Distance-Based Functional Diversity Measures and Their Decomposition: A Framework Based on Hill Numbers

Hill numbers (or the “effective number of species”) are increasingly used to characterize species diversity of an assemblage. This work extends Hill numbers to incorporate species pairwise functional distances calculated from species traits. We derive a parametric class of functional Hill numbers, which quantify “the effective number of equally abundant and (functionally) equally distinct species” in an assemblage. We also propose a class of mean functional diversity (per species), which quantifies the effective sum of functional distances between a fixed species to all other species. The product of the functional Hill number and the mean functional diversity thus quantifies the (total) functional diversity, i.e., the effective total distance between species of the assemblage. The three measures (functional Hill numbers, mean functional diversity and total functional diversity) quantify different aspects of species trait space, and all are based on species abundance and species pairwise functional distances. When all species are equally distinct, our functional Hill numbers reduce to ordinary Hill numbers. When species abundances are not considered or species are equally abundant, our total functional diversity reduces to the sum of all pairwise distances between species of an assemblage. The functional Hill numbers and the mean functional diversity both satisfy a replication principle, implying the total functional diversity satisfies a quadratic replication principle. When there are multiple assemblages defined by the investigator, each of the three measures of the pooled assemblage (gamma) can be multiplicatively decomposed into alpha and beta components, and the two components are independent. The resulting beta component measures pure functional differentiation among assemblages and can be further transformed to obtain several classes of normalized functional similarity (or differentiation) measures, including N-assemblage functional generalizations of the classic Jaccard, Sørensen, Horn and Morisita-Horn similarity indices. The proposed measures are applied to artificial and real data for illustration.     

Sampled‐based estimation of diversity sensu stricto by transforming Hurlbert diversities into effective number of species

A theoretical framework based on Hill numbers has recently been advocated to measure and partition diversity sensu stricto. Hill numbers can be interpreted intuitively as effective number of species (ENS). They conform to the so‐called replication principle allowing a mathematically coherent multiplicative partitioning of diversity. They form a family of ENS defined by the parameter q which controls the weight attributed to rare species. Despite its advantages, this framework was developed without considering its robustness when treating community samples. In this study, we first show that Hurlbert diversity indices (expected number of species among k individuals) can be transformed into ENS that conform asymptotically to the replication principle while controlling the weight given to rare species through parameter k. We investigate the statistical properties of Hill and Hurlbert ENS using simulated communities with contrasted diversity. The properties of multiplicative beta diversity estimators based on ENS are also characterized by simulating communities with different levels of differentiation. We show that Hurlbert ENS provides a better statistical performance than Hill numbers when dealing with small sample sizes. By contrast, Hill numbers and their estimators suffer from substantial bias except when rare species have a low weight (q= 2). An estimator of ENS estimating both Hill numbers for q= 2 and Hurlbert ENS for k= 2 is shown to give the best performance and is recommended for processing real datasets when rare species receive low weight. In order to better take account of rare species, current estimators of Hill numbers are not recommended when sample size is too low while Hurlbert’s ENS performs reliably. In conclusion, while Hill numbers possess some interesting mathematical properties that are not shared by Hurlbert’s ENS, the latter outperforms Hill numbers in terms of statistical properties and is well suited to processing community samples, as illustrated on a real dataset.     

Partitioning diversity for conservation analyses

Aim  Differentiation of sites or communities is often measured by partitioning regional or gamma diversity into additive or multiplicative alpha and beta components. The beta component and the ratio of within-group to total diversity (alpha/gamma) are then used to infer the compositional differentiation or similarity of the sites. There is debate about the appropriate measures and partitioning formulas for this purpose. We test the main partitioning methods, using empirical and simulated data, to see if some of these methods lead to false conclusions, and we show how to resolve the problems that we uncover.
Location  South America, Ecuador, Orellana province, Rio Shiripuno.
Methods  We construct sets of real and simulated tropical butterfly communities that can be unambiguously ranked according to their degree of differentiation. We then test whether beta and similarity measures from the different partitioning approaches rank these datasets correctly.
Results  The ratio of within-group diversity to total diversity does not reflect compositional similarity, when the Gini–Simpson index or Shannon entropy are used to measure diversity. Additive beta diversity based on the Gini–Simpson index does not reflect the degree of differentiation between N sites or communities.
Main conclusions  The ratio of within-group to total diversity (alpha/gamma) should not be used to measure the compositional similarity of groups, if diversity is equated with Shannon entropy or the Gini–Simpson index. Conversion of these measures to effective number of species solves these problems. Additive Gini–Simpson beta diversity does not directly reflect the differentiation of N samples or communities. However, when properly transformed onto the unit interval so as to remove the dependence on alpha and N , additive and multiplicative beta measures yield identical normalized measures of relative similarity and differentiation.     

