Hill Numbers Calculators

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Hill numbers (also called effective number of species or true diversity) are a mathematically unified family of diversity measures parameterized by an order q, which determines the weight given to common vs. rare species. Proposed by Mark Hill in 1973, they express diversity as the number of equally-common species that would produce the same diversity value as the observed community. At q = 0, Hill number = species richness (S). At q = 1, Hill number = e^Shannon (exponential of Shannon-Wiener entropy). At q = 2, Hill number = 1/Simpson's D. Hill numbers are increasingly recommended over raw Shannon H' and Simpson's D because they are expressed in biologically intuitive units (effective number of species) and respond linearly to diversity changes.

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Hill Number Formula

⁰D = S (species richness, q=0)

¹D = exp(H') = exp(−Σpᵢ ln pᵢ) (Shannon effective diversity, q=1)

²D = 1/D = 1/Σpᵢ² (Simpson's reciprocal, q=2)

General form: qD = (Σpᵢ^q)^(1/(1−q)) for q ≠ 0, 1

Interpretation: qD is the number of equally abundant species that would give the same diversity value at order q.

Effect of Order q

  • q = 0: All species count equally regardless of abundance — pure species richness; most sensitive to rare species
  • q = 1: Species weighted by their proportional abundance — balances richness and evenness
  • q = 2: Species weighted by their abundance squared — most sensitive to dominant species; least sensitive to rare species
  • q → ∞: Only the most dominant species matters

Why Hill Numbers Are Preferred

Traditional diversity indices (H', D) have inconsistent units and don't respond proportionally to diversity changes. Hill numbers are: expressed as effective number of species (intuitive); doubly partitioned (alpha × beta = gamma holds exactly); sensitive to q (can profile the diversity structure across q values); and linear — doubling the number of equally common species doubles qD.

Diversity Profiles

Plotting qD vs. q for two communities reveals which is more diverse at different points of the species-abundance distribution — a more complete diversity comparison than any single index.

Glossary

Hill Numbers (qD)
A family of diversity measures parameterized by order q: qD = (Σpᵢ^q)^(1/(1−q)); q=0 = species richness; q=1 = exp(Shannon H'); q=2 = 1/Simpson D; expressed as effective number of species.
Diversity Order (q)
The parameter controlling sensitivity of Hill numbers to rare vs. common species: q=0 weights all species equally (most sensitive to rarities); q=2 emphasizes dominants; intermediate q balances both.
Effective Number of Species
The number of equally abundant species that would produce the same diversity value as the observed community; the unit of Hill numbers; more intuitive and comparable than raw entropy-based indices.

Frequently Asked Questions

Hill numbers (true diversity, effective number of species) are a family of diversity measures: qD = (Σpᵢ^q)^(1/(1−q)), where q controls sensitivity to rare vs. common species. At q=0: D = species richness. At q=1: D = e^Shannon-H'. At q=2: D = 1/Simpson-D. They are expressed as 'effective number of equally common species,' making them intuitively interpretable and directly comparable. They are preferred over raw H' and D because their units are consistent across the q spectrum and they respond linearly to community changes.

At q=0, all species count equally — Hill number equals species richness and is most sensitive to rare species. As q increases, more weight is given to abundant species. At q=1, species are weighted by their proportional abundance — balancing richness and evenness. At q=2, species are weighted by their squared abundance — most sensitive to dominant species and least affected by rare species. Plotting qD across q values from 0 to infinity creates a diversity profile that reveals how community structure changes with respect to rare vs. dominant species.

The q=1 Hill number = exp(H'), where H' is the Shannon-Wiener entropy. While H' (in nats or bits) lacks clear biological units, exp(H') is the effective number of equally common species that would produce the same entropy. Example: H' = 2.3 → exp(2.3) = 10 — the community has the same diversity as 10 perfectly even species. Converting H' to its Hill number makes it directly comparable to species richness and Simpson's reciprocal diversity without unit inconsistency.

A key advantage of Hill numbers is that gamma diversity = alpha diversity × beta diversity holds exactly: γD = αD × βD. This multiplicative partitioning is not possible with additive indices like H'. Beta diversity in Hill number terms: β = γD/αD, ranging from 1 (identical communities) to N (number of sites, when communities share no species). This clear partitioning facilitates comparing diversity components across studies and spatial scales. The iNEXT R package and the vegan package support Hill number calculations and diversity profiles.