Nonparametric Analysis of Clustered Multivariate Data

TL;DR

This paper extends multivariate nonparametric procedures to clustered data, providing asymptotic results, efficiency analyses, and small sample methods, with a focus on spatial sign and rank techniques.

Contribution

It introduces a general framework for nonparametric analysis of clustered multivariate data, including asymptotic theory and practical procedures for spatial sign and rank methods.

Findings

Asymptotic distributions derived for test statistics under null and alternative hypotheses.

Efficiency comparisons of spatial sign and rank scores in clustered data.

Small sample procedures based on permutation principles discussed.

Abstract

There has been a wide interest to extend univariate and multivariate nonparametric procedures to clustered and hierarchical data. Traditionally, parametric mixed models have been used to account for the correlation structures among the dependent observational units. In this work we extend multivariate nonparametric procedures for one-sample and several samples location problems to clustered data settings. The results are given for a general score function, but with an emphasis on spatial sign and rank methods. Mixed models notation involving design matrices for fixed and random effects is used throughout. The asymptotic variance formulas and limiting distributions of the test statistics under the null hypothesis and under a sequence of alternatives are derived, as well as the limiting distributions for the corresponding estimates. The approach based on a general score function also shows, for example, how $M$-estimates behave with clustered data. Efficiency studies demonstrate practical advantages and disadvantages of the use of spatial sign and rank scores, and their weighted versions. Small sample procedures based on sign change and permutation principles are discussed. Further development of nonparametric methods for cluster correlated data would benefit from the notation already familiar to statisticians working under normality assumptions.