'Size' and 'shape' in the measurement of multivariate proximity. Issue 11 (8th May 2017) - Greenacre, Michael
Summary: Ordination and clustering methods are widely applied to ecological data that are non‐negative, for example, species abundances or biomasses. These methods rely on a measure of multivariate proximity that quantifies differences between the sampling units (e.g. individuals, stations, time points), leading to results such as: (i) ordinations of the units, where interpoint distances optimally display the measured differences; (ii) clustering the units into homogeneous clusters or (iii) assessing differences between pre‐specified groups of units (e.g. regions, periods, treatment–control groups). These methods all conceal a fundamental question: To what extent are the differences between the sampling units, computed according to the chosen proximity function, capturing the 'size' in the multivariate observations, or their 'shape'? 'Size' means the overall level of the measurements: for example, some samples contain higher total abundances or more biomass, others less. 'Shape' means the relative levels of the measurements: for example, some samples have different relative abundances, i.e. different compositions. To answer this question, several well‐known proximity measures are considered and applied to two datasets, one of which is used in a simulation exercise where 'shape' differences have been eliminated by randomization. For any dataset and any proximity measure, a quantification is achieved of the proportion of 'size' variance and 'shape' variance that the measure is capturing, as well as the proportion of variance that confounds 'size' and 'shape' together. The results consistently show that the Bray–Curtis coefficient incorporates both 'size' and 'shape' differences, to varying degrees. These two components are thus always confounded by this proximity measure in the determination of ordinations, clusters, group comparisons and relations to environmental variables. There are several implications of these results, the main one being that researchers should be aware of this issue when they choose a proximity measure. They should compute the 'size' and 'shape' components for their particular datasets, as this can radically affect the interpretation of their results. It is recommended to separate these components: analysing total abundances or other measures of 'size' by univariate methods, and using multivariate analysis on the relative abundances where size has been specifically excluded.