Estimation of Non-Linear and Time-Varying Effects in Survival Data using Smoothing Splines

Abstract

The major interests of survival analysis are either to compare the failure time distribution function or to assess the effects of covariate on survival via appropriate hazards regression models. Cox‟s proportional hazards model (Cox, 1972) is the most widely used framework, the model assumes that the effect on the hazard function of a particular factor of interest remains unchanged throughout the observation period (Proportionality assumption). For a continuous prognostic factor the model further assumes linear effect on the log hazard function (Linearity assumption). Assumptions that many authors have found to be questionable when violated since they may result to biased results and conclusions and as such non-linear risk functions have been suggested as the suitable models. In this paper, we propose a flexible method that models dynamic effects in survival data within the Cox regression framework. The method is based on penalized splines. The model offers the chance to easily verify the presence of PH and time-variation. We provide a detailed analysis and derivation of the penalized splines in the context of survival data. Key Words: Non-linear, Penalized splines, Proportional Hazard, Survival analysis