The Cox proportional hazards model (Cox model) is a popular model for survival data analysis. When the sample size is small relative to the dimension of the model, the standard maximum partial likelihood inference is often problematic. In this work, we propose the Cox catalytic prior distributions for Bayesian inference on Cox models, which is an extension of a general class of prior distributions originally designed for stabilizing complex parametric models. The Cox catalytic prior is formulated as a weighted likelihood of the regression coefficients based on synthetic data and a surrogate baseline hazard constant. This surrogate hazard can be either provided by the user or estimated from the data, and the synthetic data are generated from the predictive distribution of a fitted simpler model. For point estimation, we derive an approximation of the marginal posterior mode, which can be computed conveniently as a regularized log partial likelihood estimator. We prove that our prior distribution is proper and the resulting estimator is consistent under mild conditions. In simulation studies, our proposed method outperforms standard maximum partial likelihood inference and is on par with existing shrinkage methods. We further illustrate the application of our method to a real dataset.

翻译：Cox比例风险模型（Cox模型）是生存数据分析中常用的模型。当样本量相对于模型维度较小时，标准的最大偏似然推断往往会出现问题。本文针对Cox模型提出了一种基于催化先验分布的贝叶斯推断方法，该方法是针对稳定复杂参数模型而设计的一类通用先验分布的扩展。Cox催化先验基于合成数据和替代基线风险常数，被构造为回归系数的加权似然函数。该替代风险常数可由用户预先设定或从数据中估计，合成数据则通过拟合的简化模型的预测分布生成。在点估计方面，我们推导了边缘后验众数的近似形式，该近似可通过正则化对数偏似然估计便捷计算。我们证明了所提先验分布是恰当的，且在温和条件下所得估计量具有一致性。模拟研究表明，本方法优于标准最大偏似然推断，并与现有收缩方法性能相当。我们进一步通过实际数据集展示了该方法的实际应用。