Kernel Estimation of the Baseline Function in the Cox Model

Abstract

Survival analysis is the analysis of time-to-event data. Two important functions in the analysis of survival data are the survival function and the hazard function. The Kaplan-Meier method is widely used to estimate the survival function. One of the objectives of the analysis of survival data might be to examine whether survival times are related to other features. A popular regression model for the analysis of survival data is the Cox proportional hazard regression model. The most commonly used approaches, for the baseline survival function, are the Breslow and Kalbfleisch-Prentice methods. These methods provide a step function estimate of the survivor function and in many instances a continuous estimate would be preferable. For these reason, in this paper we proposed a kernel smoothing technique for baseline estimator, based on Kalbfleisch-Prentice method. We start with kernel smoothing of baseline hazard function, based on Kalbfleisch-Prentice estimator and epanechnikov kernel, than we use it to calculate the baseline survival function. To evaluate the usefulness of the kernel estimator of the baseline function, in the case of right censoring, based on KalbfleischPrentice estimator we conduct simulation studies across a range of conditions, by varying the sample size and censoring rate. We compare it with the smoothing of the Breslow estimator regarding bias.