FITTING MULTIVARIATE MODELS TO COMMUNITY DATA: A COMMENT ON DISTANCE-BASED REDUNDANCY ANALYSIS

TLDR

Ecological multivariate analysis using permutation tests must both partition variability according to complex designs and rely on suitable multivariate distance measures, yet prior nonparametric methods have addressed only one of these challenges. The proposed distance‑based redundancy analysis (db‑RDA) achieves this by computing principal coordinates and correcting negative eigenvalues through a constant added to squared distances, with an ecological example illustrating the differences between this approach and direct partitioning. The authors show that the eigenvalue correction is unnecessary; a direct partitioning of the distance matrix yields accurate type‑1 error rates, whereas the corrected db‑RDA fails to maintain the nominal significance level.

Abstract

Nonparametric multivariate analysis of ecological data using permutation tests has two main challenges: (1) to partition the variability in the data according to a complex design or model, as is often required in ecological experiments, and (2) to base the analysis on a multivariate distance measure (such as the semimetric Bray-Curtis measure) that is reasonable for ecological data sets. Previous nonparametric methods have succeeded in one or other of these areas, but not in both. A recent contribution to Ecological Monographs by Legendre and Anderson, called distance-based redundancy analysis (db-RDA), does achieve both. It does this by calculating principal coordinates and subsequently correcting for negative eigenvalues, if they are present, by adding a constant to squared distances. We show here that such a correction is not necessary. Partitioning can be achieved directly from the distance matrix itself, with no corrections and no eigenanalysis, even if the distance measure used is semimetric. An ecological example is given to show the differences in these statistical methods. Empirical simulations, based on parameters estimated from real ecological species abundance data, showed that db-RDA done on multifactorial designs (using the correction) does not have type 1 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.