Posterior propriety and computation for the Cox regression model with applications to missing covariates
1 Department of Statistics, University of Connecticut, 215 Glenbrook Road, U-4120, Storrs, Connecticut 06269, U.S.A. mhchen{at}stat.uconn.edu, 2 Department of Biostatistics, University of North Carolina, McGavran Greenberg Hall, CB#7420, Chapel Hill, North Carolina 27599, U.S.A. ibrahim{at}bios.unc.edu, 3 Department of Mathematics, Hong Kong University of Science and Technology, Hong Kong, China maqmshao{at}ust.hk
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Abstract |
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In this paper, we carry out an in-depth theoretical investigation of Bayesian inference for the Cox regression model. We establish necessary and sufficient conditions for posterior propriety of the regression coefficient, ß, in Cox's partial likelihood, which can be obtained as the limiting marginal posterior distribution of ß through the specification of a gamma process prior for the cumulative baseline hazard and a uniform improper prior for ß. We also examine necessary and sufficient conditions for posterior propriety of the regression coefficients, ß, using full likelihood Bayesian approaches in which a gamma process prior is specified for the cumulative baseline hazard. We examine characterisation of posterior propriety under completely observed data settings as well as for settings involving missing covariates. Latent variables are introduced to facilitate a straightforward Gibbs sampling scheme in the Bayesian computation. A real dataset is presented to illustrate the proposed methodology.
Key Words: Gamma process prior; Latent variable; Markov chain Monte Carlo; Missing at random; Necessary and sufficient conditions; Partial likelihood; Proportional hazards model.
Received August 2005. Revised April 2006.