Statistical Modelling of Survival Data with Random Effects (eBook)

Il Do Ha is a full professor in the Department of Statistics at Pukyong National University in South Korea. His research interests are multivariate survival analysis using h-likelihood, inferences on random-effect models, clinical trials and financial statistics. Dr. Ha received his Ph.D. degree in statistics from Seoul National University. He has served as an Associate Editor of Computational Statistics until 2008-2012 and has been a fellow of the Royal Statistical Society (RSS) since 2006. Jong-Hyeon Jeong is a full professor in the Department of Biostatistics at University of Pittsburgh in USA. His research interests are in survival analysis, including competing risks, quantile residual life, empirical likelihood, h-likelihood, frailty model and clinical trials. He has published his first book with Springer: Jeong, J.-H. (2014) Statistical Inference on Residual Life, New York: Springer. Dr. Jeong received his Ph.D. degree in statistics from University of Rochester. He has been a fellow of the American Statistical Association (ASA) since 2017 as well as an elected member of the international Statistical Institute (ISI) since 2007. Dr. Jeong is also serving on the editorial board for the journal “Lifetime Data Analysis”. Youngjo Lee is a full professor in the Department of Statistics at Seoul National University in South Korea and also an adjunct professor of Karolinska Institutet in Sweden. His research interests are extension, application, theory and software development for hierarchical GLM (HGLM) and multivariate survival models using h-likelihood. He has published a HGLM book with Chapman and Hall: Lee, Y., Nelder, J. A. and Pawitan, Y. (2017) Generalized Linear Models with Random Effects: Unified Analysis via H-likelihood, 2nd edition, Boca Raton: Chapman and Hall. Dr. Lee received his Ph.D. degree in statistics from Iowa State University. He has been a fellow of the Royal Statistical Society (RSS) since 1996 as well as the American Statistical Association (ASA) since 2013.

Preface 6
Contents 8
Abbreviations 13
1 Introduction 15
1.1 Goals 15
1.2 Motivating Examples 16
1.2.1 Kidney Infection Data 17
1.2.2 Litter-Matched Rat Data 17
1.2.3 Chronic Granulomatous Disease (CGD) Nested Recurrent Data 18
1.2.4 Bladder Cancer Multicenter Data 18
1.2.5 Lung Cancer Multicenter Data 19
1.2.6 Breast Cancer Competing-Risks Data 19
2 Classical Survival Analysis 20
2.1 Hazard and Survival Function 22
2.1.1 Parametric Distributions for Survival Times 23
2.1.2 Nonparametric Estimation of Basic Quantities 24
2.2 Basic Likelihood Inference 33
2.3 Cox's Proportional Hazards Models 35
2.4 Accelerated Failure Time Models 42
2.5 Discussion 44
2.6 Appendix 45
2.6.1 Construction of Likelihoods of Various Types 45
2.6.2 Derivations of Breslow's Likelihood and Cumulative Hazard Estimator 46
2.6.3 Proof of Theorem 2.1 47
2.6.4 Fitting Cox PH Model via a Poisson GLM 49
3 H-Likelihood Approach to Random-Effect Models 50
3.1 Three Paradigms of Statistical Inference 50
3.1.1 Bayesian Approach 51
3.1.2 Fisher Likelihood Approach 52
3.1.3 Extended Likelihood Approach 55
3.2 H-Likelihood 59
3.3 Hierarchical Generalized Linear Models 63
3.3.1 Inferences on the Fixed Unknowns 65
3.3.2 Inferences on the Unobservables 70
3.4 A Practical Example: Epilepsy Seizure Count Data 74
3.5 Appendix 76
3.5.1 Proof of Approximation in Poisson-Gamma HGLM 76
4 Simple Frailty Models 79
4.1 Features of Correlated Survival Data 79
4.2 The Model and H-Likelihood 81
4.2.1 Univariate Frailty Model 81
4.2.2 H-Likelihood and Related Likelihoods 83
4.3 Inference Procedures Using R 87
4.3.1 Review of Estimation Procedures 87
4.3.2 Fitting Algorithm and Inference 90
4.3.3 Implementation Using R 93
4.3.4 Illustration 94
4.4 Model Selection 99
4.4.1 Basic Concept of Akaike Information 99
4.4.2 Three AICs for the Frailty Models 100
4.5 Interval Estimation of the Frailty 103
4.5.1 Confidence Interval for the Frailty 104
4.5.2 Illustration 105
4.6 Discussion 107
4.7 Appendix 109
4.7.1 Proof of Remark 4.1 109
4.7.2 Derivation of the H-Likelihood for Frailty Model 110
4.7.3 Equivalence of Both Estimators of ? Under the Gamma Frailty Model and the EM Estimating Equation of the Frailty Parameter ? 111
4.7.4 Proof of Joint Score Equations in (4.12) 113
4.7.5 Computation of PREML Equation for Frailty Parameter ? 114
4.7.6 Construction of CI of the Frailty in (4.20) 116
5 Multicomponent Frailty Models 117
5.1 Formulation of the Multicomponent Frailty Models 117
