Daniels M.J., Hogan J.W. Missing Data in Longitudinal Studies: Strategies for Bayesian Modeling and Sensitivity Analysis
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xCONTENTS
1.7 Equivalence trial of competing doses of AZT in HIV-infected
children: Protocol 128 of the AIDS Clinical Trials Group 13
1.7.1 Study and data 13
1.7.2 Questions of interest 14
1.7.3 Missing data 14
1.7.4 Data analyses 14
2Regression Models 15
2.1 Overview 15
2.2 Preliminaries 15
2.2.1 Longitudinal data 15
2.2.2 Regression models 17
2.2.3 Full vs. observed data 18
2.2.4 Additional notation 19
2.3 Generalized linear models 19
2.4 Conditionally specified models 20
2.4.1 Random effects models based on GLMs 21
2.4.2 Random effects models for continuous response 22
2.4.3 Random effects models for discrete responses 23
2.5 Directly specified (marginal) models 25
2.5.1 Multivariate normal and Gaussian process models 26
2.5.2 Directly specified modelsfordiscretelongitudinal
responses 28
2.6 Semiparametric regression 31
2.6.1 Generalized additive models based on regression splines 32
2.6.2 Varying coefficient models 34
2.7 Interpreting covariate effects 34
2.7.1 Assumptions regarding time-varying covariates 35
2.7.2 Longitudinal vs. cross-sectional effects 36
2.7.3 Marginal vs. conditional effects 37
2.8 Further reading 38
3Methods of Bayesian Inference 39
3.1 Overview 39
3.2 Likelihood and posterior distribution 39
3.2.1 Likelihood 39
3.2.2 Score function and information matrix 41
3.2.3 The posteriordistribution 42
3.3 Prior Distributions 43
3.3.1 Conjugate priors 43
3.3.2 Noninformative priors 46
3.3.3 Informative priors 49
3.3.4Identifiability and incomplete data 50
CONTENTS xi
3.4 Computation of the posterior distribution 51
3.4.1 The Gibbs sampler 52
3.4.2 The Metropolis-Hastings algorithm 54
3.4.3 Data augmentation 55
3.4.4 Inference using the posterior sample 58
3.5 Model comparisons and assessing model fit 62
3.5.1 Deviance Information Criterion (DIC) 63
3.5.2 Posterior predictive loss 65
3.5.3 Posterior predictive checks 67
3.6 Nonparametric Bayes 68
3.7 Further reading 69
4WorkedExamplesusing Complete Data 72
4.1 Overview 72
4.2 Multivariate normal model: Growth Hormone study 72
4.2.1 Models 72
4.2.2 Priors 73
4.2.3 MCMC details 73
4.2.4 Model selection and fit 73
4.2.5 Results 74
4.2.6 Conclusions 75
4.3 Normal random effects model: Schizophrenia trial 75
4.3.1 Models 76
4.3.2 Priors 77
4.3.3 MCMC details 77
4.3.4 Results 77
4.3.5 Conclusions 78
4.4 Models for longitudinal binary data: CTQ I Study 79
4.4.1 Models 80
4.4.2 Priors 81
4.4.3 MCMC details 81
4.4.4 Model selection 81
4.4.5 Results 82
4.4.6 Conclusions 83
4.5 Summary 84
5Missing Data Mechanisms and Longitudinal Data 85
5.1 Introduction 85
5.2 Full vs. observed data 86
5.2.1 Overview 86
5.2.2 Data structures 87
5.2.3 Dropout and other processes leading to missing re-
sponses 87
xii CONTENTS
5.3 Full-data models and missing data mechanisms 89
5.3.1 Targets of inference 89
5.3.2 Missing data mechanisms 90
5.4 Assumptions about missing data mechanism 91
5.4.1 Missing completely at random (MCAR) 91
5.4.2 Missing at random (MAR) 93
5.4.3 Missing not atrandom (MNAR) 93
5.4.4Auxiliary variables 94
5.5 Missing at random applied to dropout processes 96
5.6 Observed data posterior of full-data parameters 98
5.7The ignorability assumption 99
5.7.1Likelihood and posterior under ignorability 99
5.7.2 Factored likelihood with monotone ignorable missing-
ness 101
5.7.3The practical meaning of ‘ignorability’ 102
5.8Examples of full-data models under MAR 103
5.9Full-data models under MNAR 106
5.9.1 Selection models 107
5.9.2 Mixture models 109
5.9.3 Shared parameter models 112
5.10 Summary 114
5.11 Further reading 114
6Inference about Full-Data Parameters under Ignorability 115
6.1 Overview 115
6.2 General issues in model specification 116
6.2.1 Mis-specification of dependence 116
6.2.2 Orthogonal parameters 118
6.3 Posterior sampling using data augmentation 121
6.4 Covariance structures for univariate longitudinal processes 124
6.4.1 Serial correlation models 124
