Distance‐Based Tests for Homogeneity of Multivariate Dispersions
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.
Integrity signals
Registry
Cites
Statistics
Claims the paper makes
Claims other papers make about this one
Each is checked against this paper’s own text: does it say what the citing paper says it does?
Statistics that exploit pairwise sequence similarity can be used as alternatives to classical summary statistic measures of sequence diversity differences between groups.
“Alternatively, statistics that exploit pairwise sequence similarity can be used65 as alternatives to classical summary statistic measures of sequence diversity differences between groups.”
From Statistical analysis strategies for association studies involving rare variants (2010) · cites this paper as [65]
Cites
Non‐parametric multivariate analyses of changes in community structure
Clarke, K. R. (1993) · Australian Journal of Ecology · doi:10.1111/j.1442-9993.1993.tb00438.x · Cited by 14,196 · see citing works
A new method for non-parametric multivariate analysis of variance
Anderson, M. J. (2001) · Austral Ecology · doi:10.1111/j.1442-9993.2001.01070.pp.x · Cited by 13,828 · see citing works
Self-citationAn Ordination of the Upland Forest Communities of Southern Wisconsin
Bray, J. R., & Curtis, J. T. (1957) · Ecological Monographs · doi:10.2307/1942268 · Cited by 11,736 · see citing works
Ecologically meaningful transformations for ordination of species data
Legendre, P., & Gallagher, E. D. (2001) · Oecologia · doi:10.1007/s004420100716 · Cited by 5,556 · see citing works
Some Distance Properties of Latent Root and Vector Methods Used in Multivariate Analysis
Gower, J. C. (1966) · Biometrika · doi:10.1093/biomet/53.3-4.325 · Cited by 4,274 · see citing works
Randomization, Bootstrap and Monte Carlo Methods in Biology
Manly, B. F. J. (2018) · Chapman & Hall/CRC eBooks · doi:10.1201/9781315273075 · Cited by 4,233 · see citing works
Robust tests for equality of variances
Levene, H. (1961) · Medical Entomology and Zoology · Cited by 3,271 · see citing works
Robust Tests for the Equality of Variances
Brown, M. B., & Forsythe, A. B. (1974) · Journal of the American Statistical Association · doi:10.1080/01621459.1974.10482955 · Cited by 3,216 · see citing works
Legendre, P., & Anderson, M. J. (1999) · Ecological Monographs · doi:10.1890/0012-9615(1999)069[0001:dbratm]2.0.co;2 · Cited by 2,761 · see citing works
Self-citationAggregation, Variance and the Mean
TAYLOR, L. R. (1961) · Nature · doi:10.1038/189732a0 · Cited by 2,496 · see citing works
The Abuse of Power: The Pervasive Fallacy of Power Calculations for Data Analysis
Hoenig, J. M., & Heisey, D. M. (2001) · The American Statistician · doi:10.1198/000313001300339897 · Cited by 2,046 · see citing works
Edgington, E. S., Edgington, E., & Onghena, P. (1964) · Encyclopedia of Statistical Sciences · doi:10.1002/0471667196.ess2169 · Cited by 2,034 · see citing works
Randomization, Bootstrap and Monte Carlo Methods in Biology.
