Frailties and copulas, not two of a kind

Abstract

Bivariate survival data occur in diverse disciplines such as financial mathematics [The European Journal of Finance 15 (2009), 609–618] and biostatistics [Twin Res. 4 (2001), 407–411]. Different modeling approaches have been developed. Two standard approaches, the copula model and the frailty model, provide estimates of the correlation between event times in a cluster [The Frailty Model, Springer, New York, 2008].

A unified framework is proposed here for the copula model and the different types of frailty models, i.e., the univariate, shared and correlated frailty model, in order to evaluate similarities and differences between the models.

We further investigate the frailty effect at the event time scale; frailties operate at the hazard level, and therefore influence the event times in a nonlinear way, which leads in some instances to counterintuitive findings. For instance, in a cluster with frailty smaller than one, both event times will increase, but additionally also the difference between the two event times in the cluster will increase.