The additive partitioning of species diversity: recent revival of an old idea

Ecologists have traditionally viewed the total species diversity within a set of communities as the product of the average diversity within a community (alpha) and the diversity among the communities (beta). This multiplicative concept of species diversity contrasts with the lesser known idea that α- and β-diversities sum to give the total diversity. This additive partitioning of species diversity is nearly as old as the multiplicative concept, yet ecologists are just now beginning to use additive partitioning to examine patterns of species diversity. In this review we discuss why additive partitioning remained "hidden" until just a few years ago. The rediscovery of additive partitioning has expanded the way in which ecologists define and measure β-diversity. Beta diversity is no longer relegated to describing change only along an environmental gradient. Through additive partitioning, β-diversity is explicitly an average amount of diversity just as is α-diversity. We believe that the additive partitioning of diversity into α and β components will continue to become more widely used because it allows for a direct comparison of α- and β-diversities. It also has particular relevance for testing ecological theory concerned with the determinants of species diversity at multiple spatial scales and potential applications in conservation biology.     

The partitioning of diversity: showing Theseus a way out of the labyrinth

A methodology for partitioning of biodiversity into α, β and γ components has long been debated, resulting in different mathematical frameworks. Recently, use of the Rao quadratic entropy index has been advocated since it allows comparison of various facets of diversity (e.g. taxonomic, phylogenetic and functional) within the same mathematical framework. However, if not well implemented, the Rao index can easily yield biologically meaningless results and lead into a mathematical labyrinth. As a practical guideline for ecologists, we present a critical synthesis of diverging implementations of the index in the recent literature and a new extension of the index for measuring β‐diversity. First, we detail correct computation of the index that needs to be applied in order not to obtain negative β‐diversity values, which are ecologically unacceptable, and elucidate the main approaches to calculate the Rao quadratic entropy at different spatial scales. Then, we emphasize that, similar to other entropy measures, the Rao index often produces lower‐than‐expected β‐diversity values. To solve this, we extend a correction based on equivalent numbers, as proposed by Jost (2007), to the Rao index. We further show that this correction can be applied to additive partitioning of diversity and not only its multiplicative form. These developments around the Rao index open up an exciting avenue to develop an estimator of turnover diversity across different environmental and temporal scales, allowing meaningful comparisons of partitioning across species, phylogenetic and functional diversities within the same mathematical framework. We also propose a set of R functions, based on existing developments, which perform different key computations to apply this framework in biodiversity science.     

Computing parametric beta diversity with unequal plot weights: a solution based on resampling methods

Jost (Ecology, 88:2427–2439, 2007) recently showed that the Shannon diversity is the only standard diversity measure that can be partitioned into meaningful independent alpha and beta components when plot weights are unequal. This conclusion is very disappointing if one wants to calculate the beta diversity of unequal weighted plots using a parametric measure with varying sensitivities to the occurrence of rare and abundant species. To overcome this impasse, at least partially, in this paper, I propose a parametric measure of beta diversity that is based on the combination of Shannon’s entropy with Hurlbert’s ‘expected species diversity’. Unlike most parametric measures of diversity, the proposed index has a clear probabilistic interpretation, allowing at the same time a multiplicative partition of diversity into independent alpha and beta components for unequally weighted plots.     

