On assessing the strength of dependency between failure time variates

The correlation rho between cumulative hazard variates is considered as a measure of dependence between pairs of failure time variates. Such correlation is readily estimated in the absence of censorship, but modelling assumptions are required for its estimation in the presence of independent right censorship. A semiparametric class of bivariate failure time models, in which a dependence parameter theta is in one-to-one correspondence with the cumulative hazard variate correlation rho, is considered for this purpose. A simple estimating function, consisting of a weighted sum over the grid formed by the observed failure times of differences between nonparametric and model-restricted estimates of covariance rates, is proposed for estimation and corresponding asymptotic distribution theory is given. This estimation procedure is shown to lack robustness under conditions of moderate to heavy censorship, motivating consideration of dependency measures that can be defined on the support of the observed failure times. Two such measures, the correlation between marginal martingales, and a cumulative cross ratio statistic, are discussed and corresponding nonparametric estimators are proposed.