Hill Numbers and Their Phylogenetic Generalizations

Hill Numbers and the Replication Principle

Pioneering work by Kimura and Crow (1964) in genetics and MacArthur (1965) in ecology showed that the Shannon and Gini-Simpson measures can be easily converted to “effective number of species” (i.e., the number of equally abundant species that are needed to give the same value of the diversity measure), which use the same units as species richness. Shannon entropy can be converted by taking its exponential, and the Gini-Simpson index can be converted by the formula 1/(1−HGS). Hill (1973) integrated species richness and the converted Shannon and Gini-Simpson measures into a class of diversity measures called “Hill numbers” of order q, or the “effective number of species”, defined as

This measure is undefined for q = 1, but its limit as q tends to 1 exists and gives

The relationship between Hill number of order q (q ≠ 1) and the generalized entropy can be expressed as

When q = 0, the species abundances do not count at all and 0D = S is obtained. When q = 1, the species are weighed in proportion to their frequencies, and the measure 1D (in Eq. (3b)) can be interpreted as the effective number of common or “typical” species (i.e., species with typical abundances) in the assemblage. When q = 2, abundant species are favored and rare species are discounted; the measure 2D becomes the inverse Simpson concentration. The measure 2D can be interpreted as the effective number of dominant or very abundant species in the assemblage. In general, if qD = x, then the diversity of order q of this community is the same as that of an idealized reference community with x equally abundant species. All Hill numbers are in units of “species”. It is thus possible to plot them on a single graph as a continuous function of the parameter q. This diversity profile characterizes the species-abundance distribution of an assemblage and provides complete information about its diversity. The steepness of its slope graphically illustrates the degree of dominance in the assemblage. An example is given in section “An example”.

Hill numbers differ fundamentally from Shannon entropy and the Gini-Simpson index in that they obey the replication principle. Hill (1973) proved a weak version of the doubling property: if two completely distinct assemblages (i.e., no species in common) have identical relative abundance distributions, then the Hill number doubles if the assemblages are combined with equal weights. Chiu et al. (2014, their Appendix B) recently proved a strong version of the doubling property: if two completely distinct assemblages have identical Hill numbers of order q (relative abundance distributions may be different, unlike the weak version), then the Hill number of the same order doubles if the two assemblages are combined with equal weights. Species richness is a Hill number (with q = 0) and obeys both versions of the doubling property, but most other diversity indices do not obey even the weak version. Because Hill numbers obey this replication principle, changes in their magnitude have simple interpretations, and the ratio of alpha diversity to gamma diversity accurately reflects the compositional similarity of the communities. The replication principle is best known in economics, where it has long been recognized as an important property of concentration and diversity measures (Hannah and Kay 1977). In ecology, the doubling property has been extensively discussed by many authors (MacArthur 1965, 1972; Hill 1973; Whittaker 1972; Routledge 1979; Peet 1974; Jost 2006, 2007, 2009; Ricotta and Szeidl 2009; Jost et al. 2010) and has been extended to phylogenetic measures (Chao et al. 2010); see below.