The shape of the relative frailty variance induced by discrete random effect distributions in univariate and multivariate survival models

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.