Survival analysis:models and applications

Preface
1 Introduction
1.1 What is survival analysis and how is it applied?
1.2 The history of survival analysis and its progress
1.3 General features of survival data structure
1.4 Censoring
1.4.1 Mechanisms of right censoring
1.4.2 Left censoring,interval censoring,and left truncation
1.5 Time scale and the origin of time
1.5.1 Observational studies
1.5.2 Biomedical studies
1.5.3 Health care utilization
1.6 Basic lifetime functions
1.6.1 Continuous lifetime functions
1.6.2 Discrete lifetime functions
1.6.3 Basic likelihood functions for right,left,and interval censoring
1.7 Organization of the book and data used for illustrations
1.8 Criteria for performing survival analysis
2 Descriptive approaches of survival analysis
2.1 The Kaplan-Meier(product-limit)and Nelson-Aalen estimators
2.1.1 Kaplan-Meier estimating procedures with or without censoring
2.1.2 Formulation of the Kaplan-Meier and Nelson-Aalen estimators
2.1.3 Variance and standard error of the survival function
2.1.4 Confidence intervals and confidence bands of the survival function
2.2 Life table methods
2.2.1 Life table indicators
2.2.2 Multistate life tables
2.2.3 Illustration: Life table estimates for older Americans
2.3 Group comparison of survival functions
2.3.1 Logrank test for survival curves of two groups
2.3.2 The Wilcoxon rank sum test on survival curves of two groups
2.3.3 Comparison of survival functions for more than two groups
2.3.4 Illustration: Comparison of survival curves between married
and unmarried persons
2.4 Summary
3 Some popular survival distribution functions
3.1 Exponential survival distribution
3.2 The Weibull distribution and extreme value theory
3.2.1 Basic specifications of the Weibull distribution
3.2.2 The extreme value distribution
3.3 Gamma distribution
3.4 Lognormal distribution
3.5 Log-logistic distribution
3.6 Gompertz distribution and Gompertz-type hazard models
3.7 Hypergeometric distribution
3.8 Other distributions
3.9 Summary
4 Parametric regression models of survival analysis
4.1 General specifications and inferences of parametric regression models
4.1.1 Specifications of parametric regression models on the hazard function
4.1.2 Specifications of accelerated failure time regression models
4.1.3 Inferences of parametric regression models and likelihood functions
4.1.4 Procedures of maximization and hypothesis testing on ML estimates
4.2 Exponential regression models
4.2.1 Exponential regression model on the hazard function
4.2.2 Exponential accelerated failure time regression model
4.2.3 Illustration: Exponential regression model on marital status and survival among older Americans
4.3 Weibull regression models
4.3.1 Weibull hazard regression model
4.3.2 Weibull accelerated failure time regression model
4.3.3 Conversion of Weibull proportional hazard and AFT parameters
4.3.4 Illustration: A Weibull regression model on marital status and survival among older Americans
4.4 Log-logistic regression models
4.4.1 Specifications of the log-logistic AFT regression model
4.4.2 Retransformation of AFT parameters to untransformed log-logistic parameters
4.4.3 Illustration: The log-logistic regression model on marital status and survival among the oldest old Americans
4.5 Other parametric regression models
4.5.1 The lognormal regression model
4.5.2 Gamma distributed regression models
4.6 Parametric regression models with interval censoring
4.6.1 Inference of parametric regression models with interval censoring
4.6.2 Illustration: A parametric survival model with independent interval censoring
4.7 Summary
5 The Cox proportional hazard regression model and advances
5.1 The Cox semi-parametric hazard model
5.1.1 Basic specifications of the Cox proportional hazard model
5.1.2 Partial likelihood
5.1.3 Procedures of maximization and hypothesis testing on partial likelihood
5.2 Estimation of the Cox hazard model with tied survival times
5.2.1 The discrete-time logistic regression model
5.2.2 Approximate methods handling ties in the proportional hazard model
5.2.3 Illustration on tied survival data: Smoking cigarettes and the mortality of older Americans
