Contents

Preface xvii

1 Descriptionof Motivating Examples 1

1.1 Overview 1

1.2 Dose-finding trial of an experimental treatment for schizophrenia
2

1.2.1 Study and data 2

1.2.2 Questions of interest 2

1.2.3 Missing data 2

1.2.4 Data analyses 2

1.3 Clinical trial of recombinant human growth hormone (rhGH)
for increasing muscle strength in the elderly 4

1.3.1 Study and data 4

1.3.2 Questions of interest 4

1.3.3 Missing data 4

1.3.4 Data analyses 5

1.4 Clinical trials of exercise as an aid to smoking cessation in
women: the Commit to Quit studies 6

1.4.1 Studies and data 6

1.4.2 Questions of interest 6

1.4.3 Missing data 7

1.4.4 Data analyses 8

1.5 Natural history of HIV infection in women: HIV Epidemiology
Research Study (HERS) cohort 9

1.5.1 Study and data 9

1.5.2 Questions of interest 9

1.5.3 Missing data 9

1.5.4 Data analyses 10

1.6 Clinical trial of smoking cessation among substance abusers:
OASIS study 11

1.6.1 Study and data 11

1.6.2 Questions of interest 11

1.6.3 Missing data 12

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

2 Regression 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 models for discrete longitudinal
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

3 Methods 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 posterior distribution 42

3.3 Prior Distributions 43

3.3.1 Conjugate priors 43

3.3.2 Noninformative priors 46

3.3.3 Informative priors 49

3.4 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

4 WorkedExamplesusing 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

5 Missing 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 responses
87

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 at random (MNAR) 93

5.4.4 Auxiliary variables 94

5.5 Missing at random applied to dropout processes 96

5.6 Observed data posterior of full-data parameters 98

5.7 The ignorability assumption 99

5.7.1 Likelihood and posterior under ignorability 99

5.7.2 Factored likelihood with monotone ignorable missingness
101

5.7.3 The practical meaning of ‘ignorability’ 102

5.8 Examples of full-data models under MAR 103

5.9 Full-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

6 Inference 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

6.6.1 Continuous response and continuous auxiliary covariate 135

6.6.2 Binary response and binary auxiliary covariate 137

6.6.3 Binary response and continuous auxiliary covariate 138

6.7 Model selection and model fit under ignorability 138

6.7.1 Deviance information criterion (DIC) 139

6.8 Further reading 143

7 CaseStudies: 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

7.5 Joint 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