Nonparametric procedures for survival data

Abstract

In this chapter we give a brief outline of the most important fully non-parametric tools for the analysis of survival data. The non-parametric techniques have established themselves as important tools of survival analysis due to their simplicity and the fact that their properties are well studied and understood. 4.1 The Kaplan-Meier estimator When studying the lifetimes of a population, one often has data that are incomplete, typically in form of a right-censored versions of the survival times. It turns out that even though one does not fully observe the survival times, one can still estimate the distribution of the survival times as well as the cumulative hazard function. We here describe the Nelson-Aalen and Kaplan-Meier estimator in the situation of right-censored survival data. Let T * be a survival time with survival distribution S(t) = P (T * > t) and hazard function α(t) and let C be a right-censoring time that leads to independent censoring. We thus observe T = T * ∧C and the censoring indicator ∆ = I(T * ≤ C). Denote the n independent observation form this generic model by (T i , ∆ i), i = 1,. .. , n. In the one-sample case it is often of interest to estimate the survivor function S(t) = P (T * i > t),