Estimations in homoscedastic linear regression models with censored data: an empirical process approach

Abstract

Pertaining to the estimating equations proposed by Tsiatis, based on the linear rank test, we show the existence of local confounding between the baseline hazard function and the covariates. Due to the local confounding, an estimating equation in Tsiatis' family with a larger time point of truncation could contain less information about the regression parameter than the estimating equation with a smaller time point of truncation. This phenomenon further indicates significant loss of efficiency of Tsiatis' estimating equations as well as the power loss of log-rank type tests when the baseline hazard function goes up and down along the time scale. To take care of this local confounding without using nonparametric estimates of the derivative of the baseline hazard function, we propose the empirical process approach (EPA) based on an empirical process constructed from Tsiatis' log-rank estimating equation by varying its truncating time point. The EPA will provide very tractable estimations of the regression parameters as well as Pearson's chi-squared statistics for testing the model's assumptions. Specifically, the performance of the EPA estimator is shown to be very close to the best estimator in Tsiatis' family.