Kemp, A. W., & Manly, B. F. J. (1997) · Biometrics · doi:10.2307/2533527 · Cited by 1,941 · see citing works
Compositional dissimilarity as a robust measure of ecological distance
Faith, D. P., Minchin, P. R., & Belbin, L. (1987) · Plant Ecology · doi:10.1007/bf00038687 · Cited by 1,586 · see citing works
Compositional dissimilarity as a robust measure of ecological distance
Faith, D. P., Minchin, P. R., & Belbin, L. (1987) · Springer eBooks · doi:10.1007/978-94-009-4061-1_6 · Cited by 1,465 · see citing works
Some Practical Guidelines for Effective Sample Size Determination
Lenth, R. (2001) · The American Statistician · doi:10.1198/000313001317098149 · Cited by 1,200 · see citing works
A GENERAL DISTRIBUTION THEORY FOR A CLASS OF LIKELIHOOD CRITERIA
Box, G. E. P. (1949) · Biometrika · doi:10.1093/biomet/36.3-4.317 · Cited by 1,046 · see citing works
Conover, W. J., Johnson, M. E., & Johnson, M. M. (1981) · Technometrics · doi:10.1080/00401706.1981.10487680 · Cited by 1,008 · see citing works
Metric and Euclidean properties of dissimilarity coefficients
Gower, J. C., & Legendre, P. (1986) · Journal of Classification · doi:10.1007/bf01896809 · Cited by 937 · see citing works
Mathematics and the Picturing of Data
Tukey, J. W. (1975) · Medical Entomology and Zoology · Cited by 733 · see citing works
On choosing a test statistic in multivariate analysis of variance
Olson, C. L. (1976) · Psychological Bulletin · doi:10.1037/0033-2909.83.4.579 · Cited by 649 · see citing works
Contributions to Probability and Statistics
Gleser, L. J., Perlman, M. D., Press, S. J., & Sampson, A. R. (1989) · Springer eBooks · doi:10.1007/978-1-4612-3678-8 · Cited by 612 · see citing works
Permutation Tests for Linear Models
Anderson, M. J., & Robinson, J. (2001) · Australian & New Zealand Journal of Statistics · doi:10.1111/1467-842x.00156 · Cited by 555 · see citing works
Self-citationComparative Robustness of Six Tests in Multivariate Analysis of Variance
Olson, C. L. (1974) · Journal of the American Statistical Association · doi:10.1080/01621459.1974.10480224 · Cited by 507 · see citing works
Increased variability as a symptom of stress in marine communities
Warwick, R. M., & Clarke, K. R. (1993) · Journal of Experimental Marine Biology and Ecology · doi:10.1016/0022-0981(93)90098-9 · Cited by 491 · see citing works
The Multivariate Poisson-Log Normal Distribution
AITCHISON, J., & Ho, C.-H. (1989) · Biometrika · doi:10.1093/biomet/76.4.643 · Cited by 360 · see citing works
Multivariate statistical inference and applications
(1998) · Choice Reviews Online · doi:10.5860/choice.35-5714 · Cited by 358 · see citing works
Multivariate Statistical Methods: A Primer
Nemeth, M. A. (1997) · Technometrics · doi:10.1080/00401706.1997.10485166 · Cited by 328 · see citing works
Multi-response permutation procedures for a priori classifications
Mielke, P. W., Berry, K. J., & Johnson, E. S. (1976) · Communication in Statistics- Theory and Methods · doi:10.1080/03610927608827451 · Cited by 311 · see citing works
Multivariate Statistical Inference and Applications
Booth, D. E., & Rencher, A. C. (1998) · Technometrics · doi:10.2307/1270542 · Cited by 299 · see citing works
Anderson, M. J., & Millar, R. B. (2004) · Journal of Experimental Marine Biology and Ecology · doi:10.1016/j.jembe.2003.12.011 · Cited by 296 · see citing works
Self-citationVariation in the Size of Animal Populations: Patterns, Problems and Artefacts
McArdle, B. H., Gaston, K. J., & Lawton, J. H. (1990) · Journal of Animal Ecology · doi:10.2307/4873 · Cited by 277 · see citing works
Chapman, M. G., Underwood, A. J., & Skilleter, G. A. (1995) · Journal of Experimental Marine Biology and Ecology · doi:10.1016/0022-0981(95)00017-l · Cited by 224 · see citing works
On randomization testing in vegetation science: multifactor comparisons of relevé groups
Pillar, V. D., & Orlóci, L. (1996) · Journal of Vegetation Science · doi:10.2307/3236308 · Cited by 224 · see citing works
The Randomization Theory of Experimental Inference*
Kempthorne, O. (1955) · Journal of the American Statistical Association · doi:10.1080/01621459.1955.10501979 · Cited by 216 · see citing works
Statistical Uses of the Spatial Median
Brown, B. M. (1983) · Journal of the Royal Statistical Society Series B (Statistical Methodology) · doi:10.1111/j.2517-6161.1983.tb01226.x · Cited by 210 · see citing works
Miller, R. G. (1968) · The Annals of Mathematical Statistics · doi:10.1214/aoms/1177698418 · Cited by 207 · see citing works
Warwick, R. M., Clarke, K. R., & Suharsono, S. (1990) · Coral Reefs · doi:10.1007/bf00265008 · Cited by 165 · see citing works
Note on the median of a multivariate distribution
Haldane, J. B. S. (1948) · Biometrika · doi:10.1093/biomet/35.3-4.414 · Cited by 162 · see citing works
"The Coefficient of Racial Likeness" and the Future of Craniometry.
