Theory and Inference for the Cox Model with Missing Covariates

• Activity Number: 216
• Type: Invited
• Date/Time: Tuesday, August 9, 2005 : 8:30 AM to 10:20 AM
• Sponsor: Section on Bayesian Statistical Science
• Author(s): Joseph G. Ibrahim*+ and Ming-Hui Chen and Qi-Man Shao
• Companies: University of North Carolina, Chapel Hill and University of Connecticut and University of Oregon
• Keywords: Missing at random (MAR) ; Monte Carlo EM algorithm ; Gamma process prior ; Latent variable ; Partial Likelihood ; Posterior propriety

Abstract

In this paper, we carry out an in-depth theoretical investigation for inference with missing covariate data for the Cox regression model (Cox 1972, 1975). Specifically, we establish necessary and sufficient conditions for existence of the maximum partial likelihood estimate (MPLE) for completely observed data (i.e., no missing data) settings as well as for existence of the maximum likelihood estimate (MLE) for survival data with missing covariates via the profile likelihood method. We also establish necessary and sufficient conditions for posterior propriety of the regression coefficients in Cox's partial likelihood, which can be obtained through a gamma process prior for the cumulative baseline hazard and a uniform improper prior for the regression coefficients. We examine characterizations of posterior propriety under completely observed data settings as well as for missing covariates. Latent variables are introduced to facilitate a straightforward Gibbs sampling scheme in the Bayesian computation. Several theorems under both the frequentist and Bayesian paradigms are given to establish these necessary and sufficient conditions.