Missing data in longitudinal studies : strategies for bayesian modeling and sensitivity analysis

Drawing from the authors' own work and from the most recent developments in the field, Missing Data in Longitudinal Studies: Strategies for Bayesian Modeling and Sensitivity Analysis describes a comprehensive Bayesian approach for drawing inference from incomplete data in longitudinal studies. To illustrate these methods, the authors employ several data sets throughout that cover a range of study designs, variable types, and missing data issues.

The book first reviews modern approaches to formulate and interpret regression models for longitudinal data. It then discusses key ideas in Bayesian inference, including specifying prior distributions, computing posterior distribution, and assessing model fit. The book carefully describes the assumptions needed to make inferences about a full-data distribution from incompletely observed data. For settings with ignorable dropout, it emphasizes the importance of covariance models for inference about the mean while for nonignorable dropout, the book studies a variety of models in detail. It concludes with three case studies that highlight important features of the Bayesian approach for handling nonignorable missingness.

With suggestions for further reading at the end of most chapters as well as many applications to the health sciences, this resource offers a unified Bayesian approach to handle missing data in longitudinal studies.

Author Notes

Michael J. Daniels University of Florida Gainesville, U.S.A.
Joseph W. Hogan Brown University Providence, Rhode Island, U.S.A.

Preface	p. xvii
1 Description of Motivating Examples	p. 1
1.1 Overview	p. 1
1.2 Dose-finding trial of an experimental treatment for schizophrenia	p. 2
1.2.1 Study and data	p. 2
1.2.2 Questions of interest	p. 2
1.2.3 Missing data	p. 2
1.2.4 Data analyses	p. 2
1.3 Clinical trial of recombinant human growth hormone (rhGH) for increasing muscle strength in the elderly	p. 4
1.3.1 Study and data	p. 4
1.3.2 Questions of interest	p. 4
1.3.3 Missing data	p. 4
1.3.4 Data analyses	p. 5
1.4 Clinical trials of exercise as an aid to smoking cessation in women: the Commit to Quit studies	p. 6
1.4.1 Studies and data	p. 6
1.4.2 Questions of interest	p. 6
1.4.3 Missing data	p. 7
1.4.4 Data analyses	p. 8
1.5 Natural history of HIV infection in women: HIV Epidemiology Research Study (HERS) cohort	p. 9
1.5.1 Study and data	p. 9
1.5.2 Questions of interest	p. 9
1.5.3 Missing data	p. 9
1.5.4 Data analyses	p. 10
1.6 Clinical trial of smoking cessation among substance abusers: OASIS study	p. 11
1.6.1 Study and data	p. 11
1.6.2 Questions of interest	p. 11
1.6.3 Missing data	p. 12
1.6.4 Data analyses	p. 12
1.7 Equivalence trial of competing doses of AZT in HIV-infected children: Protocol 128 of the AIDS Clinical Trials Group	p. 13
1.7.1 Study and data	p. 13
1.7.2 Questions of interest	p. 14
1.7.3 Missing data	p. 14
1.7.4 Data analyses	p. 14
2 Regression Models	p. 15
2.1 Overview	p. 15
2.2 Preliminaries	p. 15
2.2.1 Longitudinal data	p. 15
2.2.2 Regression models	p. 17
2.2.3 Full vs. observed data	p. 18
2.2.4 Additional notation	p. 19
2.3 Generalized linear models	p. 19
2.4 Conditionally specified models	p. 20
2.4.1 Random effects models based on GLMs	p. 21
2.4.2 Random effects models for continuous response	p. 22
2.4.3 Random effects models for discrete responses	p. 23
2.5 Directly specified (marginal) models	p. 25
2.5.1 Multivariate normal and Gaussian process models	p. 26
2.5.2 Directly specified models for discrete longitudinal responses	p. 28
2.6 Semiparametric regression	p. 31
2.6.1 Generalized additive models based on regression splines	p. 32
2.6.2 Varying coefficient models	p. 34
2.7 Interpreting covariate effects	p. 34
2.7.1 Assumptions regarding time-varying covariates	p. 35
2.7.2 Longitudinal vs. cross-sectional effects	p. 36
2.7.3 Marginal vs. conditional effects	p. 37
2.8 Further reading	p. 38
3 Methods of Bayesian Inference	p. 39
3.1 Overview	p. 39
3.2 Likelihood and posterior distribution	p. 39