Fisher, R. A. (1936) · The Journal of the Royal Anthropological Institute of Great Britain and Ireland · doi:10.2307/2844116 · Cited by 138 · see citing works
Valen, L. V. (2005) · Elsevier eBooks · doi:10.1016/b978-012088777-4/50005-3 · Cited by 123 · see citing works
The Randomization Theory of Experimental Inference
Kempthorne, O. (1955) · Journal of the American Statistical Association · doi:10.2307/2281178 · Cited by 112 · see citing works
Bootstrap Methods for Testing Homogeneity of Variances
Boos, D. D., & Brownie, C. (1989) · Technometrics · doi:10.1080/00401706.1989.10488477 · Cited by 107 · see citing works
Algorithm AS 78: The Mediancentre
Gower, J. C. (1974) · Journal of the Royal Statistical Society Series C (Applied Statistics) · doi:10.2307/2347150 · Cited by 93 · see citing works
Permutation Tests for Least Absolute Deviation Regression
Cade, B. S., & Richards, J. D. (1996) · Biometrics · doi:10.2307/2533050 · Cited by 82 · see citing works
High-dimensional computation of the deepest location
Struyf, A., & Rousseeuw, P. J. (2000) · Computational Statistics & Data Analysis · doi:10.1016/s0167-9473(99)00112-7 · Cited by 76 · see citing works
Gower, J. C., & Krzanowski, W. J. (1999) · Journal of the Royal Statistical Society Series C (Applied Statistics) · doi:10.1111/1467-9876.00168 · Cited by 72 · see citing works
Smith, E. P., Pontasch, K. W., & Cairns, J. (1990) · Water Research · doi:10.1016/0043-1354(90)90236-y · Cited by 70 · see citing works
Robust Procedures for Testing Equality of Covariance Matrices
O’Brien, P. C. (1992) · Biometrics · doi:10.2307/2532347 · Cited by 57 · see citing works
Resampling from centered data in the two-sample problem
Boos, D. D., Janssen, P., & Veraverbeke, N. (1989) · Journal of Statistical Planning and Inference · doi:10.1016/0378-3758(89)90051-7 · Cited by 37 · see citing works
Permutational tests for correlation matrices
Krzanowski, W. J. (1993) · Statistics and Computing · doi:10.1007/bf00146952 · Cited by 31 · see citing works
A Monte Carlo comparison of tests for equality of convariance matrices
Layard, M. W. J. (1974) · Biometrika · doi:10.1093/biomet/61.3.461 · Cited by 30 · see citing works
Testing the equality of variance-covariance matrices the robust way
TIKU, M. L., & Balakrishnan, N. (1985) · Communication in Statistics- Theory and Methods · doi:10.1080/03610928508829093 · Cited by 16 · see citing works
Bootstrap calibration to improve the reliability of tests to compare sample means and variances
Francis, R. I. C. C., & Manly, B. F. J. (2001) · Environmetrics · doi:10.1002/env.494 · Cited by 15 · see citing works
A Monte Carlo Comparison of Tests for Equality of Covariance Matrices
Layard, M. W. J. (1974) · Biometrika · doi:10.2307/2334727 · Cited by 15 · see citing works
Flagged first, then by citations
Cited by
Sorted by citations
Distinct soil microbial diversity under long-term organic and conventional farming
