Piecewise proportional hazards models with interval-censored data

We consider the piecewise proportional hazards (PWPH) model with interval-censored (IC) relapse times under the distribution-free set-up. The partial likelihood approach is not applicable for IC data, and the generalized likelihood approach has not been studied in the literature. It turns out that under the PWPH model with IC data, the semi-parametric MLE (SMLE) of the covariate effect under the standard generalized likelihood may not be unique and may not be consistent. In fact, the parameter under the PWPH model with IC data is not identifiable unless the identifiability assumption is imposed. We propose a modification to the likelihood function so that its SMLE is unique. Under the identifiability assumption, our simulation study suggests that the SMLE is consistent. We apply the method to our cancer relapse time data and conclude that the bone marrow micrometastasis does not have a significant prognostic factor.