School of Computing Sciences, University of East Anglia, Norwich, UK.
Stat Med. 2023 Aug 15;42(18):3114-3127. doi: 10.1002/sim.9760. Epub 2023 May 15.
The Cox regression, a semi-parametric method of survival analysis, is extremely popular in biomedical applications. The proportional hazards assumption is a key requirement in the Cox model. To accommodate non-proportional hazards, we propose to parameterize the shape parameter of the baseline hazard function using the additional, separate Cox-regression term which depends on the vector of the covariates. This parametrization retains the general form of the hazard function over the strata and is similar to one in Devarajan and Ebrahimi (Comput Stat Data Anal. 2011;55:667-676) in the case of the Weibull distribution, but differs for other hazard functions. We call this model the double-Cox model. We formally introduce the double-Cox model with shared frailty and investigate, by simulation, the estimation bias and the coverage of the proposed point and interval estimation methods for the Gompertz and the Weibull baseline hazards. For real-life applications with low frailty variance and a large number of clusters, the marginal likelihood estimation is almost unbiased and the profile likelihood-based confidence intervals provide good coverage for all model parameters. We also compare the results from the over-parametrized double-Cox model to those from the standard Cox model with frailty in the case of the scale-only proportional hazards. The model is illustrated on an example of the survival after a diagnosis of type 2 diabetes mellitus. The R programs for fitting the double-Cox model are available on Github.
Cox 回归是一种生存分析的半参数方法，在生物医学应用中非常流行。比例风险假设是 Cox 模型的关键要求。为了适应非比例风险，我们建议使用依赖协变量向量的额外、单独的 Cox 回归项来参数化基线风险函数的形状参数。这种参数化在层间保留了风险函数的一般形式，与 Devarajan 和 Ebrahimi（Comput Stat Data Anal. 2011;55:667-676）在 Weibull 分布情况下的模型类似，但对于其他风险函数则不同。我们称这个模型为双 Cox 模型。我们正式引入了具有共享脆弱性的双 Cox 模型，并通过模拟研究了提出的 Gompertz 和 Weibull 基线风险的点估计和区间估计方法的估计偏差和覆盖范围。对于脆弱性方差低且聚类数多的实际应用，边际似然估计几乎无偏，基于轮廓似然的置信区间为所有模型参数提供了良好的覆盖范围。我们还在仅比例风险脆弱性的情况下，将过参数化的双 Cox 模型的结果与标准 Cox 模型的结果进行了比较。该模型在 2 型糖尿病诊断后生存的实例中进行了说明。适用于拟合双 Cox 模型的 R 程序可在 Github 上获得。