5.1.1 Multilevel and Time-Dependent Frailties 118
5.1.2 Correlated Frailties 119
5.2 H-Likelihood Procedures for the Multicomponent Models 121
5.3 Examples 122
5.3.1 Mammary Tumor Data 122
5.3.2 CGD Data 124
5.3.3 Bladder Cancer Data 126
5.4 Software and Examples Using R 127
5.4.1 Mammary Tumor Data: AR(1) Frailty Model 127
5.4.2 CGD Data: Univariate, Multilevel and AR(1) Frailty Models 128
5.5 Discussion 130
5.6 Appendix 131
5.6.1 H-Likelihood Procedure in the Multicomponent Models 131
5.6.2 Computation of -2 P?(hp)/?2 133
6 Competing Risks Frailty Models 136
6.1 Classical Competing-Risk Models 136
6.1.1 Cause-Specific Hazard Function and Cumulative Incidence Function 137
6.1.2 Subdistribution Hazard Function 139
6.1.3 Relationship Between Two Hazard Functions 139
6.1.4 Regression Models Based on Two Hazard Functions 140
6.2 Cause-Specific Hazard Frailty Models 141
6.2.1 Models 141
6.2.2 H-Likelihood Under the Cause-Specific Hazard Frailty Model 144
6.2.3 Partial H-Likelihood via Profiling 146
6.2.4 Fitting Procedure 147
6.3 Subdistribution Hazard Frailty Models 149
6.3.1 Models 149
6.3.2 H-Likelihood Under the Subhazard Frailty Model 150
6.4 Examples 155
6.4.1 Cause-Specific Frailty Model for Breast Cancer Data 155
6.4.2 Subhazard Frailty Model for Breast Cancer Data 162
6.5 Software and Examples Using R 166
6.5.1 A Simulated Data Set 166
6.5.2 Bladder Cancer Data 171
6.6 Discussion 175
6.7 Appendix 176
6.7.1 Calculation of the Gradient Vector and Elements for the Information Matrix from the Partial Likelihood 176
6.7.2 Derivation of the Gradient Vector and Elements for the Information Matrix from the Partial Restricted Likelihood 178
6.7.3 Proof of Estimating Equations in (6.23) 182
7 Variable Selection for Frailty Models 183
7.1 Variable Selection 183
7.2 Implied Penalty Functions from the Frailty Models 184
7.3 Variable Selection via the H-Likelihood 187
7.3.1 Penalty Function for Variable Selection 187
7.3.2 Penalized Partial H-Likelihood Procedure 189
7.4 Examples 191
7.5 Variable Selection for the Competing-Risks Frailty Models 201
7.6 Discussion 204
7.7 Appendix 204
7.7.1 Derivation of Score Equations (7.9) for Variable Selection 204
7.7.2 Derivation of the Standard Error Formula (7.12) 205
7.7.3 Variable Selection via the Penalized Marginal Likelihood 206
8 Mixed-Effects Survival Models 208
8.1 Linear Mixed Model with Censoring 208
8.1.1 Estimation Procedure 209
8.1.2 Comparison with Other Methods 212
8.2 Multicomponent Mixed Models with Censoring 214
8.2.1 Model and Estimation Procedure 214
8.2.2 Application to the CGD Data 216
8.3 The AFT Models with LTRC 217
8.3.1 The Swedish Twin Survival Data with LTRC 217
8.3.2 The Model 219
8.3.3 Estimation Procedure Under LTRC 220
8.3.4 Application 223
8.4 Software and Examples Using R 227
8.4.1 Skin Grafts Data: LMM with Censoring 227
8.4.2 CGD Data: Multilevel LMM with Censoring 228
8.5 Discussion 228
8.6 Appendix 229
8.6.1 Proof of the Expectation Identity in (8.2) 229
8.6.2 Proofs of the IWLS Equations (8.7) 230
8.6.3 Proofs of the Two Dispersion Estimators in (8.11) 231
8.6.4 H-Likelihood Procedure for Fitting the Multicomponent LMM 233
8.6.5 Derivation of Model (8.16) 234
8.6.6 Derivations of the Score Equations in (8.21) and (8.22), and Computation of Variance of ?"0362? 235
9 Joint Model for Repeated Measures and Survival Data 237
9.1 Introduction 237
9.2 Joint Model for Repeated Measures and a Single Event-Time Data 238
9.2.1 Estimation Procedure 239
9.2.2 Numerical Study 241
9.3 Joint Model for Repeated Measures and Competing-Risks Data 243
9.4 Software and Examples Using R 245
9.4.1 Joint Analysis for Repeated Measures and a Single Event-Time Data: Renal Transplant Data 245
9.4.2 Joint Analysis of Repeated Measures and Competing-Risks Data: PBC Data 248
9.5 Discussion 251
10 Further Topics 252
10.1 Competing-Risks Frailty Models with Missing Causes of Failure 252
10.1.1 Example: Bladder Cancer Data with Missing Causes of Failure 253
10.2 Frailty Models for Semi-competing-Risks Data 255
10.2.1 Classical Semi-competing-Risks Model 256
10.2.2 Fitting the Semi-competing-Risks Frailty Model 257
10.2.3 Example: Breast Cancer Data 261
10.3 Discussion 263
10.4 Appendix 264
10.4.1 Marginal Likelihood Estimation Procedure 264
10.4.2 Comparison of H-Likelihood with Marginal Likelihood 265
10.4.3 Fourth-order Laplace approximation 266
Appendix Formula for Fitting Fixed and Random Effects 268
A.1 IWLS Procedures 268
A.2 ILS Procedures 268
References 271
Index 284