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

Abstract

SummaryThe 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 highdimensional, 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.

Nutrient enrichment homogenizes taxonomic and functional diversity of benthic macroinvertebrate assemblages in shallow lakes

“…Nonmetric multidimensional scaling (NMDS) was performed based on a Bray‐Curtis similarity matrix obtained from log( x + 1)‐transformed abundance data, using the function vegdist in the R package vegan (Oksanen et al ). Macroinvertebrate assemblage structures, both taxonomic and functional compositions, were compared among the three lake trophic groups using permutational multivariate analysis of variance (PERMANOVA) and pairwise comparison (Anderson et al ), based on a Bray‐Curtis resemblance matrix with 999 permutations. The Bray‐Curtis dissimilarities of taxonomic composition and functional composition among the three lake trophic groups were also calculated.…”

Predicting Parallelism and Quantifying Divergence in Experimental Evolution

“…The statistics MCD and MPD calculated on the first principal component have much lower power than the more commonly used statistic , though they overtake once additional PCs are considered. Given that the statistics were fairly similar and that MPD is used to calculate F 2 [2], we used MPD for the remaining analyses. Statistical power slightly increases with the degree of clustering, though the increment is very small for the range of clustering coefficients examined (Fig.…”

“…The second statistic is the Mean Pairwise Distance (MPD), a statistic frequently used when comparing variation within and between groups in ordination space [2]. MPD is defined as where is the k -element vector of the i th population and d () is the Euclidean distance…”

“…In this experiment Burkholderia cenocepacia with planktonic or biofilm life in environments containing with low or high concentrations of carbon [52]. The degree of evolutionary divergence was quantified using two forms of Permutational ANOVA (PERMANOVA) F statistics, a standard one ( F 1 ) and one that accounts for unequal levels of parallelism among treatments ( F 2 ) [2]. Null population-by-gene count matrices for each study were constructed for k treatments, randomized, and concatenated as .…”

A First Comprehensive Baseline of Hydrocarbon Pollution in Gulf of Mexico Fishes

“…The differences in mean biliary PAHs between regions and years were assessed using a modified permutational multivariate analysis of variance (PERMANOVA) and pair-wise modified PERMANOVA with 1,000 permutations and an α = 0.05. These are semiparametric methods that use the appropriate distribution-free permutation techniques accounting for unbalanced/asymmetrical designs and heterogenous dispersions among groups 85 . Species differences were evaluated using the Games-Howell Honestly Significant Difference (HSD) approach designed for unequal variances and sample sizes.…”