Entropies such as the Shannon–Wiener and Gini–Simpson indices are not themselves diversities. Conversion of these to effective number of species is the key to a unified and intuitive interpretation of diversity. Effective numbers of species derived from standard diversity indices share a common set of intuitive mathematical properties and behave as one would expect of a diversity, while raw indices do not. Contrary to Keylock, the lack of concavity of effective numbers of species is irrelevant as long as they are used as transformations of concave alpha, beta, and gamma entropies. The practical importance of this transformation is demonstrated by applying it to a popular community similarity measure based on raw diversity indices or entropies. The standard similarity measure based on untransformed indices is shown to give misleading results, but transforming the indices or entropies to effective numbers of species produces a stable, easily interpreted, sensitive general similarity measure. General overlap measures derived from this transformed similarity measure yield the Jaccard index, Sørensen index, Horn index of overlap, and the Morisita–Horn index as special cases.

Existing general definitions of beta diversity often produce a beta with a hidden dependence on alpha. Such a beta cannot be used to compare regions that differ in alpha diversity. To avoid misinterpretation, existing definitions of alpha and beta must be replaced by a definition that partitions diversity into independent alpha and beta components. Such a unique definition is derived here. When these new alpha and beta components are transformed into their numbers equivalents (effective numbers of elements), Whittaker's multiplicative law (alpha x beta = gamma) is necessarily true for all indices. The new beta gives the effective number of distinct communities. The most popular similarity and overlap measures of ecology (Jaccard, Sorensen, Horn, and Morisita-Horn indices) are monotonic transformations of the new beta diversity. Shannon measures follow deductively from this formalism and do not need to be borrowed from information theory; they are shown to be the only standard diversity measures which can be decomposed into meaningful independent alpha and beta components when community weights are unequal.

G(ST) and its relatives are often interpreted as measures of differentiation between subpopulations, with values near zero supposedly indicating low differentiation. However, G(ST) necessarily approaches zero when gene diversity is high, even if subpopulations are completely differentiated, and it is not monotonic with increasing differentiation. Likewise, when diversity is equated with heterozygosity, standard similarity measures formed by taking the ratio of mean within-subpopulation diversity to total diversity necessarily approach unity when diversity is high, even if the subpopulations are completely dissimilar (no shared alleles). None of these measures can be interpreted as measures of differentiation or similarity. The derivations of these measures contain two subtle misconceptions which cause their paradoxical behaviours. Conclusions about population differentiation, gene flow, relatedness, and conservation priority will often be wrong when based on these fixation indices or similarity measures. These are not statistical issues; the problems persist even when true population frequencies are used in the calculations. Recent advances in the mathematics of diversity identify the misconceptions, and yield mathematically consistent descriptive measures of population structure which eliminate the paradoxes produced by standard measures. These measures can be directly related to the migration and mutation rates of the finite-island model.

Abstract. We propose an integrated sampling, rarefaction, and extrapolation methodology to compare species richness of a set of communities based on samples of equal completeness (as measured by sample coverage) instead of equal size. Traditional rarefaction or extrapolation to equal-sized samples can misrepresent the relationships between the richnesses of the communities being compared because a sample of a given size may be sufficient to fully characterize the lower diversity community, but insufficient to characterize the richer community. Thus, the traditional method systematically biases the degree of differences between community richnesses. We derived a new analytic method for seamless coverage-based rarefaction and extrapolation. We show that this method yields less biased comparisons of richness between communities, and manages this with less total sampling effort. When this approach is integrated with an adaptive coverage-based stopping rule during sampling, samples may be compared directly without rarefaction, so no extra data is taken and none is thrown away. Even if this stopping rule is not used during data collection, coveragebased rarefaction throws away less data than traditional size-based rarefaction, and more efficiently finds the correct ranking of communities according to their true richnesses. Several hypothetical and real examples demonstrate these advantages.

Hill numbers or the effective number of species are increasingly used to quantify species diversity of an assemblage. Hill numbers were recently extended to phylogenetic diversity, which incorporates species evolutionary history, as well as to functional diversity, which considers the differences among species traits. We review these extensions and integrate them into a framework of attribute diversity (the effective number of entities or total attribute value) based on Hill numbers of taxonomic entities (species), phylogenetic entities (branches of unit-length), or functional entities (species-pairs with unit-distance between species). This framework unifies ecologists' measures of species diversity, phylogenetic diversity, and distance-based functional diversity. It also provides a unified method of decomposing these diversities and constructing normalized taxonomic, phylogenetic, and functional similarity and differentiation measures, including N-assemblage phylogenetic or functional generalizations of the classic Jaccard, Sørensen, Horn, and Morisita-Horn indexes. A real example shows how this framework extracts ecological meaning from complex data.