Using null models to disentangle variation in community dissimilarity

Using null models to disentangle variationin community dissimilarity from variation in a-diversity

JONATHAN M. CHASE,1,2,� NATHAN J. B. KRAFT,3 KEVIN G. SMITH,2 MARK VELLEND,3,4 AND BRIAN D. INOUYE5

1Department of Biology, Washington University, St. Louis, Missouri 63130 USA2Tyson Research Center, Washington University, Eureka, Missouri 63125 USA

3Biodiversity Research Centre, University of British Columbia, Vancouver, British Columbia V6T 1Z4 Canada4Departments of Botany and Zoology, University of British Colombia, Vancouver, British Columbia V6T 1Z4 Canada

5Department of Biological Science, Florida State University, Tallahassee, Florida 32306-4295 USA

Abstract. b-diversity represents the compositional variation among communities from site-to-site,

linking local (a-diversity) and regional (c-diversity). Researchers often desire to compare values of b-diversity across localities or experimental treatments, and to use this comparison to infer possible

mechanisms of community assembly. However, the majority of metrics used to estimate b-diversity,including most dissimilarity metrics (e.g., Jaccard’s and Sørenson’s dissimilarity index), can vary simply

because of changes in the other two diversity components (a or c-diversity). Here, we overview the utility

of taking a null model approach that allows one to discern whether variation in the measured dissimilarity

among communities results more from changes in the underlying structure by which communities vary, or

instead simply due to difference in a-diversity among localities or experimental treatments. We illustrate

one particular approach, originally developed by Raup and Crick (1979) in the paleontological literature,

which creates a re-scaled probability metric ranging from�1 to 1, indicating whether local communities are

more dissimilar (approaching 1), as dissimilar (approaching 0), or less dissimilar (approaching �1), thanexpected by random chance. The value of this metric provides some indication of the possible underlying

mechanisms of community assembly, in particular the degree to which deterministic processes create

communities that deviate from those based on stochastic (null) expectations. We demonstrate the utility of

this metric when compared to analyses of Jaccard’s dissimilarity index with case studies from disparate

empirical systems (coral reefs and freshwater ponds) that differ in the degree to which disturbance altered

a-diversity, as well as the selectivity by which disturbance acted on members of the community.

Key words: b-diversity null model; community assembly; determinism; species pool; stochasticity.

Received 3 October 2010; revised 3 January 2011; accepted 17 January 2011; final version received 8 February 2011;

published 28 February 2011. Corresponding Editor: R. Ptacnik.

Citation: Chase, J. M., N. J. B. Kraft, K. G. Smith, M. Vellend, and B. D. Inouye. 2011. Using null models to disentangle

variation in community dissimilarity from variation in a-diversity. Ecosphere 2(2):art24. doi:10.1890/ES10-00117.1

Copyright: � 2011 Chase et al. This is an open-access article distributed under the terms of the Creative Commons

Attribution License, which permits restricted use, distribution, and reproduction in any medium, provided the original

author and sources are credited.

� E-mail: [email protected]

INTRODUCTION

Recent interest in the patterns of species

diversity and community composition across

space has resurrected the concept of b-diversity(e.g., Whittaker 1960, 1972), which quantifies the

variation in the composition of species from site-

to-site, originally defined as the ratio between

local (a) diversity and regional (c) diversity (b ¼a/c). b-diversity can be a useful metric when

trying to understand patterns of species diversity

across spatial scales (e.g., Veech et al. 2002, Crist

and Veech 2006, Jost 2007, Tuomisto 2010a, b,

Anderson et al. 2011), and for example, can allow

inference about the relative importance of com-

munity assembly processes such as those that are

v www.esajournals.org 1 February 2011 v Volume 2(2) v Article 24

more deterministic (niche-related) relative tothose that are more stochastic (e.g., Condit et al.2002, Tuomisto et al. 2003, Gilbert and Lechowicz2004, Dornelas et al. 2006, Chase 2007, 2010).Unfortunately, a large number of the metrics andstatistical analyses used to estimate b-diversityare confounded, are not always directly compa-rable, and may not produce conceptually mean-ingful values (Jost 2007, Jurasinski et al. 2009,Tuomisto 2010a, b, Anderson et al. 2011). Forexample, one of the biggest problems associatedwith analyses of b-diversity is that using either amultiplicative (c ¼ a 3 b) or additive (c ¼ a þ b)partition, b-diversity is linked to variation in cand a by definition, thus making statisticalcomparisons of b-diversity among sites or re-gions confounded by coincident variation in a orc (e.g., Wilson and Shmida 1984, Lande 1996,Koleff et al. 2003, Jost 2007, Tuomisto 2010a, b).

