An Approach to Nonparametric Regression for Life History Data Using Local Linear Fitting

Abstract

Most hazard regression models in survival analysis specify a given functional form to describe the influence of the covariates on the hazard rate. For instance, Cox's model assumes that the covariates act multiplicatively on the hazard rate, and Aalen's additive risk model stipulates that the covariates have a linear additive effect on the hazard rate. In this paper we study a fully nonparametric model which makes no assumption on the association between the hazard rate and the covariates. We propose a class of estimators for the conditional hazard function, the conditional cumulative hazard function and the conditional survival function, and study their large sample properties. When the size of a data set is relatively large, this fully nonparametric approach may provide more accurate information than that acquired from more restrictive models. It may also be used to test whether a particular submodel gives a good fit to a given data set. Because our results are obtained under the multivariate counting process setting of Aalen, they apply to a number of models arising in survival analysis, including various censoring and random truncation models. Our estimators are related to the conditional Nelson-Aalen estimators proposed by Beran for the random censorship model.