Correlation analysis of dissimilarity matrices

Abstract

Distance-based methods have been a valuable tool for ecologists for decades. Indirectly, distance-based ordination and cluster analysis, in particular, have been widely practiced as they allow the visualization of a multivariate data set in a few dimensions. The explicitly distance-based Mantel test and multiple regression on distance matrices (MRM) add hypothesis testing to the toolbox. One concern for ecologists wishing to use these methods lies in deciding whether to combine data vectors into a compound multivariate dissimilarity to analyze them individually. For Euclidean distances on scaled data, the correlation of a pair of multivariate distance matrices can be calculated from the correlations between the two sets of individual distance matrices if one set is orthogonal, demonstrating a clear link between individual and compound distances. The choice between Mantel and MRM should be driven by ecological hypotheses rather than mathematical concerns. The relationship between individual and compound distance matrices also provides a means for calculating the maximum possible value of the Mantel statistic, which can be considerably less than 1 for a given analysis. These relationships are demonstrated with simulated data. Although these mathematical relationships are only strictly true for Euclidean distances when one set of variables is orthogonal, simulations show that they are approximately true for weakly correlated variables and Bray–Curtis dissimilarities.