Diversity partitioning of Rao’s quadratic entropy

Many applications of diversity indices are only valid if they are first transformed into their equivalent number of species. These equivalent numbers of species can be multiplicatively partitioned into independent alpha, beta and gamma components, and can be formed into mathematically consistent similarity measures. The utility of beta diversity and similarity measures that incorporate information about the degree of ecological dissimilarity between species is becoming increasingly recognized. The concept of equivalent number of species is here extended to Rao’s quadratic entropy, opening the way to methods of diversity partitioning that take into account taxonomic or ecological differences between species.     

The Leinster and Cobbold indices improve inferences about microbial diversity

An increasing number of ecological studies compare the diversity of microbial taxa along environmental gradients or between imposed treatments. Estimates are often based on analysis-of-variance of taxon-richness inferred from pyrosequencing data. We conducted a reanalysis of three 454-pyrosequencing studies on arbuscular-mycorrhizal-fungal diversity to evaluate the suitability of using the Leinster and Cobbold diversity-indices (LCdis) to assess diversity. We expected that the potential of LCdis to consider phylogenic relationships could resolve problems arising from ambiguous species-delineation in microbial-systems. Our reanalysis showed that comparisons between studies differing considerably in sequencing depth may be risky. Moreover, we show that LCdis not only reproduce the results of analyses of variance but can also resolve issues connected to variation in sequence read number, while additionally representing a less conservative metric of diversity than analysis-of-variance of taxa-richness. Based on these results we advocate the use of inclusive diversity indices in ecological studies targeting microbial communities.     

Hierarchical partitioning of ant diversity: implications for conservation of biogeographical diversity in arid and semi‐arid areas

Aim The scale of observation is important in detecting the spatial variation of biological assemblages, which should be taken into consideration for an appropriate plan of biogeographical conservation. We investigated whether (1) World Wildlife Fund’s ecoregion units are the appropriate scale for conserving ant diversity in Iran, (2) each ecoregion represents a distinct ant community composition and (3) patterns of diversity partitioning differ among four ecoregions. Location Iran, a sampling transect along four arid and semi‐arid ecoregions. Methods We applied hierarchical partitioning to data collected from a nested sampling design including four hierarchical levels: ‘local’, ‘landscape’, ‘ecoregional’ and ‘whole‐region’. Observed alpha and beta diversity components were compared with values of null distributions. Hierarchical cluster analysis was applied to evaluate similarity of ant species composition among ecoregions. Results Partitioning of whole‐region species richness showed that 85% of the species richness was generated by beta diversity among ecoregions and landscapes. The highest value of diversity was generated by beta diversity among ecoregions. Unlike whole‐region partitioning, separate partitioning within each ecoregion revealed that beta component among localities contributed to species richness of each ecoregion. Ecoregions showed different patterns of diversity partitioning. The alpha component contributed largely to the total diversity of two ecoregions, but for two other ecoregions, beta component contributed more than alpha component. Cluster analysis identified four discrete ant species compositions; however, it split landscapes of one ecoregion into two distinct groups. Main conclusions Whole‐region diversity partitioning indicates that ecoregions represent the appropriate scale for conserving ant diversity and that each ecoregion has a distinct ant fauna. However, different conservation strategies should be considered for different ecoregions owing to the differing scales of variation within them. Boundaries of ecoregions remain a subject for further studies. The influence of climate change on ecoregional boundaries should be considered and should be predicted with respect to future conservation maps.     

A metric of biodiversity that integrates abundance,phylogeny, and function

A metric of biodiversity is proposed that combines three of its key components: abundance, phylogeny, and ecological function. This metric is an expansion of the current abundance‐based metric that uses Hill numbers, the effective number of types in a sample if all types had the same mean proportional abundance. I define analogous proportional measures of phylogenetic divergence and functional distinctiveness. Phylogenetic divergence is measured as the sum of the proportional share of each species of a given branch of a phylogeny. Functional distinctiveness can be measured in two ways, as the proportional share of each species of a specified ecological function, or as the relative distance of each species based on functional trait values. Because all three aspects of biodiversity are measured in the same fashion (relative proportions) in similar units (effective numbers of species), an integrated metric can be defined. The combined metric provides understanding of covariation among the components and how management for one component may trade off against others. The metric can be partitioned into components of richness and evenness, and into subsets and variation among subsets, all of which can be related through a simple multiplicative framework. This metric is a complement to, rather than a replacement of, current metrics of phylogenetic and functional diversity. More work is needed to link this new metric to ecological theory, determine its error structure, and devise methods for its effective assessment.     

