Distance-based redundancy analysis: Testing multispecies responses in multifactorial ecological experiments

We present a new multivariate technique for testing the significance of ind ividual terms in a multifactorial analysis-of-variance model for multispeci es response variables. The technique will allow researchers to base analyse s on measures of association (distance measures) that are ecologically rele vant. In addition, unlike other distance-based hypothesis-testing technique s, this method allows tests of significance of interaction terms in a linea r 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 amo ng replicates using a distance measure of choice (e.g,, Bray-Curtis); (2) d etermining the principal coordinates (including a correction for negative e igenvalues if necessary), which preserve these distances (3) creating a mat rix of dummy variables corresponding to the design of the experiment (i.e., individual terms in a linear model); (4) analyzing thr relationship betwee n thp principal coordinates (species data) and the dummy variables (model) using RDA; and (5) implementing a test by permutation for particular statis tics corresponding to the particular terms in the model. This method has ce rtain advantages not shared by other multivariate testing procedures. We de monstrate the use of this technique with experimental ecological data from intertidal assemblages and show how the presence of significant multivariat e interactions can be interpreted. It is our view that distance-based RDA w ill be extremely useful to ecologists measuring multispecies responses to s tructured multifactorial experimental designs.