Shape-Constrained Nonparametric Estimators of the Baseline Distribution in the Cox Proportional Hazards Model

• Activity Number: 305
• Type: Contributed
• Date/Time: Tuesday, August 6, 2013 : 8:30 AM to 10:20 AM
• Sponsor: Section on Nonparametric Statistics
• Author(s): Gabriela Nane*+ and Hendrik Lopuhaa
• Companies: and Delft University of Technology
• Keywords: Cox proportional hazards model ; Shape constrained estimator ; Nonparametric maximum likelihood estimation

Abstract

Within survival analysis, the Cox proportional hazards model is one of the most acknowledged approaches to model right-censored lifetimes in the presence of covariates. Different functionals of the lifetime distribution are commonly investigated and the hazard function is of particular interest. Furthermore, in many circumstances, practitioners have strong evidence that the underlying baseline hazard is monotone. Therefore, it would be desirable to provide estimates that incorporate the shape restrictions of the baseline hazard function while preserving the flexible semiparametric setting of the Cox model. We propose two estimators of a monotone baseline hazard function. The non-parametric maximum likelihood estimator is obtained by maximizing the (log)likelihood function over the set of all distributions with monotone hazard functions. Moreover, we introduce a Grenander type estimator, which, for a non-decreasing baseline hazard function, is defined as the left-hand slope of the greatest convex minorant of the Breslow estimator. The two estimators are shown to be strongly consistent and asymptotically equivalent and we derive their common limit distribution at a fixed point.