Survival Analysis

Xian Liu, Department of Psychiatry, F. Edward Hebert School of Medicine; Uniformed Services University of the Health Sciences, Bethesda, USA.

• Preface xi1 Introduction 11.1 What is survival analysis and how is it applied? 11.2 The history of survival analysis and its progress 21.3 General features of survival data structure 31.4 Censoring 41.4.1 Mechanisms of right censoring 51.4.2 Left censoring, interval censoring, and left truncation 61.5 Time scale and the origin of time 71.5.1 Observational studies 81.5.2 Biomedical studies 91.5.3 Health care utilization 91.6 Basic lifetime functions 101.6.1 Continuous lifetime functions 101.6.2 Discrete lifetime functions 121.6.3 Basic likelihood functions for right, left, and interval censoring 141.7 Organization of the book and data used for illustrations 161.8 Criteria for performing survival analysis 172 Descriptive approaches of survival analysis 202.1 The Kaplan–Meier (product-limit) and Nelson–Aalen estimators 212.1.1 Kaplan–Meier estimating procedures with or without censoring 212.1.2 Formulation of the Kaplan–Meier and Nelson–Aalen estimators 242.1.3 Variance and standard error of the survival function 272.1.4 Confi dence intervals and confi dence bands of the survival function 292.2 Life table methods 362.2.1 Life table indicators 372.2.2 Multistate life tables 402.2.3 Illustration: Life table estimates for older Americans 442.3 Group comparison of survival functions 462.3.1 Logrank test for survival curves of two groups 482.3.2 The Wilcoxon rank sum test on survival curves of two groups 512.3.3 Comparison of survival functions for more than two groups 552.3.4 Illustration: Comparison of survival curves between married and unmarried persons 582.4 Summary 613 Some popular survival distribution functions 633.1 Exponential survival distribution 633.2 The Weibull distribution and extreme value theory 683.2.1 Basic specifi cations of the Weibull distribution 683.2.2 The extreme value distribution 723.3 Gamma distribution 733.4 Lognormal distribution 773.5 Log-logistic distribution 803.6 Gompertz distribution and Gompertz-type hazard models 833.7 Hypergeometric distribution 893.8 Other distributions 913.9 Summary 924 Parametric regression models of survival analysis 934.1 General specifi cations and inferences of parametric regression models 944.1.1 Specifi cations of parametric regression models on the hazard function 944.1.2 Specifi cations of accelerated failure time regression models 964.1.3 Inferences of parametric regression models and likelihood functions 994.1.4 Procedures of maximization and hypothesis testing on ML estimates 1014.2 Exponential regression models 1034.2.1 Exponential regression model on the hazard function 1034.2.2 Exponential accelerated failure time regression model 1064.2.3 Illustration: Exponential regression model on marital status and survival among older Americans 1084.3 Weibull regression models 1134.3.1 Weibull hazard regression model 1144.3.2 Weibull accelerated failure time regression model 1154.3.3 Conversion of Weibull proportional hazard and AFT parameters 1174.3.4 Illustration: A Weibull regression model on marital status and survival among older Americans 1214.4 Log-logistic regression models 1274.4.1 Specifi cations of the log-logistic AFT regression model 1274.4.2 Retransformation of AFT parameters to untransformed log-logistic parameters 1294.4.3 Illustration: The log-logistic regression model on marital status and survival among the oldest old Americans 1314.5 Other parametric regression models 1354.5.1 The lognormal regression model 1364.5.2 Gamma distributed regression models 1374.6 Parametric regression models with interval censoring 1384.6.1 Inference of parametric regression models with interval censoring 1384.6.2 Illustration: A parametric survival model with independent interval censoring 1394.7 Summary 1425 The Cox proportional hazard regression model and advances 1445.1 The Cox semi-parametric hazard model 1455.1.1 Basic specifi cations of the Cox proportional hazard model 1455.1.2 Partial likelihood 1475.1.3 Procedures of maximization and hypothesis testing on partial likelihood 1505.2 Estimation of the Cox hazard model with tied survival times 1545.2.1 The discrete-time logistic regression model 1545.2.2 Approximate methods handling ties in the proportional hazard model 1555.2.3 Illustration on tied survival data: Smoking cigarettes and the mortality of older Americans 1575.3 Estimation of survival functions from the Cox proportional hazard model 1615.3.1 The Kalbfl eisch–Prentice method 1625.3.2 The Breslow method 1645.3.3 Illustration: Comparing survival curves for smokers and nonsmokers among older Americans 1655.4 The hazard rate model with time-dependent covariates 1695.4.1 Categorization of time-dependent covariates 1695.4.2 The hazard rate model with time-dependent covariates 1715.4.3 Illustration: A hazard model on time-dependent marital status and the mortality of older Americans 1735.5 Stratified proportional hazard rate model 1765.5.1 Specifications of the stratifi ed hazard rate model 1775.5.2 Illustration: Smoking cigarettes and the mortality of older Americans with stratifi cation on three age groups 1785.6 Left truncation, left censoring, and interval censoring 1835.6.1 The Cox