Bayesian Inference in the Cox Model via Rank-Ordered Likelihood

2026-04-07

AI Summary

This study addresses the challenges in modeling the baseline hazard function within Bayesian Cox models and the limitations of existing approaches in handling tied event times and extending to shared frailty models. The authors propose a novel Bayesian inference framework based on the rank likelihood, which uniquely integrates the Plackett–Luce (PL) model and its generalized form (GPL) with Pólya–Gamma data augmentation to construct Gibbs sampling algorithms—termed PL-Cox and GPL-Cox—that require no posterior correction. This approach naturally accommodates tied data and extends directly to shared frailty settings. Simulation studies and real-data analyses demonstrate that PL-Cox exhibits robust performance, while GPL-Cox provides superior fit under high levels of ties; both methods achieve computational efficiency in shared frailty models. Implementations are available in the open-source R package BayesPLCox.

Abstract

In Bayesian inference for the Cox proportional hazards model, modeling the baseline hazard function is challenging. Recently, direct Bayesian inference using the partial likelihood is considered in the framework of general Bayesian inference. In terms of posterior computation, several studies have examined sampling algorithms under the Cox model. In this study, we developed a novel likelihood extension for the Cox proportional hazards model based on the modeling of rank-ordered data. Furthermore, we propose two Gibbs sampling algorithms that combine the full likelihood based on the Plackett--Luce and generalized Plackett--Luce models with Pólya--Gamma data augmentation, referred to as PL-Cox and GPL-Cox, respectively. The two proposed methods offer practical advantages, as they do not require correction of posterior samples and are readily extensible to shared frailty models. In simulation study, we considered multiple survival model settings, including continuous and discrete survival time models, as well as scenarios with varying degrees of ties, and found that the PL-Cox model exhibited relatively stable performance. In analyses of real data with many ties, the GPL-Cox model fit the dataset substantially better than the PL-Cox model. In analyses of real data incorporating shared frailty, both methods demonstrated good computational efficiency. The R package \texttt{BayesPLCox}, which implements the PL-Cox and GPL-Cox methods, is publicly available.

Problem

• Bayesian inference
• Cox model
• baseline hazard
• ties
• shared frailty

Innovation

• Bayesian inference
• Cox model
• Plackett-Luce model
• Pólya-Gamma augmentation
• rank-ordered likelihood