Frailty models and copulas: similarities and differences

Abstract

Copulas and frailty models are important tools to model bivariate survival data. Equivalence between Archimedean copula models and shared frailty models, e.g. between the Clayton-Oakes copula model and the shared gamma frailty model, has often been claimed in the literature. In this note we show that, in both the models, there is indeed a well-known equivalence between the copula functions; the modeling of the marginal survival functions, however, is quite different. The latter fact leads to different joint survival functions.

Keywords:

• bivariate survival data
• Clayton-Oakes copula
• positive stable frailty
• shared gamma frailty model

Acknowledgements

The authors thank both the referees for their insightful remarks, one of them pointed out an inaccuracy in the first version of this paper. The first author acknowledges the financial support of the Research Fund of Ghent University, Ghent, Belgium (Grant No. B/04953/01). The research of Luc Duchateau and Paul Janssen was supported by IAP Research Network Grant No. P6/03 of the Belgian Government (Belgian Science Policy).