Correlated Frailty Models in the Analysis of Dependent Failure Times

The multivariate survival distributions are used for description of dependent failure times (life spans) (Clayton and Cuzik 1985; Hougaard 1995, Yashin et al. 1995). The convenient representation of such distributions involves semi-parametric (copula) structure of multivariate survival function. Such structure is important in applications where the cause for dependence between life spans is the subject of the study (Yashin and Iachine 1994, Yashin and Iachine 1995a). It turns out that the wide class of frailty models allows for such a representation (Yashin et al. 1999). In genetic-epidemiological applications these models can be used to evaluate the presence and the magnitude of genetic influence on susceptibility to death as well as its variability in the population (Yashin and Iachine 1995b; 1997). In particular the correlated gamma frailty model applied to survival data on monozygotic (MZ) and dizygotic (DZ) twins can be used for this purpose. This model is characterized by the bivariate survival function ( ) ( ) 2 2 1 1 2 1 , , x T x T P x x S > > = [ ] 2 1 2 2 2 1 12 21 1 ) ( ) ( ) ( ) ( ) , ( 2 1 1