Fitting multivariate models to community data: A comment on distance-basedredundancy analysis

Nonparametric multivariate analysis of ecological data using permutation te sts has two main challenges: (1) to partition the variability in the data a ccording to a complex design or model, as is often required in ecological e xperiments, and (2) to base the analysis on a multivariate distance measure (such as the semimetric Bray-Curtis measure) that is reasonable for ecolog ical data sets. Previous nonparametric methods have succeeded in one or oth er of these areas, but not in both. A recent contribution to Ecological Mon ographs by Legendre and Anderson, called distance-based redundancy analysis (db-RDA), does achieve both. It does this by calculating principal coordin ates and subsequently correcting for negative eigenvalues, if they are pres ent, by adding a constant to squared distances. We show here that such a co rrection is not necessary. Partitioning can be achieved directly from the d istance matrix itself, with no corrections and no eigenanalysis, even if th e distance measure used is semimetric. An ecological example is given to sh ow the differences in these statistical methods. Empirical simulations, bas ed on parameters estimated from real ecological species abundance data, sho wed that db-RDA done on multifactorial designs (using the correction) does not have type I error consistent with the significance level chosen for the analysis (i.e., does not provide an exact test), whereas the direct method described and advocated here does.