Bivariate Survival Models Induced by Frailties

Abstract When two observed survival times depend via a proportional-hazards model on the same unobserved variable, called a frailty, this common dependence induces an association between the observed times. This article considers the class of bivariate survival distributions that can arise in this way, extends it to allow negative association, and shows that the observable bivariate distribution determines the unobserved frailty distribution up to a scale parameter. A cross-ratio function, easily estimated even from censored data, is the key to both the characterization results and inferential procedures and diagnostic plots that are introduced and illustrated by real examples. The models considered are the natural counterpart for survival data of the latent variable models that have long been used in factor analysis for continuous data or in latent structure analysis for binary data. The main distinguishing feature of survival data is the possibility of censoring, which makes the standard methods difficult to apply. The procedures developed here work well with censored data. Although the present work is restricted to problems involving a random sample from a single bivariate distribution, extensions to problems involving higher-dimensional distributions and/or additional measured covariates are undoubtedly possible.