Shared frailty models have been proposed to accommodate unmeasured cluster-specific risk factors through the inclusion of a common latent frailty term. Among possible frailty distributions, the Gamma distribution is appealing due to its non-negativity, flexibility, and algebraic tractability leading to closed-form marginal survival or hazard function expressions. Under the Bayesian paradigm, the posterior distributions of model parameters are usually explored with computationally intensive procedures relying on Markov chain Monte Carlo sampling. As an alternative, Laplacian-P-splines (LPS) provide a flexible and sampling-free alternative by relying on Gaussian approximations of the posterior target distributions. In this model class, analytical formulas are obtained for the gradient and Hessian, yielding a computationally efficient inference scheme for estimation of model parameters with a natural way of quantifying uncertainty. This article extends the LPS toolbox to the inclusion of shared Gamma frailty models for clustered time-to-event data. We assess the finite-sample performance of the LPS estimation procedure through an extensive simulation study and compare estimates with those obtained using penalized partial likelihood estimation, without specification of the baseline hazard, and with the variance of the frailty term being estimated using profile likelihood. Finally, the proposed LPS estimation method is exemplified using three publicly available biomedical datasets on: (i) recurrent infections in children, (ii) cancer prevention, and (iii) kidney transplantation.

翻译：共享脆弱模型通过包含一个共同的潜在脆弱项来容纳未测量的聚类特异性风险因素。在可能的脆弱分布中，伽马分布因其非负性、灵活性和代数易处理性而具有吸引力，这导致闭合形式的边际生存或风险函数表达式。在贝叶斯范式下，模型参数的后验分布通常通过依赖马尔可夫链蒙特卡洛采样的计算密集型程序进行探索。作为一种替代方案，拉普拉斯-P-样条通过依赖后验目标分布的高斯近似提供了一种灵活且免采样的替代方案。在该模型类中，获得了梯度和海森矩阵的解析公式，从而为模型参数的估计提供了一种计算高效的推理方案，并以自然的方式量化不确定性。本文将拉普拉斯-P-样条工具箱扩展到包容用于聚类时间至事件数据的共享伽马脆弱模型。我们通过一项广泛的模拟研究评估了拉普拉斯-P-样条估计过程的有限样本性能，并将估计结果与使用惩罚偏似然估计（无需指定基线风险）获得的结果进行比较，其中脆弱项的方差通过轮廓似然进行估计。最后，使用三个公开的生物医学数据集对所提出的拉普拉斯-P-样条估计方法进行了举例说明：(i) 儿童反复感染，(ii) 癌症预防，以及 (iii) 肾移植。