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.

We propose a parametric class of phylogenetic diversity (PD) measures that are sensitive to both species abundance and species taxonomic or phylogenetic distances. This work extends the conventional parametric species-neutral approach (based on 'effective number of species' or Hill numbers) to take into account species relatedness, and also generalizes the traditional phylogenetic approach (based on 'total phylogenetic length') to incorporate species abundances. The proposed measure quantifies 'the mean effective number of species' over any time interval of interest, or the 'effective number of maximally distinct lineages' over that time interval. The product of the measure and the interval length quantifies the 'branch diversity' of the phylogenetic tree during that interval. The new measures generalize and unify many existing measures and lead to a natural definition of taxonomic diversity as a special case. The replication principle (or doubling property), an important requirement for species-neutral diversity, is generalized to PD. The widely used Rao's quadratic entropy and the phylogenetic entropy do not satisfy this essential property, but a simple transformation converts each to our measures, which do satisfy the property. The proposed approach is applied to forest data for interpreting the effects of thinning.

There have been intense debates about the decomposition of regional diversity (gamma) into its within-community component (alpha) and between-community component (beta). Although a recent Ecology Forum achieved consensus in the use of "numbers equivalents" (Hill numbers) as the proper choice of diversity measure, three related major issues were still left unresolved. (1) What is the precise meaning of the "independence" or "statistical independence" of alpha diversity and beta diversity? (2) Which partitioning (additive vs. multiplicative) should be used for a given application? (3) What is the proper formula for alpha diversity, as there are two formulas in the literature? This paper proposes a possible resolution to each of these issues. For the first issue, we clarify the definitions of "independence" and "statistical independence" from two perspectives so that confusion about this issue can be cleared up. We also discuss the causes of dependence, so that the dependence relationship between any two diversity components in both partitioning schemes can be rigorously justified by theory and also intuitively understood by simulation. For the second issue, both multiplicative and additive beta diversities based on Hill numbers are useful measures and quantify different aspects of communities. However, neither can be directly applied to compare relative compositional similarity or differentiation across multiple regions with different numbers of communities because multiplicative beta diversity depends on the number of communities, and additive beta diversity additionally depends on alpha (equivalently, on gamma). Such dependences should be removed. We propose transformations to remove these dependences, and we show that the transformed multiplicative beta and additive beta both lead to the same classes of measures, which are always in a range of [0, 1] and thus can be used to compare relative similarity or differentiation among communities across multiple regions. These similarity measures include multiple-community generalizations of the Sørenson, Jaccard, Horn, and Morisita-Horn measures. For the third issue, we present some observations including a finding about which alpha formula produces independent alpha and beta components. These may help to resolve the choice of a proper formula for alpha diversity. Some related issues are also briefly discussed.

On the basis of the sampling data from an assemblage, estimation of species richness (observed plus undetected) is statistically difficult especially for highly diverse assemblages with many rare species. Simple counts of species richness in samples typically underestimate and strongly depend on sampling effort and sample completeness. There are two approaches to infer species richness and make fair comparisons among multiple assemblages based on possibly unequal sampling effort and incomplete samples that miss many species. (i) An asymptotic approach: this approach compares the estimated asymptotes of species accumulation curves. It is based on statistical sampling‐theory methods of estimating species richness. Both parametric and nonparametric methods are reviewed. We focus on the nonparametric estimators that are universally valid for all species abundance distributions. (ii) A nonasymptotic approach: this approach compares the estimated species richnesses of standardized samples with a common finite sample size or sample completeness. It is based on the seamless sample‐size‐ and coverage‐based rarefaction and extrapolation sampling curves. This approach aims to compare species richness estimates for equally large or equally complete samples. These two approaches allow researchers to efficiently use all data to make robust and detailed inferences about species richness. Two R packages (SpadeR and iNEXT) are applied to rainforest tree data for illustration.