (Section 7.3.3). In this algorithm, the observed marginal counts z(s) for missingness patterns s = 1, 2,..., S are randomly allocated to the cells of the full pdimensional table x under an assumed value for θ (the Istep). Then a new value of θ is drawn from its completedata Dirichlet posterior given the simulated version of x (the Pstep). Repeating the I and Psteps a large number of times eventually produces a draw of θ from its observeddata
posterior P(θ  Yobs ) . As we move from the saturated model to a loglinear model, the Istep remains unchanged, because the random allocation procedure has the same form regardless of the value of θ. The Pstep, however, must in general be carried out iteratively, because creating posterior draws of θ under a loglinear model requires multiple cycles of Bayesian IPF. True data augmentation, like a true EM algorithm, would thus require undesirable nested iterations. Suppose, however, that instead of iterating to full convergence within each Pstep, we perform only a single cycle of Bayesian IPF. The resulting algorithm, which we call data augmentationBayesian IPF (DABIPF), still converges to the observeddata posterior under the constrained Dirichlet prior. Although we are not drawing from the correct conditional distribution P(θ  x) in this modified Pstep, we are simulating one step from the transition rule of a Markov chain whose stationary distribution is P(θ  x). DABIPF is neither a true data augmentation algorithm nor a Gibbs sampler, but a hybrid algorithm of the type described in Section 3.4.5 with
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the same basic convergence properties. Combining the data augmentation Istep (Section 7.3.3) with the implementation of Bayesian IPF in Figure 8.2 produces a single iteration of DABIPF. The algorithm may be used for parameter simulation or (when combined with the imputation code in Figure 7.5) for multiple imputation of the unitlevel missing data. Factorizations for nearmonotone patterns In the last chapter, we derived a monotone data augmentation procedure for the saturated model that tends to converge faster than ordinary data augmentation for nearmonotone missingness patterns (Section 7.4). The algorithm was based on a factorization of the multinomial likelihood into a sequence of multinomial likelihoods pertaining to the distribution of each variable given the previous ones. For loglinear models, we can again exploit factorizations of the likelihood to improve the performance of DABIPF, but only in certain special cases. For many loglinear models, the parameters corresponding to the sequence of conditional distributions are not distinct, due to the loglinear restrictions, we cannot always separate the completedata inference into a sequence of independent inferences corresponding to the nearmonotone pattern in the dataset. For this reason, we will not consider further the possible monotone versions of DABIPF.
8.6 Examples 8.6.1 Protective Services Project for Older Persons Recall the data introduced in the last chapter (Section 7.3.5) regarding the impact of social services on elderly clients. In this six
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Table 8.5. Parameters in the model (ASPMG, ASPMD, GD)
variable dataset, the main question of interest pertained to the effect of the treatmentgroup indicator G on survival D , controlling for the possible confounding effects of the covariates age A, sex S, physical status P and mental status M. With only n = 164 clients in the study, some of whom had missing values for P and M, the data were too sparse to allow for estimation of individual GD associations within each of the sixteen covariate patterns. Now we will fit a loglinear model that constrains these associations to be the same. Consider the model (ASPMG, ASPMD, GD). The presence of the association ASPMG allows the distribution of G to vary freely across the sixteen ASPM covariate patterns; similarly, ASPMD allows the distribution of D to vary freely across the covariate patterns. The GD association allows G to have a direct influence on D beyond that provided by their mutual associations with A , S, P and M . The absence of any association between GD and ASPM, however, requires the conditional GD odds ratios within the sixteen covariate patterns to be equal. The number of free parameters in this model, 48, can be counted as shown in Table 8.5. Notice that the saturated model has 26 − 1 = 63 free parameters; the difference is 63 − 48 = 15 because the saturated model fits 16 conditional GD odds ratios rather than just one. By pooling information across covariate patterns, the reduced model can provide a stable estimate of a common GD effect even though the information within any single pattern may be weak. With complete data, inferences about the conditional GD association under this model would be similar to those given by the well known MantelHaenszel test (Mantel and Haenszel, 1959; Agresti, 1990). Using the tools of Section 8.5, we can draw inferences about the effect of interest in two ways. First, we can perform
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Figure 8.4. Posterior draws of GD logodds ratios under prior hyperparameters of (a) 0.1 and (b) 1.5, simulated from 5000 cycles of DABIPF.
