Multi-Sample Adjusted U-Statistics that Account for Confounding Covariates

Multi-sample U-statistics encompass a wide class of test statistics that allow the comparison of two or more distributions. U-statistics are especially powerful because they can be applied to both numeric and non-numeric data, e.g., ordinal and categorical data where a pairwise similarity or distance-like measure between categories is available. However, when comparing the distribution of a variable across two or more groups, observed differences may be due to confounding covariates. For example, in a case-control study, the distribution of exposure in cases may differ from that in controls entirely because of variables that are related to both exposure and case status and are distributed differently among case and control participants. We propose to use individually-reweighted data (i.e., using the stratification score for retrospective data or the propensity score for prospective data) to construct adjusted U-statistics that can test the equality of distributions across two (or more) groups in the presence of confounding covariates. Asymptotic normality of our adjusted U-statistics is established and a closed form expression of their asymptotic variance is presented. The utility of our approach is demonstrated through simulation studies, as well as in an analysis of data from a case-control study conducted among African-Americans, comparing whether the similarity in haplotypes (i.e., sets of adjacent genetic loci inherited from the same parent) occurring in a case and a control participant differs from the similarity in haplotypes occurring in two control participants.