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 models, there is indeed the 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.

The correlated and shared gamma frailty model for bivariate current status data: An illustration for cross‐sectional serological data

“…However, copula models, often used to assess the association between event times, in general, cannot capture the heterogeneity as frailty models can. For more details regarding this aspect we refer to the paper by Goethals et al [34]. Bivariate copula models for CS data including limiting distributions are discussed in Wang and Ding [35].…”

Copula based flexible modeling of associations between clustered event times

“…This is not the case for frailty models, although there are similarities between frailty models and Archimedean copulas (Goethals et al 2008). This is not the case for frailty models, although there are similarities between frailty models and Archimedean copulas (Goethals et al 2008).…”

Comments on: Inference in multivariate Archimedean copula models

“…, t d ) is different from the 'classical copula' where the idea is to capture dependence through the copula function but in a way that is not related to the marginal distributions. For a detailed discussion, including examples, see Goethals et al (2008). Another important difference between copula models and frailty models is that frailty models are conditional models (in fact dependence between observations in one cluster is generated through heterogeneity between clusters (the frailty density captures the heterogeneity)).…”