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For the analysis of clustered survival data, two different types of modelsthat take the association into account, are commonly used: frailty models andcopula models. Frailty models assume that conditional on a frailty term foreach cluster, the hazard functions of individuals within that cluster areindependent. These unknown frailty terms with their imposed distribution areused to express the association between the different individuals in a cluster.Copula models on the other hand assume that the joint survival function of theindividuals within a cluster is given by a copula function, evaluated in themarginal survival function of each individual. It is the copula function whichdescribes the association between the lifetimes within a cluster. A majordisadvantage of the present copula models over the frailty models is that thesize of the different clusters must be small and equal in order to set upmanageable estimation procedures for the different model parameters. Wedescribe in this manuscript a copula model for clustered survival data wherethe clusters are allowed to be moderate to large and varying in size byconsidering the class of Archimedean copulas with completely monotonegenerator. We develop both one- and two-stage estimators for the differentcopula parameters. Furthermore we show the consistency and asymptotic normalityof these estimators. Finally, we perform a simulation study to investigate thefinite sample properties of the estimators. We illustrate the method on a dataset containing the time to first insemination in cows, with cows clustered inherds.

Extending the Archimedean copula methodology to model multivariate survival data grouped in clusters of variable size