Efficient Gibbs Sampling in Cox Regression Models Using Composite Partial Likelihood and P\'olya-Gamma Augmentation

Abstract

The Cox regression models and their Bayesian extensions are widely used for time-to-event analysis. However, standard Bayesian approaches typically require baseline hazard modeling, and their full conditional distributions lack closed-form expressions, resulting in computational inefficiency and increased vulnerability to bias from baseline hazard misspecification. To address these issues, we propose GS4Cox, a fully Gibbs sampler for Bayesian Cox regression models with four elements: (i) generalized Bayesian framework for avoiding baseline hazard specification, (ii) composite partial likelihood and (iii) P\'olya-Gamma augmentation for closed-form expressions of full conditional distributions, and (iv) affine posterior calibration via the open-faced sandwich adjustment for location and scale adjustment of the posterior distribution. We prove asymptotic unbiasedness of the generalized Bayes estimator under composite partial likelihood and propose an affine posterior transformation that yields higher-order asymptotic agreement with the maximum partial likelihood estimator, while the posterior covariance matches the asymptotic target covariance. We demonstrated that GS4Cox consistently outperformed existing sampling methods through numerical and real-data experiments.