Hill numbers Individual-based rarefaction ENS rarefaction
Symbol \({}^{q}D\) \(S_{n}\) \(E_{n}\)
Formula \[\ {}^{q}D=\left(\sum_{i=1}^{S}{p_{i}}^{q}\right)^{\frac{1}{1-q}}\] Eqn 1 \[S_{n}=S-\sum_{X_{i}\geq 1}\frac{\begin{pmatrix}N-X_{i}\\ n\\ \end{pmatrix}}{\begin{pmatrix}N\\ n\\ \end{pmatrix}}\] Eqn 2 \[S_{n}=\ E_{n}\left(1-\left(1-\frac{1}{E_{n}}\right)^{n}\right)\] Eqn 3
Range 1,N 1,n 1,∞
ENS Yes No Yes
Estimation bias downward bias for q<2 Unbiased Unbiased
Description ENS transformation (“true diversity”) of any diversity index that is a function of \(\sum_{i=1}^{S}{}{p_{i}}^{q}\) (e.g. Richness (q=0), Shannon (q=1), Simpson (q=2)); Defined as the species richness of a hypothetical perfectly even community that has the same diversity index value as the sample. The expected species richness of a sample of n individuals (n<N). ENS transformation of \(S_{n}\). Defined as the species richness of a hypothetical community that has the same rarefied richness (\(S_{n}\)) as the sample and infinitely many individuals.
Influence of relative abundances The higher q, the lower the influence of rare species. The higher n, the higher the influence of rare species. The higher n, the higher the influence of rare species.
References Hill, 1973; Jost, 2006 Hurlbert,1971; Gotelli and Colwell, 2001 Dauby and Hardy, 2012