5.3 Estimation of survival functions from the Cox proportional hazard model
5.3.1 The Kalbfleisch-Prentice method
5.3.2 The Breslow method
5.3.3 Illustration: Comparing survival curves for smokers and nonsmokers among older Americans
5.4 The hazard rate model with time-dependent covariates
5.4.1 Categorization of time-dependent covariates
5.4.2 The hazard rate model with time-dependent covariates
5.4.3 Illustration: A hazard model on time-dependent marital status and the mortality of older Americans
5.5 Stratified proportional hazard rate model
5.5.1 Specifications of the stratified hazard rate model
5.5.2 Illustration: Smoking cigarettes and the mortality of older Americans with stratification on three age groups
5.6 Left truncation,left censoring,and interval censoring
5.6.1 The Cox model with left truncation,left censoring,and interval censoring
5.6.2 Illustration: Analyzing left truncated survival data on smoking cigarettes and the mortality of unmarried older Americans
5.7 Qualitative factors and local tests
5.7.1 Qualitative factors and scaling approaches
5.7.2 Local tests
5.7.3 Illustration of local tests: Educational attainment and the mortality of older Americans
5.8 Summary
6 Counting processes and diagnostics of the Cox model
6.1 Counting processes and the martingale theory
6.1.1 Counting processes
6.1.2 The martingale theory
6.1.3 Stochastic integrated processes as martingale transforms
6.1.4 Martingale central limit theorems
6.1.5 Counting process formulation for the Cox model
6.2 Residuals of the Cox proportional hazard model
6.2.1 Cox-Snell residuals
6.2.2 Schoenfeld residuals
6.2.3 Martingale residuals
6.2.4 Score residuals
6.2.5 Deviance residuals
6.2.6 Illustration: Residual analysis on the Cox model of smoking cigarettes and the mortality of older Americans
6.3 Assessment of proportional hazards assumption
6.3.1 Checking proportionality by adding a time-dependent variable
6.3.2 The Andersen plots for checking proportionality
6.3.3 Checking proportionality with scaled Schoenfeld residuals
6.3.4 The Arjas plots
6.3.5 Checking proportionality with cumulative sums of martingale-based residuals
6.3.6 Illustration: Checking the proportionality assumption in the
Cox model for the effect of age on the mortality of older Americans
6.4 Checking the functional form of a covariate
6.4.1 Checking model fit statistics for different link functions
6.4.2 Checking the functional form with cumulative sums of martingale-based residuals
6.4.3 Illustration: Checking the functional form of age in the Cox model on the mortality of older Americans
6.5 Identification of influential observations in the Cox model
6.5.1 The likelihood displacement statistic approximation
6.5.2 LMAX statistic for identification of influential observations
6.5.3 Illustration: Checking influential observations in the Cox model on the mortality of older Americans
6.6 Summary
7 Competing risks models and repeated events
7.1 Competing risks hazard rate models
7.1.1 Latent failure times of competing risks and model specifications
7.1.2 Competing risks models and the likelihood function without covariates
7.1.3 Inference for competing risks models with covariates
7.1.4 Competing risks model using the multinomial logit regression
7.1.5 Competing risks model with dependent failure types
7.1.6 Illustration of competing risks models: Smoking cigarettes and the mortality of older Americans from three causes of death
7.2 Repeated events
7.2.1 Andersen and Gill model(AG)
7.2.2 PWP total time and gap time models(PWP-CP and PWP-GT)
7.2.3 The WLW model and extensions
7.2.4 Proportional rate and mean functions of repeated events
7.2.5 Illustration: The effects of a medical treatment on repeated patient visits
7.3 Summary
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8 Structural hazard rate regression models
9 Special topics
Appendix A The delta method
Appendix B Approximation of the variance-covariance matrix for the predicted probabilities from results of the multinomial logit model
Appendix C Simulated patient data on treatment of PTSD(n = 255)
Appendix D SAS code for derivation of ψ estimates in reduced-form equations
Appendix E The analytic result of κ*(x)
References
Index