In statistical models for the analysis of time-to-event data, individual heterogeneity is usually accounted for by means of one or more random effects, also known as frailties. In the vast majority of the literature, the random effect is assumed to follow a continuous probability distribution. However, in some areas of application, a discrete frailty distribution may be more appropriate. We investigate and compare various existing families of discrete univariate and shared frailty models by taking as our focus the variance of the relative frailty distribution in survivors. The relative frailty variance (RFV) among survivors provides a readily interpretable measure of how the heterogeneity of a population, as represented by a frailty model, evolves over time. We explore the shape of the RFV for the purpose of model selection and review available discrete random effect distributions in this context. We find non-monotone trajectories of the RFV for discrete univariate and shared frailty models, which is a rare property. Furthermore, we proof that for discrete time-invariant univariate and shared frailty models with (without) an atom at zero, the limit of the RFV approaches infinity (zero), if the support of the discrete distribution can be arranged in ascending order. Through the one-to-one relationship of the RFV with the cross-ratio function in shared frailty models, which we generalize to the higher-variate case, our results also apply to patterns of association within a cluster. Extensions and contrasts to discrete time-varying frailty models and contrasts to correlated discrete frailty models are discussed.

翻译：在事件时间数据分析的统计模型中，个体异质性通常通过一个或多个随机效应（又称脆弱性）来刻画。现有文献中，绝大多数情况下假设随机效应服从连续概率分布。然而，在某些应用领域，离散脆弱性分布可能更为适宜。本文以存活者中相对脆弱性分布的方差为焦点，研究并比较了现有的各类离散单变量与共享脆弱性模型族。存活者中的相对脆弱性方差（RFV）提供了易于解释的度量，反映由脆弱性模型所表征的群体异质性随时间演化的过程。为模型选择之目的，我们探究了RFV的形态，并在此背景下评述了可用的离散随机效应分布。我们发现，离散单变量与共享脆弱性模型的RFV轨迹呈现非单调性，这是一项罕见性质。此外，我们证明：对于具有（或不具有）零点原子的离散时不变单变量与共享脆弱性模型，若离散分布的支撑集可按升序排列，则RFV的极限趋近于无穷（零）。通过RFV与共享脆弱性模型中交叉比函数的一一对应关系（我们将其推广至高维情形），我们的结果同样适用于集群内部关联模式。本文还讨论了与离散时变脆弱性模型的拓展与对比，以及与相关离散脆弱性模型的对比。