美洲森林群落beta多样性的纬度梯度性

Beta多样性度量不同时空尺度物种组成的变化,是生物多样性的重要组成部分;理解其地理格局和形成机制已成为当前生物多样性研究的热点问题。基于Alwyn H. Gentry在美洲收集的131个森林样方数据,采用倍性和加性分配方法度量群落beta多样性,检验beta多样性随纬度的变化趋势,并分析其形成机制。研究表明:(1) 美洲森林群落beta多样性随纬度增加显著下降,热带和亚热带地区beta多样性高于温带地区;此格局可由物种分布范围的纬度梯度性和不同粒度(grain)下物种丰富度与纬度回归斜率的差异推论得出;(2) 加性分配方法表明beta多样性对各个温度带森林群落gamma多样性的相对贡献率平均为78.2%,并且随纬度升高而降低;(3) 美洲南半球森林群落beta多样性高于其北半球,这可能反映了区域间物种进化和环境变迁历史的差异。此外,还探讨了不同beta多样性计算方法的适用情景,首次证实了森林生态系统群落水平beta多样性的纬度梯度性,这对研究生物多样性的形成机制和生物多样性保护都具有重要的意义。     

On hierarchical diversity decomposition

Abstract. Question: According to Whittaker's proposal, ecologists have traditionally viewed β‐diversity as the ratio between γ‐diversity and average α‐diversity. More recently, an alternative way of partitioning diversity has been ‘rediscovered’for which β‐diversity is obtained as the difference between γ‐diversity and average α‐diversity. This additive way of partitioning diversity has rapidly become a very popular framework for hierarchical diversity decomposition at various spatial scales. The question for this study is: Can we highlight any relation between these two ways of partitioning diversity, or do these methods really capture different facets of spatial turnover in species composition? Methods: First the properties that a diversity measure should possess for enabling additive decomposition into α‐, β‐, and γ‐components are reviewed. Next, attention is drawn to the relationships between additive and multiplicative diversity decomposition. Results: It is shown that the additive model is closely related to its multiplicative counterpart through a simple logarithmic transformation. Conclusions: Contrary to the current assumption, both methods for partitioning diversity are not as different as they appear. Hence, the supposed superiority of additive diversity partition over multiplicative diversity decomposition is largely unjustified.     

Biodiversity assessment in incomplete inventories: leaf litter ant communities in several types of Bornean rain forest

Biodiversity assessment of tropical taxa is hampered by their tremendous richness, which leads to large numbers of singletons and incomplete inventories in survey studies. Species estimators can be used for assessment of alpha diversity, but calculation of beta diversity is hampered by pseudo-turnover of species in undersampled plots. To assess the impact of unseen species, we investigated different methods, including an unbiased estimator of Shannon beta diversity that was compared to biased calculations. We studied alpha and beta diversity of a diverse ground ant assemblage from the Southeast Asian island of Borneo in different types of tropical forest: diperocarp forest, alluvial forest, limestone forest and heath forests. Forests varied in plant composition, geology, flooding regimes and other environmental parameters. We tested whether forest types differed in species composition and if species turnover was a function of the distance between plots at different spatial scales. As pseudo-turnover may bias beta diversity we hypothesized a large effect of unseen species reducing beta diversity. We sampled 206 ant species (25% singletons) from ten subfamilies and 55 genera. Diversity partitioning among the four forest types revealed that whereas alpha species richness and alpha Shannon diversity were significantly smaller than expected, beta-diversity for both measurements was significantly higher than expected by chance. This result was confirmed when we used the unbiased estimation of Shannon diversity: while alpha diversity was much higher, beta diversity differed only slightly from biased calculations. Beta diversity as measured with the Chao-Sørensen or Morisita-Horn Index correlated with distance between transects and between sample points, indicating a distance decay of similarity between communities. We conclude that habitat heterogeneity has a high influence on ant diversity and species turnover in tropical sites and that unseen species may have only little impact on calculation of Shannon beta diversity when sampling effort has been high.     