Hartmann, M., Frey, B., Mayer, J., Mäder, P., & Widmer, F. (2015) · The ISME Journal · doi:10.1038/ismej.2014.210 · Cited by 1,496 · see citing works
Global patterns of bacterial beta-diversity in seafloor and seawater ecosystems
Zinger, L., Amaral-Zettler, L. A., Fuhrman, J. A., Horner-Devine, M. C., Huse, S. M., Welch, D. B. M., Martiny, J. B. H., Sogin, M., Boetius, A., & Ramette, A. (2011) · PLoS ONE · doi:10.1371/journal.pone.0024570 · Cited by 643 · see citing works
Statistical analysis strategies for association studies involving rare variants
Bansal, V., Libiger, O., Torkamani, A., & Schork, N. J. (2010) · Nature Reviews Genetics · doi:10.1038/nrg2867 · Cited by 455 · see citing works
Hall, D. A., Voigt, R. M., Cantu-Jungles, T. M., Hamaker, B., Engen, P. A., Shaikh, M., Raeisi, S., Green, S. J., Naqib, A., Forsyth, C. B., et al. (2023) · Nature Communications · doi:10.1038/s41467-023-36497-x · Cited by 124 · see citing works
The Progressive Loss of Syntactical Structure in Bird Song along an Island Colonization Chain
Lachlan, R. F., Verzijden, M. N., Bernard, C. S., Jonker, P.-P., Koese, B., Jaarsma, S., Spoor, W., Slater, P. J. B., & ten Cate, C. (2013) · Current Biology · doi:10.1016/j.cub.2013.07.057 · Cited by 114 · see citing works
Hardinge, J., Harvey, E. S., Saunders, B. J., & Newman, S. J. (2013) · Journal of Experimental Marine Biology and Ecology · doi:10.1016/j.jembe.2013.09.018 · Cited by 93 · see citing works
Trustworthy-Looking Face Meets Brown Eyes
Kleisner, K., Priplatova, L., Frost, P., & Flegr, J. (2013) · PLoS ONE · doi:10.1371/journal.pone.0053285 · Cited by 85 · see citing works
Tet2 deficiency drives liver microbiome dysbiosis triggering Tc1 cell autoimmune hepatitis
Pandey, S. P., Bender, M. J., McPherson, A. C., Phelps, C. M., Sanchez, L. M., Rana, M., Hedden, L., Sangani, K. A., Chen, L., Shapira, J. H., et al. (2022) · Cell Host & Microbe · doi:10.1016/j.chom.2022.05.006 · Cited by 72 · see citing works
Wittkuhn, R. S., McCaw, L., Wills, A. J., Robinson, R., Andersen, A. N., Van Heurck, P., Farr, J., Liddelow, G., & Cranfield, R. (2011) · Forest Ecology and Management · doi:10.1016/j.foreco.2010.10.037 · Cited by 54 · see citing works
Kang, D.-W., DiBaise, J. K., Ilhan, Z. E., Crowell, M. D., Rideout, J. R., Caporaso, J. G., Rittmann, B. E., & Krajmalnik-Brown, R. (2015) · Anaerobe · doi:10.1016/j.anaerobe.2015.01.005 · Cited by 55 · see citing works
Popova, M., McGovern, E., McCabe, M. S., Martin, C., Doreau, M., Arbre, M., Meale, S. J., Morgavi, D. P., & Waters, S. M. (2017) · Frontiers in Microbiology · doi:10.3389/fmicb.2017.00937 · Cited by 48 · see citing works
Demircan, T., Ovezmyradov, G., Yıldırım, B., Keskin, İ., İlhan, A. E., Fesçioğlu, E. C., Öztürk, G., & Yıldırım, S. (2018) · Scientific Reports · doi:10.1038/s41598-018-29373-y · Cited by 45 · see citing works
Wang, Y., Liu, Y., Wang, J., Luo, T., Zhang, R., Sun, J., Zheng, Q., & Jiao, N. (2019) · Marine Pollution Bulletin · doi:10.1016/j.marpolbul.2019.03.035 · Cited by 45 · see citing works
