Estimation of survival function subject to time-dependent covariates and left truncation

Satten et al. [Satten, G. A., Datta, S., Robins, J. M. (2001). Estimating the marginal survival function in the presence of time dependent covariates. Statis. Prob. Lett. 54:397-403] proposed an estimator [denoted by (S) over cap (t)] of survival function of failure times that is in the class of survival function estimators proposed by Robins [Robins, J. M. (1993). Information recovery and bias adjustment in proportional hazards regression analysis of randomized trials using surrogate markers. In: Proceedings of the American Statistical Association-Biopharmaceutical Section. Alexandria, VA: ASA, pp. 24-33]. The estimator is appropriate when data are subject to dependent censoring. In this article, it is demonstrated that the estimator S(t) can be extended to estimate the survival function when data are subject to dependent censoring and left truncation. In addition, we propose an alternative estimator of survival function [denoted by S-w(t)] that is represented as an inverse-probability-weighted average Satten and Datta [Satten, G. A., Datta, S. (2001). The Kaplan-Meier estimator as an inverse-probability-of-censoring weighted average. Amer. Statist. Ass. 55:207-210]. Simulation results show that when truncation is not severe the mean squared error of S(t) is smaller than that of S-w(t), except for the case when censoring is light. However, when truncation is severe, S-w(t) has the advantage of less bias and the situation can be reversed.