Modelling failure-time associations in data with multiple levels of clustering

In recent years substantial research has been devoted to developing failure-time methodology which accounts for possible dependency between observations. An example is the univariate frailty model (Vaupel, Manton & Stallard, 1979), which incorporates an exchangeable dependence structure by the inclusion of cluster-specific random effects. In some studies it may be reasonable to expect more than one level of within-cluster association: for instance, association between a parent and child versus that between two siblings in studies of familial disease aggregation, or association between two village residents who live in different households versus that between residents of the same household in intervention studies. We propose a family of distributional models for failure-time data that accounts for multiple levels of clustering and reduces in the case of a single clustering level to a univariate frailty model. The resulting survival functions are constructed by a recursive nesting of univariate frailty-type distributions through which archimedean copula forms are determined for all bivariate margins. Properties of the proposed model are developed, illustrated and briefly contrasted with multivariate frailty model properties. In conclusion, we outline the application of our model to marginal risk regression problems.