Inventory, differentiation, and proportional diversity: a consistent terminology for quantifying species diversity

Almost half a century after Whittaker (Ecol Monogr 30:279–338, 1960) proposed his influential diversity concept, it is time for a critical reappraisal. Although the terms alpha, beta and gamma diversity introduced by Whittaker have become general textbook knowledge, the concept suffers from several drawbacks. First, alpha and gamma diversity share the same characteristics and are differentiated only by the scale at which they are applied. However, as scale is relative––depending on the organism(s) or ecosystems investigated––this is not a meaningful ecological criterion. Alpha and gamma diversity can instead be grouped together under the term “inventory diversity.” Out of the three levels proposed by Whittaker, beta diversity is the one which receives the most contradictory comments regarding its usefulness (“key concept” vs. “abstruse concept”). Obviously beta diversity means different things to different people. Apart from the large variety of methods used to investigate it, the main reason for this may be different underlying data characteristics. A literature review reveals that the multitude of measures used to assess beta diversity can be sorted into two conceptually different groups. The first group directly takes species distinction into account and compares the similarity of sites (similarity indices, slope of the distance decay relationship, length of the ordination axis, and sum of squares of a species matrix). The second group relates species richness (or other summary diversity measures) of two (or more) different scales to each other (additive and multiplicative partitioning). Due to that important distinction, we suggest that beta diversity should be split into two levels, “differentiation diversity” (first group) and “proportional diversity” (second group). Thus, we propose to use the terms “inventory diversity” for within-sample diversity, “differentiation diversity” for compositional similarity between samples, and “proportional diversity” for the comparison of inventory diversity across spatial and temporal scales. Electronic supplementary material  The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

Phylogenetic diversity measures based on Hill numbers

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.     

How dung beetles respond to a human‐modified variegated landscape in Mexican cloud forest: a study of biodiversity integrating ecological and biogeographical perspectives

Aim To analyse how the dung beetles (Scarabaeinae) respond to a modified, variegated landscape, taking into account the biogeographical peculiarities of the Mexican Transition Zone. Location This study covers cloud forest (CF) of the Sierra Norte de Puebla mountain range and part of the Sierra Madre Oriental mountain range (Mexico). Methods We applied proportional sampling based on the landscape variegation model with Scarabaeinae as the indicator group, and using two approaches: structural units (vegetation type) and spatial units (windows). We used two measures – richness and Shannon diversity – and applied multiplicative diversity partitioning to obtain independent alpha and beta diversities for the landscape, windows and vegetation types. We grouped species by biogeographical distribution pattern for the biogeographical analysis and by whether they were originally from CF. Results The transformation of CF into secondary forest, pastures and other types of vegetation increases the Scarabaeinae diversity of the landscape, in vegetation types and windows. This increase is the result of species arriving from the tropical lowlands. However, the original dung beetle community of the CF dominates at different scales in the number of species, abundance and biomass. With increasing habitat modification, beta diversity increases in the windows, and species with the Tropical Palaeoamerican distribution pattern increase in abundance in vegetation types and windows. Main conclusions The variegated character of the landscape explains well the distribution and diversity of this dung beetle community. The peculiar characteristics of the Mexican Transition Zone have an effect owing to the overlap of fauna with different biogeographical origins. The conversion of fragmented landscapes to variegated landscapes could be a conservation goal in human‐modified mountain landscapes. Sampling proportional to the area of different types of vegetation and the use of windows offer an alternative experimental design in variegated landscapes.     