3.2.1 Likelihood	p. 39
3.2.2 Score function and information matrix	p. 41
3.2.3 The posterior distribution	p. 42
3.3 Prior Distributions	p. 43
3.3.1 Conjugate priors	p. 43
3.3.2 Noninformative priors	p. 46
3.3.3 Informative priors	p. 49
3.3.4 Identifiability and incomplete data	p. 50
3.4 Computation of the posterior distribution	p. 51
3.4.1 The Gibbs sampler	p. 52
3.4.2 The Metropolis-Hastings algorithm	p. 54
3.4.3 Data augmentation	p. 55
3.4.4 Inference using the posterior sample	p. 58
3.5 Model comparisons and assessing model fit	p. 62
3.5.1 Deviance Information Criterion (DIC)	p. 63
3.5.2 Posterior predictive loss	p. 65
3.5.3 Posterior predictive checks	p. 67
3.6 Nonparametric Bayes	p. 68
3.7 Further reading	p. 69
4 Worked Examples using Complete Data	p. 72
4.1 Overview	p. 72
4.2 Multivariate normal model: Growth Hormone study	p. 72
4.2.1 Models	p. 72
4.2.2 Priors	p. 73
4.2.3 MCMC details	p. 73
4.2.4 Model selection and fit	p. 73
4.2.5 Results	p. 74
4.2.6 Conclusions	p. 75
4.3 Normal random effects model: Schizophrenia trial	p. 75
4.3.1 Models	p. 76
4.3.2 Priors	p. 77
4.3.3 MCMC details	p. 77
4.3.4 Results	p. 77
4.3.5 Conclusions	p. 78
4.4 Models for longitudinal binary data: CTQ I Study	p. 79
4.4.1 Models	p. 80
4.4.2 Priors	p. 81
4.4.3 MCMC details	p. 81
4.4.4 Model selection	p. 81
4.4.5 Results	p. 82
4.4.6 Conclusions	p. 83
4.5 Summary	p. 84
5 Missing Data Mechanisms and Longitudinal Data	p. 85
5.1 Introduction	p. 85
5.2 Full vs. observed data	p. 86
5.2.1 Overview	p. 86
5.2.2 Data structures	p. 87
5.2.3 Dropout and other processes leading to missing responses	p. 87
5.3 Full-data models and missing data mechanisms	p. 89
5.3.1 Targets of inference	p. 89
5.3.2 Missing data mechanisms	p. 90
5.4 Assumptions about missing data mechanism	p. 91
5.4.1 Missing completely at random (MCAR)	p. 91
5.4.2 Missing at random (MAR)	p. 93
5.4.3 Missing not at random (MNAR)	p. 93
5.4.4 Auxiliary variables	p. 94
5.5 Missing at random applied to dropout processes	p. 96
5.6 Observed data posterior of full-data parameters	p. 98
5.7 The ignorability assumption	p. 99
5.7.1 Likelihood and posterior under ignorability	p. 99
5.7.2 Factored likelihood with monotone ignorable missingness	p. 101
5.7.3 The practical meaning of 'ignorability'	p. 102
5.8 Examples of full-data models under MAR	p. 103
5.9 Full-data models under MNAR	p. 106
5.9.1 Selection models	p. 107
5.9.2 Mixture models	p. 109
5.9.3 Shared parameter models	p. 112
5.10 Summary	p. 114
5.11 Further reading	p. 114
6 Inference about Full-Data Parameters under Ignorability	p. 115
6.1 Overview	p. 115
6.2 General issues in model specification	p. 116
6.2.1 Mis-specification of dependence	p. 116
6.2.2 Orthogonal parameters	p. 118
6.3 Posterior sampling using data augmentation	p. 121
6.4 Covariance structures for univariate longitudinal processes	p. 124
6.4.1 Serial correlation models	p. 124
6.4.2 Covariance matrices induced by random effects	p. 128
6.4.3 Covariance functions for misaligned data	p. 129
6.5 Covariate-dependent covariance structures	p. 130
6.5.1 Covariance/correlation matrices	p. 130
6.5.2 Dependence in longitudinal binary models	p. 134
6.6 Joint models for multivariate processes	p. 134
6.6.1 Continuous response and continuous auxiliary covariate	p. 135
6.6.2 Binary response and binary auxiliary covariate	p. 137
6.6.3 Binary response and continuous auxiliary covariate	p. 138
6.7 Model selection and model fit under ignorability	p. 138
6.7.1 Deviance information criterion (DIC)	p. 139
6.7.2 Posterior predictive checks	p. 141
6.8 Further reading	p. 143
7 Case Studies: Ignorable Missingness	p. 145
7.1 Overview	p. 145
7.2 Structured covariance matrices: Growth Hormone study	p. 145
7.2.1 Models	p. 145
7.2.2 Priors	p. 146
7.2.3 MCMC details	p. 146
7.2.4 Model selection and fit	p. 147
7.2.5 Results and comparison with completers-only analysis	p. 147
7.2.6 Conclusions	p. 149
7.3 Normal random effects model: Schizophrenia trial	p. 149
7.3.1 Models and priors	p. 149
7.3.2 MCMC details	p. 150
7.3.3 Model selection	p. 150
