Biased richness and evenness relationships within Shannon–Wiener index values

The purpose of this analysis was to empirically model and graphically illustrate the numerical relationships between richness (S, 4–35 species) and evenness (E) with respect to Shannon–Wiener index (H′, log_e-based) values. Thirty-two richness-based third-order polynomial regression models (R > 0.99, P < 0.001, n = 28–71) were constructed to characterize these relationships. A composite diagram showed richness varied curvilinearly, with steepness increasing and the spacing between curves decreasing with greater evenness and H′. Maximum H′ values for each richness curve were equal to log_e S (when E = 1), whereas minima were approximated by evenness values of ～1/S (when H′ = 0). It was concluded from multiple and polynomial regression analyses that: (i) evenness contributed more than richness (E:S ≥3:1) to determining H′, based on standardized partial beta-coefficients; (ii) the differential in E:S ratios increased with greater richness; (iii) the patterns of H′ sample variation between maximum unevenness and perfect evenness was convexo-concave shaped; and (iv) richness as an explanatory variable of H′ was likely an artifact of evenness (0–1 scale) being rescaled according to individual H′ maxima. H′ was redefined as a logarithm-weighted measure of evenness at a given level of richness, which means H′ is either an imperfect index of diversity or a biased measure of evenness. It was also found that the fundamental components of the Shannon–Wiener index measure dominance concentration rather than evenness, with the reversal in emphasis due to multiplication of the H′ equation by ?1. H′-derived effective species numbers (exp H′, D) increasingly deviated from those of the diversity model D = S × E in response to increasing richness (up to 69% for 35 species), particularly when evenness was between 0.15 and 0.40. Of two cross-validated H′ prediction methods (P < 0.001, n = 325), the collective use of individual richness-based polynomial regression equations (r = 0.954) was better than a single multiple regression model that incorporated a broad spectrum of richness levels (r = 0.882). A simple graphic model was constructed to illustrate patterns of evenness variation as a function of changing richness and H′ values. Based on the identified biases, particularly E:S ratios, it was recommended that use of H′ be discontinued as a basis for assessing diversity in ecological research or, at the very least, accompanied by independent analyzes of richness and evenness.