Decision trees for instantaneous hazard rates with adjustment for covariate effects

We investigate the problem of finding confidence sets for a threshold in the baseline hazard function of the Cox proportional hazards model. The aim is to find an effective way of condensing information in the baseline into a small number of estimable parameters. A binary decision tree is used as a working model, with the jump point (threshold) representing a time at which baseline risk changes abruptly between two levels. We find the asymptotic distribution of estimators of best-fitting parameters under an arbitrary misspecification of the working model. The estimators 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. Running title: Decision trees for instantaneous hazards