Bootstrapping in survival analysis

Abstract

In this survey we consider the model of right random censorship in which the quantity of primary interest is the lifetime distribution function F. Kaplan and Meier (1958) derived a nonparametric maximum likelihood estimator F-n for F, which is a natural generalization of the empirical distribution function in the complete data case. It was Efron (1981) who first proposed procedures for the construction of bootstrap observations, which then produce a bootstrapped version F-n* of F-n. We discuss the properties of F-n and F-n(*) which lead to the important consistency result: almost surely, as n --> infinity, the process n(1/2)(F-n* - F-n) converges weakly to the same Gaussian limit as the original Kaplan-Meier process n(1/2)(F-n - F). We also deal with the analogous properties for the corresponding quantile processes. Moreover, we consider the question of accuracy of the bootstrap approximation. Finally, we look into the situation of regression models which study the effect of covariates on the distributions of the lifetimes. We consider bootstrap procedures for the regression parameter in Cox's proportional hazards model and for Beran's distribution function estimator in a fixed design nonparametric regression model.