Distance‐Based Tests for Homogeneity of Multivariate Dispersions

TLDR

Traditional likelihood‑based tests for multivariate dispersion are limited by sensitivity to nonnormality, inability to handle high‑dimensional data, and inadequacy for skewed, zero‑inflated ecological datasets that require non‑Euclidean dissimilarities. The study proposes distance‑based tests for homogeneity of multivariate dispersions that can use any dissimilarity measure. The tests use the rotational invariance of the multivariate centroid or spatial median to compute spread along principal coordinate axes, extending Levene’s test with P‑values from an F‑distribution or permutations of least‑squares or LAD residuals. Illustrative examples and Monte Carlo simulations demonstrate that the distance‑based tests are robust, powerful, and useful for detecting real differences in dispersion across ecological and biological studies.

Abstract

Summary The traditional likelihood‐based test for differences in multivariate dispersions is known to be sensitive to nonnormality. It is also impossible to use when the number of variables exceeds the number of observations. Many biological and ecological data sets have many variables, are highly skewed, and are zero‐inflated. The traditional test and even some more robust alternatives are also unreasonable in many contexts where measures of dispersion based on a non‐Euclidean dissimilarity would be more appropriate. Distance‐based tests of homogeneity of multivariate dispersions, which can be based on any dissimilarity measure of choice, are proposed here. They rely on the rotational invariance of either the multivariate centroid or the spatial median to obtain measures of spread using principal coordinate axes. The tests are straightforward multivariate extensions of Levene's test, with P ‐values obtained either using the traditional F ‐distribution or using permutation of either least‐squares or LAD residuals. Examples illustrate the utility of the approach, including the analysis of stabilizing selection in sparrows, biodiversity of New Zealand fish assemblages, and the response of Indonesian reef corals to an El Niño. Monte Carlo simulations from the real data sets show that the distance‐based tests are robust and powerful for relevant alternative hypotheses of real differences in spread.