A Comparative Study of Reciprocal Averaging and Other Ordination Techniques

Abstract

SUMMARY Reciprocal averaging is a technique of indirect ordination, related both to weighted averages and to principal components analysis and other eigenvector techniques. A series of tests with simulated community gradients (coenoclines), simulated community patterns (coenoplanes), and sets of vegetation samples was used to compare ordination performance of reciprocal averaging (RA) with non-standardized and standardized principal components analysis (PCA) and polar or Bray-Curtis ordination (PO). Of these, non-standardized PCA is most vulnerable to effects of beta diversity, giving distorted ordinations of sample sets with three or more half-changes. PO and RA give good ordinations to five or more half-changes, and standardized PCA is intermediate. Sample errors affect all these techniques more at low than at high beta diversity, but PCA is most vulnerable to effects of sample errors. All three techniques could ordinate well a small (1-5 x 1-5 half-changes) simulated community pattern; and PO and RA could ordinate larger patterns (4 5 x 4-5 half-changes) well. PCA distorts larger community patterns into complex surfaces. Given a rectangular pattern (1-5 x 4-5 halfchanges), RA distorts the major axis of sample variation into an arch in the second axis of ordination. Clusters of samples tend to distort PCA ordinations in rather unpredictable ways, but they have smaller effects on RA, and none on PO. Outlier samples do not affect PO (unless used as endpoints), but can cause marked deterioration in RA and PCA ordinations. RA and PO are little subject to the involution of axis extremes that affects nonstandardized PCA. Despite the arch effect, RA is much superior to PCA at high beta diversities and on the whole preferable to PCA at low beta diversities. Second and higher axes of PCA and RA may express ecologically meaningless, curvilinear functions of lower axes. When curvilinear displacements are combined with sample error, axis interpretation is difficult. None of the techniques solves all the problems for ordination that result from the curvilinear relationships characteristic of community data. For applied ordination research consideration of sample set properties, careful use of supporting information to evaluate axes, and comparison of results of RA or PCA with PO and direct ordination are suggested.