Definition
PERMDISP (Anderson 2006) is a distance-based, permutation test for whether the multivariate dispersion (spread) of observations differs among groups — the multivariate generalization of Levene's test and the dispersion counterpart to PERMANOVA (which tests differences in location). It measures spread as the average distance of observations to their group centroid (or spatial median) under any chosen dissimilarity measure, and tests group differences in that spread by permutation.
Key Ideas
- Dispersion, not location. Where PERMANOVA (Anderson 2001) asks whether group centroids differ, PERMDISP asks whether group spreads differ. Both are needed: a significant location test can be an artefact of unequal dispersions, so the two are interpreted together.
- Distance-to-centroid deviations. Embed the dissimilarity matrix in Euclidean space via principal coordinates (PCO; Gower 1966); for each observation compute its distance zij to its group centroid (least-squares) or spatial median (robust to outliers). The z's are the univariate response.
- Multivariate Levene's test. Apply a one-way ANOVA F-statistic to the z deviations — exactly Levene's (1960) test (Brown-Forsythe median version for robustness) lifted to the multivariate/dissimilarity setting.
- Permutation inference. Because the z's are neither independent nor normal, P-values are obtained by permuting the least-squares or LAD residuals, rather than (or in addition to) relying on the F-distribution.
- Any dissimilarity. The test works with Euclidean, Bray-Curtis, Jaccard, or any distance — the defining flexibility for ecological community data.
Why It Matters
- Works where classical tests fail. Box's M (the likelihood-ratio test for equal covariance matrices) assumes multivariate normality and requires more observations than variables; PERMDISP handles high-dimensional (p>n), skewed, zero-inflated data and non-Euclidean geometry.
- Standard ecology toolkit. Implemented as
betadisper in R's vegan package, it is routinely paired with PERMANOVA to analyse community structure, beta-diversity, and multivariate ecological responses.
- A reusable pattern. "Reduce a multivariate question to distances-to-centroid, then test by permutation" is a broadly applicable nonparametric device.
Open Questions
- Confounding of location and dispersion. A rejection in PERMANOVA may reflect dispersion rather than location differences; disentangling the two requires running and interpreting both.
- Choice of centre and dissimilarity. Centroid vs. spatial median, and the choice of dissimilarity measure, affect power and can change conclusions; there is no fully data-adaptive rule.
- Balance and permutation scheme. Unbalanced designs and complex/nested structures complicate the correct exchangeable units for permutation.
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