Simultaneous marginal survival estimators when doubly censored data is present

Abstract

A doubly censoring scheme occurs when the lifetimes T being measured, from a well-known time origin, are exactly observed within a window [L, R] of observational time and are otherwise censored either from above (right-censored observations) or below (left-censored observations). Sample data consists on the pairs (U, δ) where U = min{R, max{T, L}} and δ indicates whether T is exactly observed (δ = 0), right-censored (δ = 1) or left-censored (δ = −1). We are interested in the estimation of the marginal behaviour of the three random variables T, L and R based on the observed pairs (U, δ). We propose new nonparametric simultaneous marginal estimators ${\hat S_{T}, \hat S_{L}}$ and ${\hat S_{R}}$ for the survival functions of T, L and R, respectively, by means of an inverse-probability-of-censoring approach. The proposed estimators ${\hat S_{T}, \hat S_{L}}$ and ${\hat S_{R}}$ are not computationally intensive, generalize the empirical survival estimator and reduce to the Kaplan-Meier estimator in the absence of left-censored data. Furthermore, ${\hat S_{T}}$ is equivalent to a self-consistent estimator, is uniformly strongly consistent and asymptotically normal. The method is illustrated with data from a cohort of drug users recruited in a detoxification program in Badalona (Spain). For these data we estimate the survival function for the elapsed time from starting IV-drugs to AIDS diagnosis, as well as the potential follow-up time. A simulation study is discussed to assess the performance of the three survival estimators for moderate sample sizes and different censoring levels.