PROTEST: A PROcrustean Randomization TEST of community environment concordance

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Abstract
A multivariate measure of the concordance or association between matrices of species abundances and environmental variables was generally lacking in ecology until recently. Traditional statistical procedures comparing such relationships are often unsuitable because of non-linearity among species and/or environmental data. To address these problems, I propose a randomization test based on Procrustes analysis. One matrix is subject to reflection, rigid rotation, translation, and dilation to minimize the sum of the squared residual deviations between points for each observation and the identical observation in the target matrix. This is a classical Procrustes approach to matrix analysis. To assess the significance of this measure of matrix concordance, I use a randomization test to determine whether the sum of residual deviations is less than that expected by chance. The PROcrustean randomization TEST (PROTEST) may be used with either raw data matrices or with multivariate summaries of the original data (i.e. both direct or indirect gradient analysis). I provide examples of PROTEST analyses with benthic invertebrate communities, lake-water chemistry, lake morphology, and lake geographic position. Significant concordance between the benthic community and both lake-water chemistry and geographic position were found. PROTEST results differed from Mantel test results as the choice of distance measure with Mantel tests will influence the level of significance obtained.