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，标准的极大偏似然推断往往存在问题。本文提出用于Cox模型贝叶斯推断的Cox催化先验分布，该方法是专为稳定复杂参数模型而设计的通用先验分布类别的扩展。Cox催化先验被构造为基于合成数据与代理基线风险常数的回归系数加权似然函数。该代理风险可由用户提供或从数据中估计得到，而合成数据则通过拟合更简单模型的预测分布生成。对于点估计，我们推导了边际后验众数的近似形式，该近似可便捷地计算为正则化对数偏似然估计量。我们证明了所提先验分布是适定的，且在温和条件下所得估计量具有相合性。在模拟研究中，我们提出的方法优于标准极大偏似然推断，并与现有收缩方法性能相当。我们进一步通过真实数据集展示了该方法的应用。