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 multiplica- tively 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. 1.