U-statistics are a set of non-parametric test statistics to compare a distribution of outcomes in two or more groups. When the number of groups are two, the most commonly used statistic is the Wilcoxon statistic to determine whether the samples are from the same population. For multi-group comparisons, a number of non-parametric statistics can be used corresponding to different situations. The article discusses two such test statistics: the Kruskal-Wallis statistic (Ref. 1) and the Jonckheere-Terpstra statistic (Ref. 2). The Kruskal-Wallis statistic can be used to test if the null hypothesis in the outcome distributions in K(> 2) groups are equal against the alternative hypothesis that at least one outcome differs from others. The Jonckheere-Terpstra test is to find whether the outcome distribution in group (g + 1) is stochastically greater than that in group g = 1,2,... ,K - 1 where the outcome distributions are increasingly ordered. The Wilcoxon statistic-based test and the other two are centered around expectation functionals and U-statistics are unbiased estimators of the expected functionals. U-statistics can be used for comparing time-to-event outcomes in more than two groups, but the estimators are biased due to rightcensoring. To overcome this, an inverse probability of censoring weighted (IPCW) U-statistics was proposed by Datta (Ref. 3) under independent and covariate dependent censoring. The authors extend this approach with a novel martingale representation, the asymptotic distribution for the IPCW U-statistics with kernel of arbitrary order. The U-statistic that accounts for confounding covariates were proposed by Satten et al. (Ref. 4). This extension standardized the empirical cumulative distribution function in each group. This balances the distribution of confounding covariates in the group after adjustment.
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