model with left truncation, left censoring, and interval censoring 1845.6.2 Illustration: Analyzing left truncated survival data on smoking cigarettes and the mortality of unmarried older Americans 1855.7 Qualitative factors and local tests 1915.7.1 Qualitative factors and scaling approaches 1915.7.2 Local tests 1935.7.3 Illustration of local tests: Educational attainment and the mortality of older Americans 1955.8 Summary 1996 Counting processes and diagnostics of the Cox model 2016.1 Counting processes and the martingale theory 2026.1.1 Counting processes 2026.1.2 The martingale theory 2046.1.3 Stochastic integrated processes as martingale transforms 2076.1.4 Martingale central limit theorems 2086.1.5 Counting process formulation for the Cox model 2116.2 Residuals of the Cox proportional hazard model 2136.2.1 Cox–Snell residuals 2136.2.2 Schoenfeld residuals 2146.2.3 Martingale residuals 2166.2.4 Score residuals 2186.2.5 Deviance residuals 2196.2.6 Illustration: Residual analysis on the Cox model of smoking cigarettes and the mortality of older Americans 2206.3 Assessment of proportional hazards assumption 2226.3.1 Checking proportionality by adding a time-dependent variable 2256.3.2 The Andersen plots for checking proportionality 2276.3.3 Checking proportionality with scaled Schoenfeld residuals 2286.3.4 The Arjas plots 2296.3.5 Checking proportionality with cumulative sums of martingale-based residuals 2306.3.6 Illustration: Checking the proportionality assumption in the Cox model for the effect of age on the mortality of older Americans 2326.4 Checking the functional form of a covariate 2366.4.1 Checking model fit statistics for different link functions 2366.4.2 Checking the functional form with cumulative sums of martingale-based residuals 2376.4.3 Illustration: Checking the functional form of age in the Cox model on the mortality of older Americans 2396.5 Identifi cation of infl uential observations in the Cox model 2436.5.1 The likelihood displacement statistic approximation 2446.5.2 LMAX statistic for identifi cation of infl uential observations 2476.5.3 Illustration: Checking influential observations in the Cox model on the mortality of older Americans 2486.6 Summary 2537 Competing risks models and repeated events 2557.1 Competing risks hazard rate models 2567.1.1 Latent failure times of competing risks and model specifications 2567.1.2 Competing risks models and the likelihood function without covariates 2597.1.3 Inference for competing risks models with covariates 2617.1.4 Competing risks model using the multinomial logit regression 2637.1.5 Competing risks model with dependent failure types 2667.1.6 Illustration of competing risks models: Smoking cigarettes and the mortality of older Americans from three causes of death 2687.2 Repeated events 2827.2.1 Andersen and Gill model (AG) 2837.2.2 PWP total time and gap time models (PWP-CP and PWP-GT) 2867.2.3 The WLW model and extensions 2887.2.4 Proportional rate and mean functions of repeated events 2917.2.5 Illustration: The effects of a medical treatment on repeated patient visits 2947.3 Summary 3088 Structural hazard rate regression models 3108.1 Some thoughts about the structural hazard regression models 3108.2 Structural hazard rate model with retransformation of random errors 3138.2.1 Model specification 3148.2.2 The estimation of the full model 3178.2.3 The estimation of reduced-form equations 3188.2.4 Decomposition of causal effects on hazard rates and survival functions 3238.2.5 Illustration: The effects of veteran status on the mortality of older Americans and its pathways 3278.3 Summary 3449 Special topics 3479.1 Informative censoring 3479.1.1 Selection model 3489.1.2 Sensitivity analysis models 3519.1.3 Comments on current models handling informative censoring 3529.2 Bivariate and multivariate survival functions 3529.2.1 Inference of the bivariate survival model 3539.2.2 Estimation of bivariate and multivariate survival models 3559.2.3 Illustration of marginal models handling multivariate survival data 3599.3 Frailty models 3599.3.1 Hazard models with individual frailty 3609.3.2 The correlated frailty model 3649.3.3 Illustration of frailty models: The effect of veteran status on the mortality of older Americans revisited 3669.4 Mortality crossovers and the maximum life span 3769.4.1 Basic specifications 3789.4.2 Relative acceleration of the hazard rate and timing of mortality crossing 3819.4.3 Mathematical conditions for maximum life span and mortality crossover 3839.5 Survival convergence and the preceding mortality crossover 3849.5.1 Mathematical proofs for survival convergence and mortality crossovers 3859.5.2 Simulations 3879.5.3 Explanations for survival convergence and the preceding mortality crossover 3939.6 Sample size required and power analysis 3989.6.1 Calculation of sample size required 3999.6.2 Illustration: Calculating sample size required 4019.7 Summary 403Appendix A The delta method 405Appendix B Approximation of the variance–covariance matrix for the predicted probabilities from results of the multinomial logit model 407Appendix C Simulated patient data on treatment of PTSD (n = 255) 410Appendix D SAS code for derivation of φ estimates in reduced-form equations 417Appendix E The analytic result of κ*(x) 422References 424Index 438