Using functional diversity patterns to explore metacommunity dynamics: a framework for understanding local and regional influences on community structure

The metacommunity concept, describing how local and regional scale processes interact to structure communities, has been successfully applied to patterns of taxonomic diversity. Functional diversity has proved useful for understanding local scale processes, but has less often been applied to understanding regional scale processes. Here, we explore functional diversity patterns within a metacommunity context to help elucidate how local and regional scale processes influence community assembly. We detail how each of the four metacommunity perspectives (species sorting, mass effects, patch dynamics, neutral) predict different patterns of functional beta‐ and alpha‐diversity and spatial structure along two key gradients: dispersal limitation and environmental conditions. We then apply this conceptual model to a case study from alpine tundra plant communities. We sampled species composition in 17 ‘sky islands’ of alpine tundra in the Colorado Rocky Mountains, USA that differed in geographic isolation and area (key factors related to dispersal limitation) and temperature and elevation (key environmental factors). We quantified functional diversity in each site based on specific leaf area, leaf area, stomatal conductance, plant height and chlorophyll content. We found that colder high elevation sites were functionally more similar to each other (decreased functional beta‐diversity) and had lower functional alpha‐diversity. Geographic isolation and area did not influence functional beta‐ or alpha‐diversity. These results suggest a strong role for environmental conditions structuring alpine plant communities, patterns consistent with the species sorting metacommunity perspective. Incorporating functional diversity into metacommunity theory can help elucidate how local and regional factors structure communities and provide a framework for observationally examining the role of metacommunity dynamics in systems where experimental approaches are less tractable.     

The relationship between species replacement,dissimilarity derived from nestedness,and nestedness

Aim Beta diversity can be partitioned into two components: dissimilarity due to species replacement and dissimilarity due to nestedness ( Baselga, 2010 , Global Ecology and Biogeography, 19 , 134–143). Several contributions have challenged this approach or proposed alternative frameworks. Here, I review the concepts and methods used in these recent contributions, with the aim of clarifying: (1) the rationale behind the partitioning of beta diversity into species replacement and nestedness‐resultant dissimilarity, (2) how, based on this rationale, numerators and denominators of indices have to match, and (3) how nestedness and nestedness‐resultant dissimilarity are related but different concepts. Innovation The rationale behind measures of species replacement (turnover) dictates that the number of species that are replaced between sites (numerator of the index) has to be relativized with respect to the total number of species that could potentially be replaced (denominator). However, a recently proposed partition of Jaccard dissimilarity fails to do this. In consequence, this partition underestimates the contribution of species replacement and overestimates the contribution of richness differences to total dissimilarity. I show how Jaccard dissimilarity can be partitioned into meaningful turnover and nestedness components, and extend these new indices to multiple‐site situations. Finally the concepts of nestedness and nestedness‐resultant dissimilarity are discussed. Main conclusions Nestedness should be assessed using consistent measures that depend both on paired overlap and matrix filling, e.g. NODF, whereas beta‐diversity patterns should be examined using measures that allow the total dissimilarity to be separated into the components of dissimilarity due to species replacement and dissimilarity due to nestedness. In the case of multiple‐site dissimilarity patterns, averaged pairwise indices should never be used because the mean of the pairwise values is unable to accurately reflect the multiple‐site attributes of dissimilarity.     

The alpha–beta–regional relationship: providing new insights into local–regional patterns of species richness and scale dependence of diversity components

Ecologists frequently regress local species richness on regional species richness to draw inferences about the processes that structure local communities. A more promising approach is to quantify the contributions of alpha and beta diversity to regional diversity (the ABR approach) using additive partitioning. We applied this approach to four local–regional relationships based on data from 583 arboreal beetle species collected in a hierarchically nested sampling design. All four local–regional relationships exhibited proportional sampling, yet the ABR approach indicated that each was produced by a different combination of alpha and beta richness. Using the results of the ABR analysis, we also analysed the scale dependence of alpha and beta using a hierarchical linear model. Alpha diversity contributed less than expected to regional diversity at the finest spatial scale and more than expected at the broadest spatial scale. A switch in relative dominance from beta to alpha diversity with increasing spatial scale suggested scale transitions in ecological processes. Analysing the scale dependence of diversity components using the ABR approach furthers our understanding about the additivity of species diversity in biological communities.