Comparison of multidimensional fuzzy set ordination with CCA and DB-RDA

Multidimensional fuzzy set ordination (MFSO) is an ordination algorithm based on the principles of fuzzy set theory that employs a dissimilarity matrix and environmental or experimental variables directly in the calculation of ordination coordinates. The objective is similar to constrained ordinations such as canonical correspondence analysis (CCA) and distance-based redundancy analysis (DB-RDA), although MFSO employs a different conceptual and algorithmic basis. Results from MFSO, CCA, and DB-RDA are compared on four data sets to determine the relative ability of the methods to identify and quantify environmental variable effect sizes on community variability. Models were fit by best all-subsets solutions and tested for significance by permutation tests. Methods were compared on the basis of variables selected and dimensionality of results and were evaluated on the basis of the correlation of the pair-wise distances of the derived ordination solutions to a Bray-Curtis dissimilarity matrix of the taxon data, and by cross-comparisons of the inertia explained by the fitted or derived values substituted into DB-RDA and CCA analyses. MFSO and DB-RDA achieved the highest correlation with the Bray-Curtis dissimilarity matrix on one data set each, and they tied for highest correlation on the two other data sets (differences <1%). CCA tied for best on one data set, tied for second on another, and achieved the lowest correlation on two data sets. When substituting MFSO- and DB-RDA-derived values into CCA, MFSO results achieved higher inertia explained on two of the four data sets; MFSO and DB-RDA achieved equivalent results on the other two. Substituting MFSO- or CCA-derived or fitted values into DB-RDA, MFSO results achieved higher inertia explained on all four data sets. While goodness-of-fit statistics were often similar across methods, the methods sometimes chose solutions of different dimensionality or employed different variables. In general, all algorithms performed well, finding relatively low-dimensional solutions that achieved high correlation with the Bray-Curtis dissimilarity matrix and explained significant amounts of inertia in the eigenanalyses. Considering both the correlation and substitution tests, MFSO achieved the best results of the three methods. DB-RDA results were nearly as good as MFSO, however, and clearly superior to CCA.