## Copula Models

Copula models are an attractive possibility to model clustered survival data. However, the application of such models requires unique cluster size for all clusters in the study population. This situation is illustrated in the twin data examples. Here the cluster size is always two. Other bivariate examples are event times in paired organs such as eyes and kidneys. Duchateau and Janssen (2008) give an interesting four-dimensional example with udder data. There are exactly four udder quarters for each cow, which is considered to be the cluster. Researcher are interested in the time until the occurrence of an infection. Typical examples of unique cluster size are family studies with fixed family size, for example, studies with kernel families of size three (father, mother, child). The copula C is a function defined on [0, 1]n and taking values in [0, 1]. The

copula establishes the link between the marginal survival functions to generate the joint survival function. Let Sj(tj) (j = 1, . . . , n) denote the marginal survival functions in clusters of size n. Then the joint survival function is given by S(t1, . . . , tn) = C(S1(t1), . . . , Sn(tn)). The existence of the copula C follows from the theorem by Sklar (1959). For continuous marginal survival functions, a unique copula exists. For a comprehensive overview of copula models see Nelsen (2006). For survival data, the family of copulas is often restricted to the class of Archimedian copulas, described in detail by Genest and MacKay (1986):

S(t1, . . . , tn) = p ( q(S1(t1)) + . . .+ q(Sn(tn))

) ,

where p is a decreasing function defined on nonnegative numbers taking values in [0, 1] and p(0) = 0. Furthermore, it is assumed that p(t) has a positive second derivative for all t > 0 and q denotes the inverse of p. A detailed discussion about the relation between Archimedian copulas and the shared frailty model can be found in Goethals et al. (2008) and Duchateau and Janssen (2008). In the following we will restrict our discussion to the bivariate case for ease of presentation. Higher-dimensional extensions are possible but they extend the length of the formulas. The copula of the shared gamma frailty model (the Clayton copula) will serve as a specific example to explain the basic ideas in Section 6.1. It will be helpful to derive the correlated gamma frailty copula in Section 6.2. Section 6.3 deals with a general correlated frailty

Clayton (1978), Cox and Oakes (1984), and Yashin and Iachine (1999a) pointed out that the bivariate survival function in the shared gamma frailty model (4.3) can also be derived using a radically different approach. In the next paragraph we will outline this approach and discuss its interpretational consequences in more detail. Denote two possibly dependent event times by T1, T2, and let the expression

S(t1, t2) = P(T1 > t1, T2 > t2) be their bivariate joint survival function that is absolutely continuous with margins S1(t1) = S(t1, 0) and S2(t2) = S(0, t2). Consequently, the conditional survival function of T1 given T2 > t2 is

S(t1|T2 > t2) = S(t1, t2) S2(t2)

and that of T1 given T2 = t2 is

S(t1|T2 = t2) = âˆ‚S(t1,t2)

.