6.4.2 Covariance matrices induced by random effects 128
6.4.3 Covariance functions for misaligned data 129
6.5 Covariate-dependent covariance structures 130
6.5.1 Covariance/correlation matrices 130
6.5.2 Dependence in longitudinal binary models 134
6.6 Joint models for multivariate processes 134
6.6.1Continuous response and continuous auxiliary covariate 135
6.6.2Binary response and binary auxiliary covariate 137
6.6.3Binary response and continuous auxiliary covariate 138
6.7Model selection and model fit under ignorability 138
6.7.1 Deviance information criterion (DIC) 139
6.7.2 Posterior predictive checks 141
CONTENTS xiii
6.8 Further reading 143
7CaseStudies: Ignorable Missingness 145
7.1 Overview 145
7.2 Structured covariance matrices: Growth Hormone study 145
7.2.1 Models 145
7.2.2 Priors 146
7.2.3 MCMC details 146
7.2.4 Model selection and fit 147
7.2.5 Results and comparison with completers-only analysis 147
7.2.6 Conclusions 149
7.3 Normal random effects model: Schizophrenia trial 149
7.3.1 Models and priors 149
7.3.2 MCMC details 150
7.3.3 Model selection 150
7.3.4 Results and comparison with completers-only analysis 150
7.3.5 Conclusions 151
7.4 Marginalized transition model: CTQ I trial 151
7.4.1 Models 152
7.4.2 MCMC details 153
7.4.3 Model selection 153
7.4.4 Results 154
7.4.5 Conclusions 154
7.5Joint modeling with auxiliary variables: CTQ II trial 155
7.5.1 Models 156
7.5.2 Priors 157
7.5.3 Posterior sampling 157
7.5.4 Model selection and fit 157
7.5.5 Results 158
7.5.6 Conclusions 159
7.6 Bayesian p-spline model: HERS CD4 data 159
7.6.1 Models 160
7.6.2 Priors 161
7.6.3 MCMC details 161
7.6.4 Model selection 161
7.6.5 Results 161
7.7 Summary 162
8ModelsforHandling NonignorableMissingness 165
8.1 Overview 165
8.2 Extrapolation factorization and sensitivity parameters 166
8.3 Selection models 167
8.3.1 Background and history 167
xiv CONTENTS
8.3.2 Absence of sensitivity parameters in the missing data
mechanism 168
8.3.3 Heckman selection model for a bivariate response 171
8.3.4 Specification of the missing data mechanism for longi-
tudinal data 173
8.3.5 Parametric selection models for longitudinal data 174
8.3.6Feasibility of sensitivity analysis for parametric selection
models 175
8.3.7 Semiparametricselection models 176
8.3.8 Posterior sampling strategies 180
8.3.9 Summary of pros and cons of selection models 181
8.4 Mixture models 181
8.4.1Background, specification, and identification 181
8.4.2 Identification strategies for mixture models 183
8.4.3 Mixture models with discrete-time dropout 188
8.4.4 Mixture models with continuous-time dropout 198
8.4.5 Combinations of MAR and MNAR dropout 201
8.4.6 Mixture models or selection models? 202
8.4.7 Covariate effects in mixture models 203
8.5 Shared parameter models 206
8.5.1 General structure 206
8.5.2 Pros and cons of shared parameter models 207
8.6 Model selection and model fitinnonignorable models 209
8.6.1 Deviance information criterion (DIC) 209
8.6.2 Posterior predictive checks 213
8.7 Further reading 215
9InformativePriorsandSensitivity Analysis 216
9.1 Overview 216
9.1.1 General approach 216
9.1.2 Global vs. local sensitivity analysis 217
9.2 Some principles 219
9.3 Parameterizing the full-data model 220
9.4 Specifying priors 222
9.5 Pattern mixture models 224
9.5.1 General parameterization 224
9.5.2 Using model constraints to reduce dimensionality of
sensitivity parameters 225
9.6 Selection models 226
9.7 Further reading 231
10 Case Studies: Nonignorable Missingness 233
10.1 Overview 233
CONTENTS xv
10.2 Growth Hormone study: Pattern mixture models and sensitiv-
ity analysis 234
10.2.1 Overview 234
10.2.2Multivariate normal model under ignorability 234
10.2.3 Pattern mixture model specification 235
10.2.4 MAR constraints for pattern mixture model 235
10.2.5 Parameterizing departures from MAR 236
10.2.6 Constructing priors 238
10.2.7 Analysis using point mass MAR prior 238
10.2.8 Analyses using MNAR priors 239
10.2.9 Summary of pattern mixture analysis 246
10.3 OASIS Study: Selection models, mixture models, and elicited
priors 248
10.3.1 Overview 248