7.3.4 Results and comparison with completers-only analysis	p. 150
7.3.5 Conclusions	p. 151
7.4 Marginalized transition model: CTQ I trial	p. 151
7.4.1 Models	p. 152
7.4.2 MCMC details	p. 153
7.4.3 Model selection	p. 153
7.4.4 Results	p. 154
7.4.5 Conclusions	p. 154
7.5 Joint modeling with auxiliary variables: CTQ II trial	p. 155
7.5.1 Models	p. 156
7.5.2 Priors	p. 157
7.5.3 Posterior sampling	p. 157
7.5.4 Model selection and fit	p. 157
7.5.5 Results	p. 158
7.5.6 Conclusions	p. 159
7.6 Bayesian p-spline model: HERS CD4 data	p. 159
7.6.1 Models	p. 160
7.6.2 Priors	p. 161
7.6.3 MCMC details	p. 161
7.6.4 Model selection	p. 161
7.6.5 Results	p. 161
7.7 Summary	p. 162
8 Models for Handling Nonignorable Missingness	p. 165
8.1 Overview	p. 165
8.2 Extrapolation factorization and sensitivity parameters	p. 166
8.3 Selection models	p. 167
8.3.1 Background and history	p. 167
8.3.2 Absence of sensitivity parameters in the missing data mechanism	p. 168
8.3.3 Heckman selection model for a bivariate response	p. 171
8.3.4 Specification of the missing data mechanism for longitudinal data	p. 173
8.3.5 Parametric selection models for longitudinal data	p. 174
8.3.6 Feasibility of sensitivity analysis for parametric selection models	p. 175
8.3.7 Semiparametric selection models	p. 176
8.3.8 Posterior sampling strategies	p. 180
8.3.9 Summary of pros and cons of selection models	p. 181
8.4 Mixture models	p. 181
8.4.1 Background, specification, and identification	p. 181
8.4.2 Identification strategies for mixture models	p. 183
8.4.3 Mixture models with discrete-time dropout	p. 188
8.4.4 Mixture models with continuous-time dropout	p. 198
8.4.5 Combinations of MAR and MNAR dropout	p. 201
8.4.6 Mixture models or selection models?	p. 202
8.4.7 Covariate effects in mixture models	p. 203
8.5 Shared parameter models	p. 206
8.5.1 General structure	p. 206
8.5.2 Pros and cons of shared parameter models	p. 207
8.6 Model selection and model fit in nonignorable models	p. 209
8.6.1 Deviance information criterion (DIC)	p. 209
8.6.2 Posterior predictive checks	p. 213
8.7 Further reading	p. 215
9 Informative Priors and Sensitivity Analysis	p. 216
9.1 Overview	p. 216
9.1.1 General approach	p. 216
9.1.2 Global vs. local sensitivity analysis	p. 217
9.2 Some principles	p. 219
9.3 Parameterizing the full-data model	p. 220
9.4 Specifying priors	p. 222
9.5 Pattern mixture models	p. 224
9.5.1 General parameterization	p. 224
9.5.2 Using model constraints to reduce dimensionality of sensitivity parameters	p. 225
9.6 Selection models	p. 226
9.7 Further reading	p. 231
10 Case Studies: Nonignorable Missingness	p. 233
10.1 Overview	p. 233
10.2 Growth Hormone study: Pattern mixture models and sensitivity analysis	p. 234
10.2.1 Overview	p. 234
10.2.2 Multivariate normal model under ignorability	p. 234
10.2.3 Pattern mixture model specification	p. 235
10.2.4 MAR constraints for pattern mixture model	p. 235
10.2.5 Parameterizing departures from MAR	p. 236
10.2.6 Constructing priors	p. 238
10.2.7 Analysis using point mass MAR prior	p. 238
10.2.8 Analyses using MNAR priors	p. 239
10.2.9 Summary of pattern mixture analysis	p. 246
10.3 OASIS Study: Selection models, mixture models, and elicited priors	p. 248
10.3.1 Overview	p. 248
10.3.2 Selection model specification	p. 249
10.3.3 Selection model analyses under MAR and MNAR	p. 251
10.3.4 Pattern mixture model specification	p. 252
10.3.5 MAR and MNAR parameterizations	p. 252
10.3.6 Pattern mixture analysis under MAR	p. 255
10.3.7 Pattern mixture analysis under MNAR using elicited priors	p. 255
10.3.8 Summary: selection vs. pattern mixture approaches	p. 259
10.4 Pediatric AIDS trial: Mixture of varying coefficient models for continuous dropout	p. 261
10.4.1 Overview	p. 261
10.4.2 Model specification: CD4 counts	p. 263
10.4.3 Model specification: dropout times	p. 265
10.4.4 Summary of analyses under MAR and MNAR	p. 265
10.4.5 Summary	p. 266
Distributions	p. 268
Bibliography	p. 271
Author Index	p. 292
Index	p. 298

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