Semiparametric Analysis of an Expanded Cox Proportional Hazards Model with Time-Varying Covariates

Abstract

Time-varying covariates are often encountered in survival analysis. The Cox proportional hazards model can incorporate time-varying covariates, while the interpretation of regression parameters is less straightforward. We instead propose a complementary log-log survival model. When covariates are time-independent, the proposed model reduces to the Cox proportional hazards model; however, when they are time-varying, the proposed model provides a direct interpretation of regression parameters in the survival function. We develop semiparametric estimation procedures based on estimating equations, and establish the asymptotic properties of the estimators for the regression parameters and survival functions. In addition, we include weight functions to the estimating equations to improve efficiency. We demonstrate the proposed methods by simulation studies and application to the Mayo Clinic Primary Biliary Cirrhosis data and data from a landmark HIV randomized prevention trial.