When designing and analyzing a medical study, researchers focusing on survival data must take into account the heterogeneity of the study population: due to uncontrollable variation, some members change states more rapidly than others. Survival data measures the time to a certain event or change of state. For example, the event may be death, occurrence of disease, time to an epileptic seizure, or time from response until disease relapse. Frailty is a convenient method to introduce unobserved proportionality factors that modify the hazard functions of an individual. In spite of several new research developments on the topic, there are very few books devoted to frailty models. Modeling Survival Data Using Frailty Models covers recent advances in methodology and applications of frailty models, and presents survival analysis and frailty models ranging from fundamental to advanced. Eight data on survival times with covariates sets are discussed, and analysis is carried out using the R statistical package. This book covers: Basic concepts in survival analysis, shared frailty models and bivariate frailty models Parametric distributions and their corresponding regression models Nonparametric Kaplan–Meier estimation and Cox's proportional hazard model The concept of frailty and important frailty models Different estimation procedures such as EM and modified EM algorithms Logrank tests and CUSUM of chi-square tests for testing frailty Shared frailty models in different bivariate exponential and bivariate Weibull distributions Frailty models based on L?vy processes Different estimation procedures in bivariate frailty models Correlated gamma frailty, lognormal and power variance function frailty models Additive frailty models Identifiability of bivariate frailty and correlated frailty models The problem of analyzing time to event data arises in a number of applied fields, such as medicine, biology, public health, epidemiology, engineering, economics, and demography. Although the statistical tools presented in this book are applicable to all these disciplines, this book focuses on frailty in biological and medical statistics, and is designed to prepare students and professionals for experimental design and analysis.
Author(s): David D. Hanagal
Edition: 1
Publisher: Chapman and Hall\/CRC
Year: 2011
Language: English
Commentary: index missing
Pages: 332
Modeling Survival Data Using Frailty Models......Page 2
Modeling Survival Data Using Frailty Models......Page 4
Dedication
......Page 6
Contents......Page 8
List of Tables......Page 14
List of Figures......Page 16
Preface......Page 18
Acknowledgments......Page 19
About the Author......Page 20
Part I Basic Concepts in Survival Analysis......Page 21
1.1 Introduction......Page 23
1.2 Bone Marrow Transplantation (BMT) for Leukemia......Page 24
1.3 Remission Duration from a Clinical Trial for Acute
Leukemia......Page 25
1.4 Times of Infection of Kidney Dialysis Patients......Page 27
1.7 Kidney Dialysis (HLA) Patients Data......Page 28
1.8 Diabetic Retinopathy Data......Page 30
1.9 Myeloma Data......Page 32
1.10.2 Failure (or Hazard) Rate......Page 33
Bivariate and Multivariate Survival Function......Page 35
1.11.1 Censored Type I Data......Page 36
1.11.4 Multicensored Data......Page 37
1.11.5 Separating out Failure Modes......Page 38
2.1 Introduction......Page 39
2.2 Exponential Distribution......Page 40
2.3 Weibull Distribution......Page 41
2.4 Extreme Value Distributions......Page 43
2.5 Lognormal......Page 45
2.6 Gamma......Page 46
2.7 Loglogistic......Page 49
2.8 Maximum Likelihood Estimation......Page 50
2.9 Parametric Regression Models......Page 55
2.9.1 Example 2.1......Page 60
2.9.2 Example 2.2......Page 61
3.1 Empirical Survival Function......Page 63
3.2.1 Probability Plotting......Page 64
3.2.2 Hazard and Cumulative Hazard Plotting......Page 67
3.2.3 Exponential and Weibull Hazard Plots......Page 68
Advantages of Graphical Methods of Estimation
......Page 70
Calculating Kaplan-Meier Estimates......Page 71
A General Expression for K-M Estimates......Page 72
Nelson-Aalen Estimator......Page 73
Example 3.5:......Page 74
3.5 Comparison between Two Survival Functions......Page 79
Wilcoxon Test:......Page 80
Example 3.6:......Page 81
Proportional Hazards Model Assumption......Page 82
Properties and Applications of the Proportional Hazards Model......Page 83
Example 3.7:......Page 85
Part II Univariate and Shared Frailty Models for Survival Data......Page 89
4.1 Introduction......Page 91
4.2 The Definition of Shared Frailty
......Page 93
4.3 The Implications of Frailty......Page 94
4.4 The Conditional Parametrization......Page 96
4.5 The Marginal Parametrization......Page 97
4.7 Frailty as a Model of Stochastic Hazar......Page 98
4.8 Identifi
ability of Frailty Models......Page 99
5.2 Gamma Frailty......Page 101
5.3 Positive Stable Frailty......Page 103
5.4 Power Variance Function Frailty......Page 105
5.5 Compound Poisson Frailty......Page 106
5.6 Compound Poisson Distribution with Random Scale......Page 109
5.7 Frailty Models in Hierarchical Likelihood......Page 112
5.8 Frailty Models in Mixture Distributions......Page 114
5.8.1 Gamma Frailty in Weibull Mixture......Page 115
5.8.3 PVF Frailty in Weibull Mixture......Page 116
5.9 Piecewise Gamma Frailty Model......Page 117
