Some solutions to the multivariate Behrens–Fisher problem for dissimilarity‐based analyses

• Published in: Australian & New Zealand journal of statistics Vol. 59; no. 1; pp. 57 - 79
• Main Authors: Anderson, Marti J., Walsh, Daniel C. I., Robert Clarke, K., Gorley, Ray N., Guerra‐Castro, Edlin
• Format: Journal Article
• Language: English
• Published: Hoboken Wiley Subscription Services, Inc 01.03.2017
• Subjects: Behrens–Fisher problem; bootstrap; Centroids; Change detection; Communities; Continental shelves; Dimensional analysis; Dispersion; Dispersions; dissimilarity matrix; ecological community data; Ecological monitoring; Ecology; Empirical analysis; Euclidean geometry; Heterogeneity; Invertebrates; Matrices (mathematics); Matrix methods; Null hypothesis; PERMANOVA; permutation test; Permutations; Populations; Resampling; Samples; Statistical analysis; Statistical methods; Statistics
• ISSN: 1369-1473; 1467-842X
• DOI: 10.1111/anzs.12176

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. This article describes a new multivariate test for differences in centroids in the presence of heterogeneity of dispersions on the basis of a chosen dissimilarity measure.