Anderson (2006) Distance-Based Tests for Homogeneity of Multivariate Dispersions

Summary

Anderson proposes a distance-based, permutation test for the homogeneity of multivariate dispersions (spread) across groups — the multivariate analogue of Levene's test, and the dispersion counterpart to PERMANOVA (which tests differences in location). The traditional likelihood-based test (Box's M) is sensitive to nonnormality and cannot be used when the number of variables exceeds the number of observations — a common situation in ecology, where data are high-dimensional, skewed, zero-inflated, and best compared through non-Euclidean dissimilarities. The method (later "PERMDISP", betadisper in R's vegan) computes each observation's distance to its group centroid or spatial median in principal-coordinate space and tests those distances by ANOVA with permutation-based P-values.

Key Claims

• The problem with the classical test. The likelihood-ratio (Box's M) test for equal dispersions assumes multivariate normality (fails under skew/zero-inflation) and breaks down when variables outnumber observations; it also presumes Euclidean geometry, which is often inappropriate for ecological data.
• Distance-to-centroid device. Using principal coordinates (Gower 1966) of any chosen dissimilarity matrix, compute for each observation its distance $z_{ij}$ to its group centroid (least-squares) or spatial median (robust); group differences in the mean $z$ measure differences in dispersion.
• Multivariate Levene's test. Run an ANOVA F-statistic on the $z$ deviations — a direct multivariate extension of Levene's (1960) test (Brown-Forsythe median variant for robustness).
• Permutation inference. P-values come from the F-distribution or, to sidestep the nonindependence and nonnormality of the $z$'s, from permutation of the least-squares or LAD residuals.
• Any dissimilarity. A unique feature is that dispersion can be measured under any distance/dissimilarity (Euclidean, Bray-Curtis, …), making it suitable for community/abundance data.

Concepts Introduced or Extended

• PERMDISP (Distance-Based Test for Multivariate Dispersion)

Entities Mentioned

• Marti J. Anderson

Quotes

"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 … A unique feature is that the analysis may be based on any dissimilarity measure of choice, through the use of principal coordinates."

My Take

This is the piece that lets ecologists ask a question the classical multivariate toolbox couldn't answer for their data: are two groups differently variable (not just differently centered), when you have hundreds of species, non-normal counts, and want Bray-Curtis rather than Euclidean distance? Its elegance is in reducing a hard multivariate problem to a familiar univariate one — distances-to-centroid fed into Levene's test — and then using permutation to make the inference assumption-light. Paired with PERMANOVA it forms the standard two-part answer in community ecology: PERMANOVA for location, PERMDISP for dispersion, with the important caveat (that the paper is careful about) that a "significant PERMANOVA" can be driven by dispersion differences rather than genuine location shifts, so the two tests must be read together. It sits slightly outside this wiki's econometric core, but the distance-to-centroid + permutation pattern is a broadly useful nonparametric idea.