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In this paper we extend the bivariate hazard ratio to multivariate competing risks data and show that it is equivalent to the cause-specific cross hazard ratio. Two approaches are proposed to estimate these two equivalent association measures. One extends the plug-in estimator, and the other adapts the pseudo-likelihood estimator for bivariate survival data to multivariate competing risks data. The asymptotic properties of the extended estimators are established by using empirical processes techniques. The extended plug-in and pseudo-likelihood estimators have comparable performance with the existing U-statistic when the data have no tied events. However, in many applications, there are tied events in which all the three estimators are found to produce biased results. To our best knowledge, we are not aware of any association analysis for multivariate competing risks data that has considered tied events. Hence we propose a modified U-statistic to specifically handle tied observations. The modified U-statistic clearly outperforms the other estimators when there are rounding errors. All methods are applied to the Cache County Study to examine mother–child and sibship associations in dementia among this aging population, where the event times are rounded to the nearest integers. The modified U performs consistently with our simulation results and provides more reliable results in the presence of tied events.

Piecewise Cause-Specific Association Analyses of Multivariate Untied or Tied Competing Risks Data