A note on Kendall’s Tau coefficient for gap times in presence of right censoring

In several clinical and epidemiology studies, data from events that occur successively in time in the same individual, are frequently reported. Among these, the most common are recurrent events where each subject may experience a number of failures over the course of follow-up. Examples include repeated hospitalization of patients, recurrences of tumor, recurrent infections, among others. In this work, the interest is to study the correlation between successive recurrent events, gap times, in the presence of right censoring. To measure the association between two gap times we use the Kendall’s τ correlation coefficient, by incorporating suitable bivariate estimators of the joint distribution function of the gap times and of the marginal distribution function of the second gap time, into the integrals that define the probability of concordant pairs and the probability of discordant pairs. Two of the estimators of the joint distribution function of the gap times considered in this work are already known, but we consider also estimators with Kaplan-Meier weights defined by using decision trees and random forests methodology. We conclude that all the estimators perform better in a scenario of negative association. When the association is moderately negative, the performance of the estimator with smoothed weights using random forests is superior. In the case of strong positive association, the best estimator is the presmoothed nonparametric but, in the case of moderate positive association, this estimator has identical performance as the estimator with presmoothed weights using random forests.This work was supported by Portuguese funds through the CMAT - Research Centre of Mathematics of University of Minho - within projects UIDB/00013/2020 and UIDP/00013/2020. Authors thank to referees for their careful reading and for their constructive suggestions