Nonparametric Estimation of Proportional Hazards with Monotone Baseline Hazard and Covariate Effect

Order-restricted inference has been applied to survival analysis when its hazard function is known to have a specific shape prior to data analysis. Under the proportional hazards assumption, the partial likelihood approach is commonly used to estimate a covariate effect on the distribution of survival time without specifying its baseline hazard function, but at the same time, the shape information of the baseline hazard function cannot be used in the partial liklelihood estimation procedure. In this paper, we propose a nonparametric full likelihood method for estimating the covariate effect and baseline hazard functions simultaneously under monotone shape restriction. We develop an efficient algorithm using generalized isotonic regression techniques. We extend the algorithm to model with time-dependent covariates. Simulation studies demonstrate that the proposed full likelihood method shows smaller variance than the partial likelihood approach with reduction of bias. Analysis of data from a bone marrow transplantation study illustrates the practical utility of the isotonic methodology in estimating a nonlinear and monotone hazard function.