Efficient estimation of the cox model when incorporating the subgroup restricted mean survival time

The restricted mean survival time has been widely used in the field of medical research because of its clear physical and simple clinical interpretation. In this paper, we propose an efficient estimation that incorporates the auxiliary restricted mean survival information into the estimation of the proportional hazard (PH) model. Compared to conventional models that do not incorporate available auxiliary information, the proposed method improves efficiency in estimating regression parameters by utilizing the double empirical likelihood method. We prove that the estimator asymptotically follows a multivariate normal distribution with a covariance matrix that can be consistently estimated. To address scenarios where the PH assumption is violated, we also extended the method to the stratified Cox model. In addition, simulation studies show that the proposed estimators are more efficient than those derived from the conventional partial likelihood approach. A type 2 diabetes dataset is then used to evaluate the risk of antidiabetic drugs and demonstrate the proposed method.