Estimating optimal step-function approximations to instantaneous hazard rates

Abstract

We investigate the problem of estimating the best binary decision tree approximation to the baseline hazard function in the Cox proportional hazards model. Our motivation is to find an effective way of condensing key functional information in the baseline hazard into a small number of estimable parameters. The parameters consist of a threshold and two hazard levels, one to the left of the threshold and one to the right, defined in terms of the best $L^2$ approximation to the nonparametric baseline hazard function. Estimators of these parameters are introduced and shown to converge at cube-root rate to a non-normal limit distribution. Two alternate ways of constructing confidence intervals for the threshold are compared. Results from a simulation study and an example concerning a threshold for the age of onset of schizophrenia in a large cohort study are discussed.