Size and shape in the measurement of multivariate proximity

Most methods of multivariate analysis rely on a measure of proximity between individual cases or samples to quantify inter-sample differences. The choice of this measure is fundamental to the method and its subsequent results. For example, when data are abundance counts of a set of species at several sampling locations, some approaches rely on the Bray-Curtis dissimilarity measure between samples, while other approaches rely on the chi-square distance. A set of observed species abundances at a location has both size, in the form of the overall levels of the species counts, and shape, in the form of the relative values of the counts. The aim of this report is to clarify how much the chosen proximity measure is capturing differences in size between samples as opposed to differences in shape. After motivating the idea using physical morphometric data, the study is extended to nonnegative data in general, with special focus on abundance counts and biomass estimates, which are ubiquitous in ecological research.