Distance-based Redundancy Analysis: Testing Multispecies Responses in Multifactorial Ecological Experiments

We present a new multivariate technique for testing the significance of individual terms in a multifactorial analysis-of-variance model for multispecies response variables. The technique will allow researchers to base analyses on measures of association (distance measures) that are ecologically relevant. In addition, unlike other distance-based hypothesis-testing techniques, this method allows tests of significance of interaction terms in a linear model. The technique uses the existing method of redundancy analysis (RDA) but allows the analysis to be based on Bray-Curtis or other ecologically meaningful measures through the use of principal coordinate analysis (PCoA). Steps in the procedure include: (1) calculating a matrix of distances among replicates using a distance measure of choice (e.g., Bray-Curtis); (2) determining the principal coordinates (including a correction for negative eigenvalues, if necessary), which preserve these distances; (3) creating a matrix of dummy variables corresponding to the design of the experiment (i.e., individual terms in a linear model); (4) analyzing the relationship between the principal coordinates (species data) and the dummy variables (model) using RDA; and (5) implementing a test by permutation for particular statistics corresponding to the particular terms in the model. This method has certain advantages not shared by other multivariate testing procedures. We demonstrate the use of this technique with experimental ecological data from intertidal assemblages and show how the presence of significant multivariate interactions can be interpreted. It is our view that distance-based RDA will be extremely useful to ecologists measuring multispecies responses to structured multifactorial experimental designs.