Multisample adjusted U-statistics that account for confounding covariates

Abstract

Multisample U-statistics encompass a wide class of test statistics that allow the comparison of 2 or more distributions. U-statistics are especially powerful because they can be applied to both numeric and nonnumeric data, eg, 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 2 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 (ie, 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 2 (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 (ie, 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 2 control participants.

Keywords: adjusted U-statistics; multiple group comparison; propensity score.

FIGURE 1

Comparison of empirical p-values and theoretical (uniform) p-values for the Kruskal-Wallis type test (Panel A) and Jonckheere-Terpstra type test (Panel B). Brown (long-dashed curve) corresponds to standardization to the study population, blue (dotted curve) is standardization to the group 1, red (solid curve) is the parametric model, and black (dash-dotted curve) is the naive U-statistic that does not account for confounding.

FIGURE 2

Power of the adjusted U-statistic for the Kruskal-Wallis type test (Panel A) and Jonckheere-Terpstra type test (Panel B) when the parametric model is correctly specified, and the power of the adjusted U-statistic for the Kruskal-Wallis type test (Panel C) and Jonckheere-Terpstra type test (Panel D) when the parametric model is mis-specified. Solid curve is the Wald test for the parametric model. Long-dashed and dotted curves are adjusted U-statistics that standardize to the study population and group 1, respectively. The parameter α determines strength of the association. When α = 0 (no association) the power corresponds to the size of the test. All tests have 3 groups and 2 degrees of freedom.

FIGURE 3

Expected vs. empirical p-values under null hypothesis using Kernel (12) and simulation data based on the COMPT study.