Analysis of Incomplete Multivariate Data

Transcription

1 Analysis of Incomplete Multivariate Data J. L. Schafer Department of Statistics The Pennsylvania State University USA CHAPMAN & HALL/CRC A CR.C Press Company Boca Raton London New York Washington, D.C.

2 Contents Preface xiii 1 Introduction Purpose Background The EM algorithm Markov chain Monte Carlo Why analysis by simulation? Looking ahead Scope of the rest of this book Knowledge assumed on the part of the reader Software and computational details Bibliographic notes 8 2 Assumptions The complete-data model Ignorability Missing at random Distinctness of parameters The observed-data likelihood and posterior Observed-data likelihood Examples Observed-data posterior Examining the ignorability assumption Examples where ignorability is known to hold Examples where ignorability is not known to hold Ignorability is relative General ignorable procedures A simulated example 24

3 viii CONTENTS Departures from ignorability Notes on nonignorable alternatives The role of the complete-data model Departures from the data model Inference treating certain variables as fixed 31 3 EM and data augmentation Introduction The EM algorithm Definition Examples EM for posterior modes Restrictions on the parameter space The ECM algorithm Properties of EM Stationary values Rate of convergence Example Further comments on convergence Markov chain Monte Carlo Gibbs sampling Data augmentation Examples of data augmentation The Metropolis-Hastings algorithm Generalizations and hybrid algorithms Properties of Markov chain Monte Carlo The meaning of convergence Examples of nonconvergence Rates of convergence 83 4 Inference by data augmentation Introduction Parameter simulation Dependent samples Summarizing a dependent sample Rao-Blackwellized estimates Multiple imputation Bayesianly proper multiple imputations Inference for a scalar quantity Inference for multidimensional estimands Assessing convergence Monitoring convergence in a single chain 119

4 CONTENTS ix Monitoring convergence with parallel chains Choosing scalar functions of the parameter Convergence of posterior summaries Practical guidelines Choosing a method of inference Implementing a parameter-simulation experiment Generating multiple imputations Choosing an imputation model Further comments on imputation modeling Methods for normal data Introduction Relevant properties of the complete-data model Basic notation Bayesian inference under a conjugate prior Choosing the prior hyperparameters Alternative parameterizations and sweep The EM algorithm Preliminary manipulations The E-step Implementation of the algorithm EM for posterior modes Calculating the observed-data loglikelihood Example: serum-cholesterol levels of heartattack patients Example: changes in heart rate due to marijuana use Data augmentation The I-step The P-step Example: cholesterol levels of heart-attack patients Example: changes in heart rate due to marijuana use More on the normal model Introduction Multiple imputation: example Cholesterol levels of heart-attack patients Generating the imputations Complete-data point and variance estimates 194

5 CONTENTS Combining the estimates Alternative choices for the number of imputations Multiple imputation: example Predicting achievement in foreign language study Applying the normal model Exploring the observed-data likelihood and posterior Overcoming the problem of inestimability Analysis by multiple imputation A simulation study Simulation procedures Complete-data inferences Results Fast algorithms based on factored likelihoods Monotone missingness patterns Computing alternative parameterizations Noniterative inference for monotone data Monotone data augmentation Implementation of the algorithm Uses and extensions Example 236 Methods for categorical data Introduction The multinomial model and Dirichlet prior The multinomial distribution Collapsing and partitioning the multinomial The Dirichlet distribution Bayesian inference Choosing the prior hyperparameters Collapsing and partitioning the Dirichlet Basic algorithms for the saturated model Characterizing an incomplete categorical dataset ' The EM algorithm Data augmentation Example: victimization status from the National Crime Survey Example: Protective Services Project for Older Persons 272

6 CONTENTS 7.4 Fast algorithms for near-monotone patterns Factoring the likelihood and prior density Monotone data augmentation Example: driver injury and seatbelt use Loglinear models Introduction Overview of loglinear models Definition Eliminating associations Sufficient statistics Model interpretation Likelihood-based inference with complete data Maximum-likelihood estimation Iterative proportional fitting Hypothesis testing and goodness of fit Example: misclassification of seatbelt use and injury Bayesian inference with complete data Prior distributions for loglinear models Inference using posterior modes Inference by Bayesian IPF Why Bayesian IPF works Example: misclassification of seatbelt use and injury Loglinear modeling with incomplete data Examples ML estimates and posterior modes Goodness-of-fit statistics Data augmentation and Bayesian IPF Protective Services Project for Older Persons Driver injury and seatbelt use Methods for mixed data Introduction ; The general location model Definition Complete-data likelihood Example Complete-data Bayesian inference Restricted models Reducing the number of parameters 341

7 xii CONTENTS Likelihood inference for restricted models Bayesian inference Algorithms for incomplete mixed data Predictive distributions EM for the unrestricted model Data augmentation Algorithms for restricted models Data examples St. Louis Risk Research Project Foreign Language Attitude Scale National Health and Nutrition Examination Survey ' Further topics Introduction Extensions of the normal model Restricted covariance structures Heavy-tailed distributions Interactions Semicontinuous variables Random-effects models Models for complex survey data Nonignorable methods Mixture models and latent variables Coarsened data and outlier models Diagnostics 386 Appendices A Data examples 387 B Storage of categorical data 395 C Software 399 References 401 Index 415