In survival analysis, the relationship between a survival time and a covariate is conveniently modeled with the proportional hazards regression model proposed by Cox (1972). This model usually assumes that the covariate has a log-linear effect on the hazard function.;We consider the proportional hazards regression model with a nonparametric risk effect instead of a log-linear effect. We discuss estimation of the risk function and its derivatives in two cases: when the baseline hazard function is parameterized and when it is not parameterized. In the case of a parametric baseline hazard, inference is based on a local version of the likelihood function, while in the case of the nonparametric baseline hazard, a local version of the partial likelihood is used. We establish the asymptotic normality of the resulting maximum local likelihood estimators and the maximum local partial likelihood estimators, respectively. It turns out that in a common situation, both methods have the same asymptotic bias and variance, even though the local partial likelihood uses no information about the baseline hazard function.;The methods are compared to each other via simulations, and the performance of the local partial likelihood method is explored more extensively through further simulations and through application to actual data. Methods for applying the local partial likelihood technique in the multivariate situation are also discussed.
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