On association in a copula with time transformations

We consider estimation of the cross ratio in Clayton's (1978) copula in which covariates are incorporated into the marginal distributions via semiparametric accelerated life regression models. Generalisations of Oakes' (1982, 1986) concordance estimating equations yield a closed form for the association parameter under right censoring. Joint inferences about covariate effects in the marginal models and the cross ratio are obtained with U-statistics, martingales and a resampling technique for nonsmooth estimating equations. Simulating under an exponential model, we find that our procedure may provide a more precise estimate of association than other methods for proportional hazards models. A goodness-of-fit test for the assumed copula is presented and used in an analysis of a diabetic retinopathy dataset.