10.3.2 Selection model specification 249
10.3.3Selection model analyses under MAR and MNAR 251
10.3.4 Pattern mixture model specification 252
10.3.5 MAR and MNAR parameterizations 252
10.3.6Pattern mixture analysis under MAR 255
10.3.7Pattern mixture analysis under MNAR using elicited
priors 255
10.3.8 Summary: selection vs. pattern mixture approaches 259
10.4 Pediatric AIDS trial: Mixture of varying coefficient models for
continuous dropout 261
10.4.1 Overview 261
10.4.2 Model specification: CD4 counts 263
10.4.3 Model specification: dropout times 265
10.4.4 Summary of analyses under MAR and MNAR 265
10.4.5 Summary 266
Distributions 268
Bibliography 271
Author Index 292
Index 298
Preface
Motivations
Aconsiderable amount of research in public health, medicine, and the life and
social sciences is driven by longitudinalstudies that collect repeated obser-
vations over time. An almost inevitable complication in drawing inferences is
that data will be missing. For studies with long follow up periods, the pro-
portion of individuals with missing data can be substantial.
Despite a vast and growing literature, and recent advances in statistical
software, the appropriate handling of missing data in longitudinal studies re-
mains a vexing statistical problem, and has enough complexity to make it a
worthy topic ofstudy in its own right; more than just the standard set of
longitudinal modeling and missing data tools are needed.
With repeated measures, the best information about missing responses fre-
quently comes from observed responses on the same variable; hence methods
for covariance modeling take on heightened importance. Missingness in longi-
tudinal studies is frequently a result of dropout, mortality and other processes
that are well characterized as event times; hence methods for joint modeling
of repeated measures and event times play a central role.
And as with any analysis of incomplete data, parameters or models of inter-
est cannot be inferred without resorting to assumptions that employ various
degrees of subjective judgment. All too often, the assumptions used for data
analysis are motivated more by convenience than by substance; this seems
especially true for longitudinal data, where the models or data structures are
complex enough on their own without the added complication of missing data.
Hence a principled approach is needed forformalizing subjective assumptions
in contextually interpretable ways.
The Bayesian approach
When data are incomplete, inferences about parameters of interest cannot be
carried out without the benefit of subjective assumptions about the distribu-
tion of missing responses. They are subjective because data cannot be used
to critique them. Some of these assumptions are used with such regularity
that we forget they are being made; for example, when commercial software
such as SAS or Stata is used to analyze incomplete longitudinal data using
arandom effects model, the missing at random (MAR) assumption is being
used; when the Kaplan-Meier estimator is used to summarize a survival curve
xvii
xviii PREFACE
from censored event times, non-informative censoring is being assumed. Nei-
ther assumption can be formally checked, so the validity of inferences relies
on subjective judgment.
For this reason, it is our view that analysis of incomplete data cries out
for the Bayesian approach in order to formalize the subjective component of
inference. Specifically, we argue that subjective assumptions should be formu-
lated in terms of prior distributions for parameters that index the conditional
distribution of missing data given observed data. In this framework, default
assumptions such as missing at random can be encoded using point mass pri-
ors, or assumptions can be elaborated by centering priors away from MAR,
introducing uncertainty abouttheassumptions, or both.
Ideally, the role of the statistician is to write down a model that enables
critical thinking about the missing data assumptions, based on questions like
these: Can missing data be predicted from observables using an extrapolation?
