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. WIREs Comp Stat 2011 3 457–471 DOI: 10.1002/wics.178 This article is categorized under: Statistical and Graphical Methods of Data Analysis > Multivariate Analysis