Statistical & Clinical Analysis
Rao Quadratic Entropy Q Formula: Functional & Phylogenetic Trait Diversity Index
Incorporate pairwise species trait or genetic dissimilarity d_ij into biodiversity measurement using C. R. Rao’s (1982) Quadratic Entropy Q = SUM_i SUM_j d_ij * p_i * p_j and Jost’s Effective Functional Species 1 / (1 - Q).
Why three warbler twins have FAR less functional diversity than an eagle, a heron, and a hummingbird!
Shannon and Simpson diversity treat all distinct species names as 100% equidistant (d_ij = 1 for all i != j) — rating three functionally redundant grass species identically to an oak tree, a nitrogen-fixing legume, and a fern! C. Radhakrishna Rao (1982, Theor. Popul. Biol.) generalized Simpson’s Gini-impurity by weighting every pair of randomly drawn individuals (p_i, p_j) by their symmetric pairwise functional or phylogenetic distance d_ij in [0, 1] (with d_ii = 0): Q = SUM_{i=1..S} SUM_{j=1..S} d_ij * p_i * p_j!
Notice two beautiful special cases: when all distinct species are maximally dissimilar (d_ij = 1 for i != j), Rao’s Q reduces identically to Gini-Simpson diversity 1 - SUM(p_i^2); and when d_ij is scaled in [0, 1], Lou Jost / Ricotta & Szeidl’s Numbers-Equivalent transformation N_eff,Q = 1 / (1 - Q) yields the effective number of maximally distinct (d_ij = 1) equiprobable species!
| 3-Species Assemblage (Shares p = [0.50, 0.30, 0.20]) | Pairwise Distances [d_12, d_13, d_23] | Rao Quadratic Entropy Q | Effective Distinct Species 1 / (1 - Q) |
|---|---|---|---|
| Trait-Clustered Guild | [0.20, 0.80, 0.70] | 0.3040 | 1.4368 |
| Maximally Distinct Taxa (All d_ij = 1 -> Gini-Simpson) | [1.00, 1.00, 1.00] | 0.6200 | 2.6316 |
| Functionally Identical Clones (All d_ij = 0) | [0.00, 0.00, 0.00] | 0.0000 | 1.0000 |
Rao Quadratic Entropy Q = 2 * (d_12*p_1*p_2 + d_13*p_1*p_3 + d_23*p_2*p_3)
Exploits zero diagonal d_ii = 0 and symmetry d_ij = d_ji across species counts B2:B4 and off-diagonal [0, 1] trait dissimilarities D2:D4.
=IF(OR(MIN(B2:B4) < 0, SUM(B2:B4) <= 0, MIN(D2:D4) < 0, MAX(D2:D4) > 1), "Requires Counts >= 0, Sum > 0, and Normalized Distances d_ij in [0, 1]", ROUND(2 * (D2 * (B2 / SUM(B2:B4)) * (B3 / SUM(B2:B4)) + D3 * (B2 / SUM(B2:B4)) * (B4 / SUM(B2:B4)) + D4 * (B3 / SUM(B2:B4)) * (B4 / SUM(B2:B4))), 4))Numbers-Equivalent Functional Diversity 1 / (1 - Q)
Transforms normalized Rao Q in [0, 1) into effective number of maximally distinct equiprobable functional types.
=IF(OR(MIN(B2:B4) < 0, SUM(B2:B4) <= 0, MIN(D2:D4) < 0, MAX(D2:D4) > 1), "Requires Counts >= 0, Sum > 0, and Normalized Distances d_ij in [0, 1]", ROUND(1 / (1 - 2 * (D2 * (B2 / SUM(B2:B4)) * (B3 / SUM(B2:B4)) + D3 * (B2 / SUM(B2:B4)) * (B4 / SUM(B2:B4)) + D4 * (B3 / SUM(B2:B4)) * (B4 / SUM(B2:B4)))), 4))Try the example
Use the original values or change them to test your understanding.