Survival Analysis

Preface xi 1 Introduction 1 1.1 What is survival analysis and how is it applied? 1 1.2 The history of survival analysis and its progress 2 1.3 General features of survival data structure 3 1.4 Censoring 4 1.5 Time scale and the origin of time 7 1.6 Basic lifetime functions 10 1.7 Organization of the book and data used for illustrations 16 1.8 Criteria for performing survival analysis 17 2 Descriptive approaches of survival analysis 20 2.1 The Kaplan-Meier (product-limit) and Nelson-Aalen estimators 21 2.2 Life table methods 36 2.3 Group comparison of survival functions 46 2.4 Summary 61 3 Some popular survival distribution functions 63 3.1 Exponential survival distribution 63 3.2 The Weibull distribution and extreme value theory 68 3.3 Gamma distribution 73 3.4 Lognormal distribution 77 3.5 Log-logistic distribution 80 3.6 Gompertz distribution and Gompertz-type hazard models 83 3.7 Hypergeometric distribution 89 3.8 Other distributions 91 3.9 Summary 92 4 Parametric regression models of survival analysis 93 4.1 General specifi cations and inferences of parametric regression models 94 4.2 Exponential regression models 103 4.3 Weibull regression models 113 4.4 Log-logistic regression models 127 4.5 Other parametric regression models 135 4.6 Parametric regression models with interval censoring 138 4.7 Summary 142 5 The Cox proportional hazard regression model and advances 144 5.1 The Cox semi-parametric hazard model 145 5.2 Estimation of the Cox hazard model with tied survival times 154 5.3 Estimation of survival functions from the Cox proportional hazard model 161 5.4 The hazard rate model with time-dependent covariates 169 5.5 Stratified proportional hazard rate model 176 5.6 Left truncation, left censoring, and interval censoring 183 5.7 Qualitative factors and local tests 191 5.8 Summary 199 6 Counting processes and diagnostics of the Cox model 201 6.1 Counting processes and the martingale theory 202 6.2 Residuals of the Cox proportional hazard model 213 6.3 Assessment of proportional hazards assumption 222 6.4 Checking the functional form of a covariate 236 6.5 Identifi cation of infl uential observations in the Cox model 243 6.6 Summary 253 7 Competing risks models and repeated events 255 7.1 Competing risks hazard rate models 256 7.2 Repeated events 282 7.3 Summary 308 8 Structural hazard rate regression models 310 8.1 Some thoughts about the structural hazard regression models 310 8.2 Structural hazard rate model with retransformation of random errors 313 8.3 Summary 344 9 Special topics 347 9.1 Informative censoring 347 9.2 Bivariate and multivariate survival functions 352 9.3 Frailty models 359 9.4 Mortality crossovers and the maximum life span 376 9.5 Survival convergence and the preceding mortality crossover 384 9.6 Sample size required and power analysis 398 9.7 Summary 403 Appendix A The delta method 405 Appendix B Approximation of the variance-covariance matrix for the predicted probabilities from results of the multinomial logit model 407 Appendix C Simulated patient data on treatment of PTSD (n = 255) 410 Appendix D SAS code for derivation of phi estimates in reduced-form equations 417 Appendix E The analytic result of kappa*(x) 422 References 424 Index 438