Taipale, S., Peltomaa, E., & Salmi, P. (2020) · Biomolecules · doi:10.3390/biom10040559 · Cited by 36 · see citing works
McMillan, N. A., Kunkel, K. E., Hagan, D. L., & Jachowski, D. S. (2019) · Restoration Ecology · doi:10.1111/rec.12856 · Cited by 34 · see citing works
Gut Microbiota in Systemic Lupus Erythematosus and Correlation With Diet and Clinical Manifestations
Wang, X., Shu, Q., Song, L., Liu, Q., Qu, X., & Li, M. (2022) · Frontiers in Medicine · doi:10.3389/fmed.2022.915179 · Cited by 33 · see citing works
Claes, P., Reijniers, J., Shriver, M. D., Snyders, J., Suetens, P., Nielandt, J., De Tré, G., & Vandermeulen, D. (2015) · Journal of Anatomy · doi:10.1111/joa.12252 · Cited by 32 · see citing works
Deep Nasal Sinus Cavity Microbiota Dysbiosis in Parkinson’s Disease
Pal, G., Ramirez, V., Engen, P. A., Naqib, A., Forsyth, C. B., Green, S. J., Mahdavinia, M., Batra, P. S., Tajudeen, B. A., & Keshavarzian, A. (2021) · npj Parkinson s Disease · doi:10.1038/s41531-021-00254-y · Cited by 28 · see citing works
Bates, C. T., Escalas, A., Kuang, J., Hale, L., Wang, Y., Herman, D., Nuccio, E. E., Wan, X., Bhattacharyya, A., Fu, Y., et al. (2022) · The ISME Journal · doi:10.1038/s41396-021-00916-y · Cited by 17 · see citing works
- Authors
- Anderson, M. J.
- Affiliations
- University of Auckland (NZ)
- Corresponding author
- Anderson, Marti J.
- ORCID
- Anderson, Marti J.
- Venue
- Biometrics
- Publisher
- Oxford University Press (OUP)
- Published
- 2005-10-20
- Volume
- 62
- Issue
- 1
- Pages
- 245-253
- ISSN
- 0006-341X (ISSN-L)
- 1541-0420
- OpenAlex
- W2118327902
- Cited by
- 3,304
- References
- 55
- Citations by year
- 2026: 482 · 2025: 259 · 2024: 221 · 2023: 194 · 2022: 268 · 2021: 294 · 2020: 280 · 2019: 215 · 2018: 207 · 2017: 166 · 2016: 161 · 2015: 114 · 2014: 101 · 2013: 110 · 2012: 77
- FWCI
- 12.45
- Type
- journal article
- Language
- en
- Open access
- closed
- Locations
- Biometrics (publishedVersion)
- PubMed (publishedVersion)
- RePEc: Research Papers in Economics (submittedVersion)
- Topics
- Physical Sciences › Mathematics › Statistics and Probability › Advanced Statistical Methods and Models
- Physical Sciences › Mathematics › Statistics and Probability › Statistical Methods and Applications
- Physical Sciences › Mathematics › Statistics and Probability › Statistical Distribution Estimation and Applications
- Keywords
- multivariate dispersion, dissimilarity measures, Levene's test, spatial median, rotation invariance, principal coordinates, permutation tests, robustness to non-normality, high-dimensional data, stabilizing selection, reef-building corals, El Niño, biodiversity
- MeSH terms
- Animals, Ecology, Fishes, Multivariate Analysis, Anthozoa, Biodiversity, Sparrows
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
What counts as a claim ›How we measure confidence ›How we check this ›
Are you an author, or do you see a problem with this page? Request a review
