 2017 The Authors Australian & New Zealand Journal of Statistics published by John Wiley & Sons Australia, Ltd on behalf of Statistical Society of Australia. This is an open access article under the terms of the Creative Commons Attribution-NonCommercial-NoDerivs License, which permits use and distribution in any medium, provided the original work is properly cited, the use is non-commercial and no modifications or adaptations are made. Aust. N. Z. J. Stat. 59(1), 2017, 57–79 doi: 10.1111/anzs.12176 Some solutions to the multivariate Behrens–Fisher problem for dissimilarity-based analyses Marti J. Anderson 1 *, Daniel C. I. Walsh 2 , K. Robert Clarke 3 , Ray N. Gorley 4 and Edlin Guerra-Castro 5 Massey University and PRIMER-E Limited and Universidad Nacional Aut´ onoma de M´ exico Summary The essence of the generalised multivariate Behrens–Fisher problem (BFP) is how to test the null hypothesis of equality of mean vectors for two or more populations when their dispersion matrices differ. Solutions to the BFP usually assume variables are multivariate normal and do not handle high-dimensional data. In ecology, species’ count data are often high- dimensional, non-normal and heterogeneous. Also, interest lies in analysing compositional dissimilarities among whole communities in non-Euclidean (semi-metric or non-metric) multivariate space. Hence, dissimilarity-based tests by permutation (e.g., PERMANOVA, ANOSIM) are used to detect differences among groups of multivariate samples. Such tests are not robust, however, to heterogeneity of dispersions in the space of the chosen dissimilarity measure, most conspicuously for unbalanced designs. Here, we propose a modification to the PERMANOVA test statistic, coupled with either permutation or bootstrap resampling methods, as a solution to the BFP for dissimilarity-based tests. Empirical simulations demonstrate that the type I error remains close to nominal significance levels under classical scenarios known to cause problems for the un-modified test. Furthermore, the permutation approach is found to be more powerful than the (more conservative) bootstrap for detecting changes in community structure for real ecological datasets. The utility of the approach is shown through analysis of 809 species of benthic soft-sediment invertebrates from 101 sites in five areas spanning 1960 km along the Norwegian continental shelf, based on the Jaccard dissimilarity measure. Key words: Behrens–Fisher problem; bootstrap; dissimilarity matrix; ecological community data; PERMANOVA; permutation test. 1. Introduction The Behrens–Fisher problem (BFP) is one of the oldest puzzles in problems (Behrens 1929; Fisher 1935; Welch 1938). The essence of this problem is how validly to compare the means (or multivariate mean vectors) of two or more populations when their variances (or multivariate dispersions) differ. Solutions to the BFP for univariate data generally assume *Author to whom correspondence should be addressed. 1 New Zealand Institute for Advanced Study (NZIAS), Massey University, Albany campus, Private Bag 102 904, North Shore, Auckland 0745, New Zealand e-mail: m.j.anderson@massey.ac.nz 2 Institute of Natural & Mathematical Sciences (INMS), Massey University, Albany Campus, Private Bag 102 904, North Shore, Auckland, 0745, New Zealand. 3 Plymouth Marine Laboratory, Prospect Place, The Hoe, Plymouth PL1 3DH, UK. 4 PRIMER-E Limited, c/o Plymouth Marine Laboratory, Prospect Place, The Hoe, Plymouth PL1 3DH, UK. 5 CONACYT, Unidad Multidisciplinaria de Docencia e Investigac´ on Sisal, Facultad de Ciencias, Universidad Nacional Aut´ onoma de M´ exico, Puerto de Sisal, Yucat´ an, M´ exico Acknowledgement. This work was supported by a Royal Society of New Zealand Marsden Grant. Australian & New Zealand Journal of Statistics