5.9.1 Frailty Models and Dependence Function......Page 119
5.9.2 Example: Epilepsy Data......Page 122
6.1 Introduction......Page 125
6.2 Inference for the Shared Frailty Model......Page 126
6.3 The EM Algorithm......Page 128
6.4 The Gamma Frailty Model......Page 130
6.5 The Positive Stable Frailty Model......Page 131
6.6.1 Application to Seizure Data......Page 133
6.7 Modified EM (MEM) Algorithm for Gamma Frailty
Models......Page 134
6.8 Application......Page 136
6.9 Discussion......Page 137
7.2 Analysis for Bone Marrow Transplantation (BMT) Data......Page 139
7.3 Analysis for Acute Leukemia Data......Page 142
7.4 Analysis for HLA Data......Page 145
7.5 Analysis for Kidney Infection Data......Page 149
7.6 Analysis of Litters of Rats......Page 151
7.7 Analysis for Diabetic Retinopathy Data......Page 153
8.1 Introduction......Page 157
8.2 Tests for Gamma Frailty Based on Likelihood Ratio
and Score Tests......Page 158
8.2.1 The Model and the Main Results......Page 159
8.2.2 Analysis of Diabetic Retinopathy......Page 162
8.3 Logrank Tests for Testing B=0......Page 163
8.3.1 Notations and Review......Page 164
8.3.2 Parametric Tests for Uncensored Samples......Page 166
8.3.3 Nonparametric Tests for Uncensored Samples......Page 168
8.3.4 Effect of Censoring......Page 171
8.3.5 Some Numerical Examples......Page 174
8.4 Test for Heterogeneity in Kidney Infection Data......Page 175
8.4.1 Models and Methods......Page 177
9.1 Introduction......Page 181
9.2.1 Marshall-Olkin (M-O):......Page 182
9.2.3 Freund......Page 183
9.2.4 Proschan-Sullo (P-S):......Page 184
9.3.1 Weibull Extension of BVE of Gumbel......Page 186
9.3.2 Weibull Extension of BVE of Marshall-Olkin......Page 190
9.3.3 Weibull Extension of BVE of Block-Basu......Page 191
9.3.4 Weibull Extension of BVE of Freund and Proschan-
Sullo......Page 192
9.4.1 Weibull Extension of BVE of Gumbel......Page 194
9.4.3 Weibull Extension of BVE of Block-Basu......Page 196
9.5.1 Weibull Extension of BVE Models......Page 197
9.6.1 Estimation of Parameters......Page 199
9.8 Compound Poisson (with Random Scale) Frailty in BVW Models......Page 201
9.9 Estimation and Tests for Frailty under BVW Baseline......Page 202
10.1 Introduction......Page 205
10.1.1 Biological Interpretation of Failure Rat......Page 206
10.1.2 A Model for Random Failure Rate Processes......Page 207
10.2 Levy Processes and Subordinators......Page 209
10.2.1 Standard Compound Poisson Process......Page 210
10.2.5 PVF Processes......Page 211
10.2.6 Special Cases......Page 212
10.3.1 Example 10.1: Gamma Process......Page 213
10.3.2 Example 10.2: Compound Poisson Process......Page 214
10.4 Other Frailty Process Constructions......Page 215
10.5 Hierarchical Levy Frailty Models......Page 216
10.5.1 Application to the Infant Mortality Data......Page 220
Part III Bivariate Frailty Models for Survival Data......Page 223
11.1 Introduction......Page 225
11.2 Bivariate Frailty Models and Laplace Transforms......Page 226
11.4 The Problem of Confounding......Page 227
11.5 A General Model of Covariate Dependence......Page 228
11.6 Pseudo-Frailty Model......Page 230
11.7 Likelihood Construction......Page 231
11.8 Semiparametric Representations......Page 232
11.9.2 Basegroup Estimation Method......Page 234
11.9.3 Estimation Methods Based on the EM Algorithm......Page 235
11.9.4 Profile Estimation for Frailty Models......Page 237
11.9.5 Profile Estimation for Transformation Models......Page 239
11.9.6 Conclusions......Page 241
12.1 Introduction......Page 243
12.3 Correlated Power Variance Function Frailty Model......Page 245
12.4 Genetic Analysis of Duration......Page 246
12.4.1 Correlation Coecient......Page 247
12.4.2 Six Genetic Models of Frailty......Page 249
12.4.3 Danish Twins Survival Data......Page 250
12.4.4 Results: Genetics of Frailty and Longevity......Page 251
12.5 General Bivariate Frailty Model......Page 253
12.5 General Bivariate Frailty Model......Page 255
12.5.2 Lognormal Model......Page 256
12.5.3 Estimation Strategies......Page 257
12.6 Correlated Compound Poisson Frailty for the
Bivariate Survival Lifetimes......Page 258
12.7 Applications......Page 260
13.1 Introduction......Page 263
13.2 Modeling Multivariate Survival Data Using the Frailty Model......Page 265
13.3 Correlated Frailty Model......Page 266
13.4 Relations to Other Frailty Models......Page 267
13.4.3 Twin Model......Page 268
13.4.5 Genetic Model......Page 269
13.4.7 Competing Risks......Page 270
13.4.8 Example......Page 271
13.5.1 Example 13.1......Page 275
13.5.2 Example 13.2......Page 276
13.5.3 Example 13.3......Page 277
13.5.4 Application in Danish Adoptive Register Data......Page 279
13.6 Additive Genetic Gamma Frailty for Linkage Analysis of Diseases......Page 283
13.6.1 Genetic Frailties Defined by Multiple Unlinked Disease Loci......Page 285
13.6.2 Expected Genetic Frailties over the Inheritance Vectors......Page 286
13.6.4 Conditional Hazards Ratio for Sib Pairs......Page 288
13.6.6 Breast Cancer Data Example......Page 290
14.1 Introduction......Page 293
14.2 Identifiability of Bivariate Frailty Models......Page 295
14.3 Identifiability of Correlated Frailty Models......Page 296
14.4.1 Bivariate Frailty Models with Infinite Mean
......Page 297
14.4.2 Bivariate Frailty Models with Finite Mean......Page 298
14.5 Discussion......Page 299
Appendix......Page 301
Bibliography......Page 311
