Bandeen-Roche and Liang (2002, Modelling multivariate failure time associations in the presence of a competing risk. Biometrika 89, 299-314.) tailored Oakes (1989, Bivariate survival models induced by frailties. Journal of the American Statistical Association 84, 487-493.)'s conditional hazard ratio to evaluate cause-specific associations in bivariate competing risks data. In many population-based family studies, one observes complex multivariate competing risks data, where the family sizes may be > 2, certain marginals may be exchangeable, and there may be multiple correlated relative pairs having a given pairwise association. Methods for bivariate competing risks data are inadequate in these settings. We show that the rank correlation estimator of Bandeen-Roche and Liang (2002) extends naturally to general clustered family structures. Consistency, asymptotic normality, and variance estimation are easily obtained with U-statistic theories. A natural by-product is an easily implemented test for constancy of the association over different time regions. In the Cache County Study on Memory in Aging, familial associations in dementia onset are of interest, accounting for death prior to dementia. The proposed methods using all available data suggest attenuation in dementia associations at later ages, which had been somewhat obscured in earlier analyses.