Ecology, 100(12), 2019, e02852 © 2019 by the Ecological Society of America Quantifying evenness and linking it to diversity, beta diversity, and similarity 1,3 2 ANNE CHAO AND CARLO RICOTTA 1Institute of Statistics, National Tsing Hua University, Hsin-Chu 30043 Taiwan 2Department of Environmental Biology, University of Rome, “La Sapienza,” Piazzale Aldo Moro 5, Rome 0185 Italy Citation: Chao, A., and C. Ricotta. 2019. Quantifying evenness and linking it to diversity, beta diversity, and similarity. Ecology 100(12):e02852. 10.1002/ecy.2852 Abstract. An enormous number of measures based on different criteria have been proposed to quantify evenness or unevenness among species relative abundances in an assem- blage. 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 pro- pose 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 vec- tor, where the mean vector represents the species relative abundances of a completely even assemblage. Thus, evenness among species relative abundances is measured by the correspond- ing normalized extent of closeness between these two vectors. We consider five divergence mea- sures, 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 abun- dances) 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 Sørensen-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 spe- cies’ abundances among assemblages. The contribution of a species to each dissimilarity mea- sure 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 phyloge- netic version. A real data example is used to illustrate our approach. We also discuss some cri- teria and other measures that were previously proposed in the literature. Key words: beta diversity; dissimilarity; divergence; diversity; evenness; Hill numbers; phylogenetic diversity; principle of transfer; similarity; unevenness. INTRODUCTION exponent of Shannon entropy), and Simpson diversity (q = 2, i.e., the inverse of the Simpson concentration Until fairly recently, how to quantify biodiversity was index); see Hill (1973) and Chao et al. (2014) for one of the most controversial issues in ecology (Magur- reviews. ran and McGill 2011). Surprisingly, since an Ecology Compared to diversity, quantifying evenness or forum (Ellison 2010 and the papers following it), a con- unevenness among species relative abundances is an even sensus seems to have emerged about the use of Hill num- more complicated and extensively discussed issue (Pielou bers, that is, the effective number of species (Hill 1973), 1966, Alatalo 1981, Routledge 1983, Smith and Wilson as the diversity measure of choice for quantifying species 1996, Gosselin 2001, 2006, Ricotta 2003, Olszewski diversity and decomposition. Hill numbers, parameter- 2004, Jost 2010, Tuomisto 2012, Kvalseth 2015, among ized by a diversity order q (where q controls the weights others). A number of authors (Routledge 1983, Smith of species relative abundances), encompass the three and Wilson 1996, Kvalseth 2015) discussed necessary most useful diversity measures: species richness (diver- properties for an evenness measure. Researchers from = = sity of order q 0), Shannon diversity (q 1, i.e., the different perspectives and under different criteria have developed a wider range of (un)evenness measures than diversity indices. The list of evenness measures seems Manuscript received 3 January 2019; revised 13 June 2019; accepted 8 July 2019. Corresponding Editor: Tom E. X. Miller. endless, reflecting that our concept of evenness is not 3 E-mail: [email protected] very clear and also that disagreements exist about the Article e02852; page 1 Article e02852; page 2 ANNE CHAO AND CARLO RICOTTA Ecology, Vol. 100, No. 12 requirements an evenness measure should satisfy. profiles and assessing the contribution of each species/ Indeed, before the possibility of similarly reaching a con- node to taxonomic and phylogenetic dissimilarity sensus can be considered to be more than a distant goal, measures. it is necessary to formulate a unified methodology/ approach that can encompass most useful evenness BASIC REQUIREMENTS FOR AN EVENNESS MEASURE measures. In this paper, we propose an intuitive and unified The main reason for the development of the broad mathematical framework to obtain evenness measures. spectrum of evenness measures seen today is that differ- In our framework, unevenness among species relative ent researchers require different criteria. Our proposed abundances is measured by a normalized divergence requirements for an evenness measure are systematically between the vector of species relative abundances and given separately for three cases: (1) Species richness is the mean vector, where the mean vector represents the fixed but abundance-based diversity varies. (2) Species species relative abundances of a completely even assem- richness varies but abundance-based diversity is almost blage with the same number of species. Thus, evenness fixed. (3) Species richness and/or abundance-based among species relative abundances is measured by the diversity are varying. corresponding normalized extent of closeness between these two vectors. Basic requirements for an evenness measure when species Here, we first present our perspective on some basic richness is fixed requirements for an evenness measure. Then we con- sider five divergence measures, leading to five classes Our proposed criteria when species richness is fixed of evenness indices. All divergence and evenness mea- are the same as those for the concept of diversity sures are in terms of (relative) abundance-based Hill because diversity is based on the evenness principle. numbers (i.e., q > 0) and species richness (q = 0), That is, when species richness is fixed, the more even revealing explicitly the relation between evenness and the species abundances, the higher the diversity. Here, diversity. Our measures possess all of the basic require- we first present the mathematical formulas of species ments presented later in this paper and encompass diversity, which are based on species relative abun- many previous evenness indices as special cases. When dances. Assuming there are S species in an assemblage multiple assemblages are considered, we apply our with species relative abundance vector p ¼ðp1; p2; measures to evaluate each species’ compositional even- ...; pSÞ, taxonomic/species diversity (Hill number) of ness, that is, evenness in this single species’ relative order q is defined as abundances, among assemblages, calculated for each ! 1=ð1ÀqÞ species, individually. Then the abundance-based Jac- XS q q ; ; card- and Sørensen-type dissimilarity measures (which D ¼ pi q 0 q 6¼ 1. (1a) are monotonic functions of beta diversity) can be i¼1 expressed as weighted averages of the individual spe- = ’ For q 1, we have cies compositional unevenness values. Thus, our ! framework links among the concepts of diversity, even- XS 1 q ness, beta diversity and similarity. The contribution of D ¼ lim D ¼ exp À pi log pi . (1b) q!1 a species to the Jaccard- and Sørensen-type dissimilar- i¼1 ity measures can be explicitly disentangled and quanti- p p fied in terms of this single species’ compositional When species richness S is fixed, evenness of , E( ), unevenness among multiple assemblage. should possess the same properties as an abundance- In traditional taxonomic diversity analysis, all species based diversity measure (Taillie 1979). We require two are considered to be equally distinct from one another. unequivocal shared properties: However, in an evolutionary context, species differences can be based directly on their evolutionary relationships Requirement 1a—When species richness is fixed, the in the form of phylogenetic trees. Species that are closely fundamental and most essential property required related are generally less distinct in important ecological for an evenness measure is the principle of transfer, characteristics than are distantly related species. A vast which states that when an amount of abundance is number of phylogenetic diversity metrics and related transferred from a relatively more abundant species (dis)similarity measures have been proposed; see Chao to a relatively less abundant species, evenness et al. (2014) for a review. When all species (as tip nodes) should increase. Specifically, assume that species i is in a focal assemblage are connected by a rooted ultra- more abundant than species j