Nonparametric Survival Curve Estimation

Abstract

We have seen that there are a wide variety of hazard function shapes to choose from if one models survival data using a parametric model. But which parametric model should one use for a particular application? When modeling human or animal survival, it is hard to know what parametric family to choose, and often none of the available families has sufficient flexibility to model the actual shape of the distribution. Thus, in medical and health applications, nonparametric methods, which have the flexibility to account for the vagaries of the survival of living things, have considerable advantages. In this chapter we will discuss non-parametric estimators of the survival function. The most widely used of these is the product-limit estimator, also known as the Kaplan-Meier estimatorKaplan-Meier estimator. This estimator, first proposed by Kaplan and Meier [35], is the product over the failure times of the conditional probabilities of surviving to the next failure time. Formally, it is given by {\^{S}}(t)=∏ti≤t(1−q̂i)=∏ti≤t1−dini{\$}{\$}{\backslash}displaystyle{\{}{\backslash}hat{\{}S{\}}(t) ={\backslash}prod {\backslash}limits {\_}{\{}t{\_}{\{}i{\}}{\backslash}leq t{\}}(1 -{\backslash}hat{\{}q{\}}{\_}{\{}i{\}}) ={\backslash}prod {\backslash}limits {\_}{\{}t{\_}{\{}i{\}}{\backslash}leq t{\}}{\backslash}left (1 - {\backslash}frac{\{}d{\_}{\{}i{\}}{\}} {\{}n{\_}{\{}i{\}}{\}}{\backslash}right ){\}}{\$}{\$}where niis the number of subjects at risk at time ti, and diis the number of individuals who fail at that time. The example data in Table 1.1may be used to illustrate the construction of the Kaplan-Meier estimate, as shown in Table 3.1.Table 3.1Kaplan-Meier estimatetinidiqi1 − qiSi=∏(1−qi){\$}{\$}S{\_}{\{}i{\}} ={\backslash}prod (1 - q{\_}{\{}i{\}}){\$}{\$}2610.1670.8330.8464510.2000.8000.6936310.3330.6670.497