a single degreeoffreedom test of the null model (ASPMG, ASPMD) against the (ASPMG, ASPMD, GD) alternative using a largesample approximation to ∆X 2 or ∆G2. Under both of these models, the ECM algorithm converges to ML estimates on the boundary. To move the estimates away from the boundary, we reran ECM for each model using prior hyperparameters of 1.1. Comparing the observeddata loglikelihood at the two posterior modes, we find ∆G2 = 0.826. The corresponding pvalue is 0.36, so there is essentially no evidence of any conditional GD association given A, S, P and M. A second method, which does not rely on largesample approximations, is to simulate posterior draws of the parameters under the larger model (ASPMG, ASPMD, GD) and examine the marginal distribution of the conditional GD association. Starting from the posterior mode that we obtained from ECM, we ran 5000 cycles of DABIPF after a burnin period of 100 cycles. We did this under two alternative priors, setting the prior hyperparameters to 0. 1 and 1.5, respectively; the first results in very little smoothing, whereas the second pulls the estimates rather strongly toward a uniform table. Boxplots of the simulated logodds ratios under these two priors are shown in Figure 8.4. Simulated posterior means and interval estimates for the odds ratio in (a) are 1.74 and (0.68, 3.75), respectively; the corresponding estimates for (b) are
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1.50 and (0.82, 2.51). Thus there is no evidence to suggest that enriched social services are beneficial to clients. On the contrary, we find weak evidence that membership in the experimental group (G = 1) is associated with an increased rate of mortality (D = 1). The effects are not statistically significant,’ however; simulated twotailed Bayesian pvalues are 0.26 for (a) and 0.20 for (b). Recall that when we tried to draw inferences about the conditional G D associations under the saturated model (Section 7.3.5), we encountered difficulty because the parameters were so poorly estimated. Comparing the new boxplots in Figure 8.4 to the old ones in Figure 7.7, we see that the new inferences are much more plausible, and also much less sensitive to the choice of prior hyperparameters; pooling across covariate patterns to estimate a common odds ratio was indeed helpful. 8.6.2 Driver injury and seatbelt use In the last chapter (Section 7.4.3) we examined data from a large sample of 80 084 automobile accidents and found apparently convincing evidence that seatbelt use reduces the risk of injury. Those data, however, were marred by errors in the police reports of injury and belt use. A followup study of an additional 1796 accidents provided information on error rates in the police reports. We attempted to use the followup data to calibrate the larger dataset, correcting the inference for potential biases due to misclassification. Those efforts were hindered by the complexity of the saturated model, the only model available to us at the time. Let us reexamine these data by applying a more parsimonious loglinear model to the combined sample of 81 880 accidents. Our loglinear modeling of the 1796 followup cases (Section 8.3.4) provides some insight into the misclassification mechanism. Among various models relating damage D, driver sex S, true belt use B2 and true injury I2 to the error indicators for belt use EB and injury EI, we found that
(DSB2 I 2 , B2 I 2 EB EI , DEB , DEI , SEI )
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seemed to provide a good fit. For modeling the combined dataset of 81 880 cases, however, it is more convenient to work with D, S, B2, I2 and the original policereport variables B1 and I 1 because E B and E I are not determined for the accidents not included in the followup study. Because E B is a function of B1 and B2, and E I is a function of I1 and I2, all the associations in the above model are present in
(DSB2 I 2 B1I1B2 I 2 , DB1B2 , DI1I 2 , SI1I 2 ) . Furthermore, we expect the full fourway association DSB1I1 to be well estimated because these four variables are recorded for all cases in the combined dataset. Therefore, we will fit the model
(DSB1I1, DSB2 I 2 , B1I1B2 I 2 , DB1B2 , DI1I 2 , SI1I 2 ) , which has a total of 39 free parameters. Under this model, both ECM and DABIPF converge rather slowly. This is not surprising, because B 2 and I2 are missing for about 98% of the cases in the combined dataset. Because of the slow convergence, it would be difficult to draw inferences by parameter simulation; consecutive draws from DABIPF are so highly correlated that a very large number of cycles would be needed to obtain good posterior summaries. Instead of storing draws of parameters, we created ten multiple imputations of the followup belt use B 2 and injury status I2. These imputations were created by running ten parallel chains of DABIPF for 2500 cycles each, using the ML estimate as a starting value and setting the prior hyperparameters equal to 0.5. The ten imputations are shown in Table 8.6. The imputed variables, denoted by B and I represent true belt use and injury, and the variation among the ten imputations reflects the uncertainty due to misclassification in the original data. After imputation the followup cases were removed, so that only the original 80 084 accidents are represented in the imputed data. Using these ten imputations, we calculated point and interval estimates for the odds ratios relating seatbelt use to injury. The estimation was carried out on the logarithmic
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scale, as described in Section 6.4.2. Results for the overall odds ratio, and for the conditional odds ratios within each of the four D × S cells, are summarized in Table 8.7. Over the entire population, seatbelt use appears to reduce the odds of injury by about 1 − 0.73 = 27%, and the effect is statistically significant (pvalue = 0.04). This marginal analysis, however, ignores the possible confounding effects due to the covariates D and S. Within the four D × S cells, the estimated odds ratios are all less than one, but the interval estimates are very wide; none of the effects is statistically significant. After controlling for damage and sex of driver, the evidence for any beneficial effect of seatbelt use is very weak. The fact that all four of the conditional odds ratios are less than one suggests that we may be able to strengthen our conclusions by assuming a common odds ratio across the four D × S cells; that is, by fitting the loglinear model (DSB, DSI, BI), we may be able to find a significant BI association. The loglinear model (DSB, DSI, BI) implies a logit model for I that includes main effects for D and S, a D × S interaction and a main effect
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Table 8.6. Multiple imputations of accident frequencies by damage D (1=low, 2=high), sex of driver S (1=male, 2=female), actual belt use B (1=no, 2=yes), and actual injury (1=not injured, 2=injured), reflecting errors of classification
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Table 8.7. Multipleimputation inferences for odds ratios relating to belt use to injury, overall and within cells of damage by sex of driver: estimates, intervals, pvalues and percent missing information
Table 8.8. Multipleimputation inferences for logisticregression coefficients for predicting injury, assuming a common effect of belt use across classes of damage and sex
for B. We fit this logit model to each of the imputed datasets, coding dummy variables for the main effects of D (1 if high damage, 0 otherwise), S (1 if female, 0 otherwise), B (1 if belt used, 0 otherwise) and the D × S interaction (1 if high damage and female, 0 otherwise). The results of the multiplyimputed logit analysis are shown in Table 8.8. The estimate of the common odds ratio is exp(0.19) = 0.83 and the 95% interval ranges from exp(0.51) = 0.60 to exp(0.13) = 1.14, so the evidence is still weak. Accounting for occasional errors in the police reports greatly increases our uncertainty about the relationship between belt use and injury. These results are consistent with those of Chen (1989), who reached similar conclusions using likelihoodbased methods.
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