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

Abstract 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 ( $$\text {RFV}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mtext>RFV</mml:mtext> </mml:math> ) 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 $$\text {RFV}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mtext>RFV</mml:mtext> </mml:math> for the purpose of model selection and review available discrete random effect distributions in this context. We find non-monotone trajectories of the $$\text {RFV}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mtext>RFV</mml:mtext> </mml:math> for discrete univariate and shared frailty models with multiple changes in slope over time, which is a property that seems to be absent for continuous frailty models discussed in the literature. We also show that, in contrast to continuous frailty models, the heterogeneity of data generated by a discrete time-invariant frailty distribution, such as the categorical k -point distribution, approaches either infinity or homogeneity in the long run. Through the one-to-one relationship of the $$\text {RFV}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mtext>RFV</mml:mtext> </mml:math> with the cross-ratio function in shared frailty models, our results also apply to patterns of association within a cluster. Extensions and contrasts to discrete time-varying frailty models are discussed.