Theory and Application of Kernel Estimation Methods for Covariate-Varying Effects in Survival Analysis

Abstract

Study of prognostic and predictive biomarkers play an important role in the design and analysis of clinical trials. The development of personalized medicine calls for the investigation of the biomarker effect on the treatment in survival analysis. To study the treatment-biomarker interaction, we employ kernel smoothing techniques to propose two methods. One is to use the Cox model under the proportional hazard assumption, where the treatment coefficient is modeled as a function of a biomarker without specifying the functional form. We study this model with multivariate covariates, and contribute to the theoretical development. We discuss the maximal deviation of the estimate, then construct simultaneous confidence bands for both coefficient and derivative functions. When the proportional hazard assumption is violated, we propose to use restricted mean survival time as the tool to study the association between the treatment and biomarker. We apply kernel regression to make statistical inference for the one-sample problem of the biomarker effect on the restricted mean survival time. We then study the two-sample problem with the same method. Treatment-biomarker interaction here is modeled as the difference of the restricted mean survival times on the same level of biomarker between treatment groups. For both methods, multiple confidence bands are built and associated hypothesis testing problems are discussed. We assess the proposed methods in extensive simulation studies, and find good finite sample performance. We also apply the methods to real world clinical trial datasets to demonstrate how the treatment effect is associated with the varying level of the biomarker.