Biplots for multifactorial analysis of distance

Many data sets in practice fit a multivariate analysis of variance (MANOVA) structure, but do not accord with MANOVA assumptions for their analysis. One way forward is to calculate the matrix of dissimilarities or distances between every pair of individuals, and then to conduct an analysis of distance on the resulting data. Various metric scaling plots can be used to interpret the results of the analysis. However, developments to date of this approach have focused mainly on the individuals in the sample, and little attention has been paid to the assessment of influence of the original variables on the results. The present article attempts to rectify this omission. We discuss the inclusion of biplots on all forms of metric scaling representations in the analysis of distance. Exact biplots will often be nonlinear so we propose a simple linear approximation, and contrast it with other simple linear possibilities. An example from ecology illustrates the methodology.