Survival Analysis For Left Censored Data

Abstract

A left censoring scheme is such that the random variable of interest, X, is only observed if it is greater than or equal to a left censoring variable L, otherwise L is observed. The analysis is then based on the pair of random variables (U, δ) where U = max(L, X) and δ = 1{L ≤ X}. The problem concerns the estimation of the survival function SX(t) = Pr{X > t} from a left censored sample where X is assumed to be independent of L. We derive a Left-Kaplan-Meier estimator, \(\hat{\textup{S}}_{\textup{X}}\), as a solution of a backward Doleans differential equation. It is proved that this Left-Kaplan-Meier estimator is self-consistent, thus a generalized maximum likelihood estimator. Following Efron’s (1967) technique for the case of a right-censored scheme, it is shown that the Left-Kaplan-Meier estimator is the same estimator you would obtain through a redistribution to the left algorithm. The consistency of the Left-Kaplan-Meier estimator is established. The influence curves corresponding to \(\hat{\textup{S}}_{\textup{X}}\), are calculated. This provides an alternative derivation of the asymptotic variance of \(\hat{\textup{S}}_{\textup{X}}\), (Reid, 1981). The asymptotic normality then follows through standard arguments.