Multivariate U-statistics: A tutorial with applications

U-statistics represent an important class of statistics arising from modeling quantities of interest defined by multi-subject responses such as the classic Mann-Whitney-Wilcoxon rank tests. However, classic applications of U-statistics are largely limited to univariate outcomes within a cross-sectional data setting. As longitudinal study designs become increasingly popular in today's research, it is imperative to generalize the classic theory of U-statistics to such a study setting to meet the challenges of modern clinical and translational research. In this article, we focus on applications of U-statistics to longitudinal study data. We first give a brief overview of U-statistics and then discuss how to apply this powerful class of statistics to model quantities of interest with a longitudinal data setting. In addition to generalizing U-statistics and associated inference theory to longitudinal data analysis, we also discuss a class of function response models (FRM) to bring the power of U-statistics to uncharted territory. We illustrate applications of generalized U-statistics and FRM with data from some real longitudinal studies.