Are dropouts systematically different from completers? If so, how? And what
is the degree of uncertainty about the assumptions?
This is notanewidea(see Rubin, 1977), but in our reading of the lit-
erature it is surprisingly underutilized. One possible reason is the difficulty
associated with parameterizing complex models so that missing data assump-
tions are coherently represented with priors. Another potential reason is lack
of knowledge about how to use appropriate software. With this in mind, we
have included 11 worked examples and case studies. These illustrate con-
cepts ranging from basic modeling of repeated measures to structured mod-
els for covariance to use of informative priors under nonignorable missing-
ness. Several of these examples have been carried out using WinBUGS soft-
ware, which is available at http://www.mrc-bsu.cam.ac.uk/bugs/.Code
used for analyses in the book, and datasets where available, are posted at
http://www.stat.brown.edu/~jhogan.
Audience
The book is intended for statisticians, data analysts and other scientists in-
volved in the collection and analysis of longitudinal data. Our emphasis is
on clinical and public health research, but many of the ideas will be appli-
cable to other fields. We assume familiarity with statistical inference at the
levelofCasella and Berger (2001), and with regression at the level of Kutner
et al. (2003). Familiarity with regression for longitudinal data and principles
of Bayesian inference is helpful, but these ideas are reviewed in Chapters 2
and 3, where numerous references to books and key papers are given.
Readers of this book are likely to have at least a passing familiarity with
basic methods for handling missing data, but the literature is vast and there
are distinct points of view on how to approach inference. Several outstanding
texts can be consulted to gain an appreciation, including Little and Rubin
(2002); Schafer (1997); van der Laan and Robins (2003); Tsiatis (2006); and
Molenberghs and Kenward (2007).
PREFACE xix
Although this is primarily a researchmonograph, we expect that it will
be asuitable supplemental text for graduate courses on longitudinal data,
missing data, or Bayesian inference. We have used material from the book for
our own graduate courses atUniversityofFlorida and Brown University.
Content
The book is composed of ten chapters, roughly divided intothreemain parts.
Chapters 1 through 4 make up the first part, which covers needed background
material. Chapter 1 provides detailed descriptions of motivating examples
that are used throughout the book; Chapters 2 and 3 provide background on
longitudinal data models and Bayesian inference, respectively; and Chapter 4
illustrates key concepts with worked data analysis examples.
The second part includes Chapters 5 through 7; it introduces missing data
mechanisms for longitudinal data settings, and gives in-depth treatment of in-
ference under the ignorability assumption. Chapter 5 gives detailed coverage
of Rubin’s missing data taxonomy (Rubin, 1976) applied to longitudinal data,
including MAR, ignorability, and implications for posterior inference. The no-
tion of a full-data model is introduced, and the chapter primarily describes
assumptions that are needed to infer features or parameters of the full-data
model from incomplete data. Chapter 6 discusses inference under the ignora-
bility assumption, with emphasis on the importance of covariance modeling.
In Chapter 7, several case studies are used to illustrate.
Chapters 8 through 10 make up the third part, where the focus shifts to
nonignorable missingness. In Chapters 8and9,wereiterate the central idea
that inference about a full-data distribution from incomplete data requires
untestable assumptions. We divide the full-data distribution into an observed-
data model and an extrapolation model, and introduce sensitivity parameters
that govern the untestable assumptions about the extrapolation. Chapter 8
reviews common approaches such as selection models, mixture models, and
shared parameter models, and focuses on parameterization of missing data
assumptions in each model.
In Chapter 8, the primary focus is on the likelihood (data model), whereas
Chapter 9 is concerned with formulation of priors that reflect assumptions
about missing data.InChapter 9, we show how to construct priors that reflect
specific missing data mechanisms (MAR, MNAR), how to center the models
at MAR, and give suggestions about calibrating the priors. Both mixture and
selection modeling approaches are addressed. Finally, Chapter 10 provides
three detailed case studies that illustrate analyses under MNAR.
Acknowledgments
We began this project in 2005; prior to that and since then, we have benefited
from friendship, cooperation and support of family, friends and colleagues.
Ourresearch in this area has been encouraged and positively influenced
by interactions with many colleagues in statistics and biostatistics; Alicia
Carriquiry, Peter Diggle, ConstantineGatsonis,RobKass,NanLaird, Tony