Association estimation for clustered failure time data with a cure fraction

Substantial research has been devoted to developing methodology for inferring the association of clustered failure time data. However, in the study of familial disease, there may be a proportion of patients cured or nonsusceptible to the disease. Thus, it is necessary to simultaneously consider two types of association, i.e., the association of the susceptibility of the individuals, and that of the ages at onset between the susceptible individuals. In this paper, we consider the pairwise association in both types of association to reduce the mathematical intractability and the difficulty in specifying the full correlation structure. The former association is measured by the pairwise odds ratio of the binary cure statuses, and the latter by the bivariate Clayton copula with a semiparametric marginal regression model for any pair of correlated failure times. For the marginal model, it is formulated as a fairly general semiparametric regression cure model. A two-stage estimation procedure is adopted for the association estimation. We establish the consistency and asymptotic normality of the estimators for these two types of association. Simulation studies are conducted to assess finite sample properties, and the proposed method is illustrated by a subset of the data in the Australian Twins Study.