Linear Regression model with Time-Dependent Covariates: its Semiprametric Efficiency Theory and Phenomenon of Local Confounding

Abstract

A simple integration equation is shown to accommodate a rather general class of semiparametric linear regression models with time-dependent covariates. Interestingly this equation also provides as a platform for unifying the accelerated failure time (AFT) model with a wide spectrum of transformation models, including Cox's hazard regression model. Very importantly, by reparametrization under identifiability conditions, all the members in this class can be re-expressed, at least approximately, in a form of AFT model, but not vise versa. So we only focus on a comprehensive semiparametric statistical inference pertaining to the AFT model in this talk. First the semiparametric Fisher information bound is explicitly calculated with right-censored data, and then the overidentified estimating equation (OEE) approach is shown to be semiparametric efficient for achieving the information bound. Second, it is demonstrated that almost all log-rank estimating equations can suffer severe loss of information due to the phenomenon called local confounding. This is primarily caused by the presence sign changes on the ratio of the baseline hazard rate and its derivative, which is found in both the locally asymptotic linearity and the efficient scores. Again the OEE approach is shown to alleviate the damaging effects. Numerically evaluations of this effect are reported in a simulated biological life-history example.