Quantifying evenness and linking it to diversity, beta diversity, and similarity

An enormous number of measures based on different criteria have been proposed to quantify evenness or unevenness among species relative abundances in an assemblage. However, a unified approach that can encompass most of the widely used indices is still lacking. Here, we first present some basic requirements for an evenness measure. We then propose that unevenness among species relative abundances in an assemblage can be measured by a normalized divergence between the vector of species relative abundances and the mean vector, where the mean vector represents the species relative abundances of a completely even assemblage. Thus, evenness among species relative abundances is measured by the corresponding normalized extent of closeness between these two vectors. We consider five divergence measures, leading to five classes of evenness indices. All our evenness measures are in terms of diversity (Hill number) of order q > 0 (here q controls the weighting of species relative abundances) and species richness (diversity of order q = 0). We propose quantifying evenness through a continuous profile that depicts evenness as a function of diversity order q > 0. The profiles can be easily and visually compared across multiple assemblages. Our evenness indices satisfy all the requirements presented in this paper and encompass many widely used evenness measures as special cases. When there are multiple assemblages, the abundance-based Jaccard- and Sorensen-type dissimilarity measures (which are monotonic functions of beta diversity) can be expressed as weighted averages of the individual species' compositional unevenness values; here, each individual species' compositional unevenness is calculated based on that species' abundances among assemblages. The contribution of a species to each dissimilarity measure can be clearly disentangled and quantified in terms of this single species' compositional unevenness among assemblages. Thus, our framework links the concepts of evenness, diversity, beta diversity, and similarity. Moreover, the framework can be readily extended to a phylogenetic version. A real data example is used to illustrate our approach. We also discuss some criteria and other measures that were previously proposed in the literature.