Evenness under incomplete sampling: coupled error channels, hidden compensation, and the richness-estimation bottleneck

Luca Delvecchio

1. Normalized evenness combines an abundance-diversity functional with richness, so incomplete sampling acts through two coupled errors. For \(E_{\alpha,\beta}=L_\beta(D_\alpha)/L_\beta(S)\), the log-error decomposes exactly as \(T_\beta=U_\beta+V_\beta\). With observed richness, \(V_\beta\) is nonnegative whenever species are missed, but total error can have either sign. 2. We generated 950,100 without-replacement rarefactions from 3,167 Ecological Register inventories and combined them with fixed-seed multinomial simulations of 250 known communities spanning five species-abundance families. At \(\alpha=\beta=1\) and 25% retention (median expected coverage 0.963), \(\mathrm{Var}(U)+\mathrm{Var}(V)=0.01887\), 4.17 times \(\mathrm{Var}(T)=0.00452\). The full normalized estimator was more rank-stable than either one-channel counterfactual (0.906 versus 0.804 and 0.851), while within-inventory Spearman channel dependence was exactly \(\beta\)-invariant. In known-truth simulations, channel dependence weakened with coverage in all 75 SAD×\(\alpha\)×\(\beta\) combinations tested over coverage 0.80–0.95. 3. In Shannon/Pielou evenness, Chao–Wang–Jost entropy estimation materially improved joint correction. Chao1+CWJ had median raw-scale MSE ratios 0.96, 0.86, and 0.88 relative to the traditional estimator at coverages 0.80, 0.90, and 0.95; with true richness supplied as an oracle, the ratios were 0.49, 0.36, and 0.30. A paired log-MSE decomposition showed that the joint–oracle gap was mainly bias at coverage 0.80 but increasingly variance at 0.90 and 0.95. Richness error therefore remains, on median, the principal residual bottleneck, while negative covariance can conceal much of its marginal variance. 4. A coverage-standardized estimand changed the error geometry rather than merely shrinking error. The richness channel became two-sided, with negative values in about half of admissible replicates and median community mean near zero, while negative covariance persisted. Population evenness and evenness at standardized completeness should therefore be treated as distinct estimands. A supplementary proposition gives the eventual high-coverage decoupling limit under multinomial sampling.