There has been recent debate regarding appro-priate ways to create a-independence in theevaluation of b-diversity (e.g., Baselga 2010, Jost2010, Ricotta 2010, Veech and Crist 2010a, b).Although we do not intend to engage directly inthis debate, we emphasize that because all threediversity components (a, b and c) are intercon-nected, any two of the three components will bestatistically dependent on one another, regardlessof the diversity measure (e.g., Ricotta 2010).Thus, if there is a change in some factor thatinfluences the number of species that can coexistin any given site (e.g., local disturbance, produc-tivity, predators), influencing a-diversity, it isunclear whether a change in b-diversity is due todifferences in the underlying assembly processesthat create b-diversity (e.g., deterministic versusstochastic factors), or instead due to differencesthat result because the factor of interest changedthe level of a-diversity, necessarily causing aconcomitant change in b-diversity (e.g., Vellend2004, Vellend et al. 2007, Chase 2007, 2010, Chaseet al. 2009). To illustrate with a simple example, ifthe average number of species per site (a)decreases via random local extinctions (e.g.,ecological drift due to a decrease in habitat size)without a proportional change in regional (c)diversity (e.g., if the number of localities is verylarge), a change in b will be detected even thoughthere has been no change in the ecologicaldifferences or amount of dispersal among locali-ties.

For an empiricist who wants to comparepatterns of b-diversity among sites and inferpossible mechanisms, a statistical null modelingapproach (sensu Gotelli and Graves 1996) pro-vides a straightforward and versatile way todiscern to what degree changes in observed b-diversity are influenced by random changes in a-diversity (e.g., Connor and Simberloff 1978, Raupand Crick 1979, Crist et al. 2003, Vellend 2004,Dornelas et al. 2006, Freestone and Inouye 2006,Chase 2007, 2010, Vellend et al. 2007, Belmaker etal. 2008, Leprieur et al. 2008, Chase et al. 2009,Lepori and Malmquist 2009, Smith et al. 2009). Anull modeling approach in this context essential-ly asks: ‘‘what would b-diversity look like with acompletely random assembly process, given aand c-diversity?’’ With this null-expected distri-bution in hand, one can then ask: ‘‘does theobserved b-diversity deviate from the nullexpectation, and if so, by how much?’’

In this article, we focus on an approach that isrelevant for studies that use (dis)similaritymetrics based on presence-absence data compar-isons between two communities (e.g., Jaccard’s orSørenson’s dissimilarity metrics). These dissimi-larity metrics are often used as an estimate of b-diversity (e.g., Vellend 2001, Koleff et al. 2003).We first illustrate how one of the most commonlyused metrics to estimate b-diversity, Jaccard’sdissimilarity index, is strongly influenced by thenumber of species that live in each site (e.g., a-diversity) and the number of species that live inthe regional species pool (c-diversity). As such, alarge number of studies that have compared suchdissimilarities among communities that vary ina-diversity can not discern whether the differ-ences in dissimilarity actually result from differ-ences in the compositional variation among localcommunities, or just due to differences amongthose communities in a-diversity. We next over-view one particular null modeling approach thatallows the quantification of the degree to whichpairwise community dissimilarity differs fromthat which would be expected by random chancealone. We conclude with a discussion of theutility and limitations of this approach, and pointtoward a family of related approaches that willallow a more thorough investigation of themechanisms underlying patterns of communitycompositional differentiation across sites.


CHASE ET AL.

A null expectation for community (dis)similaritywhen a-diversity varies

First, we illustrate how the value of incidence-based pairwise (dis)similarity indices such asJaccard’s index depend on a-diversity. That is,when comparing among regions (or habitattypes, experimental treatments, time periods,etc.), changes in (dis)similarity cannot be disen-tangled from changes in a-diversity among thoseregions unless this dependence is explicitlyconsidered (Fig. 1). For two samples with a1and a2 species selected with equal probabilityfrom a pool of c-species, the expected number ofshared species (SSexp) is an accelerating functionof a-diversity: SSexp ¼ a1a2/c (Connor andSimberloff 1978). From this, we can calculate anull expected Jaccard’s similarity index (Jexp) (asimilar approach can be used for other incidence-based dissimilarity metrics) by

Jexp ¼SSexp

a1 þ a2 � SSexp

:

When a1 ¼ a2 for any given c, the nullexpectation is an accelerating positive relation-ship between a and Jexp; Jexp¼ 1 when a¼ c (Fig.

1). When a1 6¼ a2, a three-dimensional surfacegives the null Jexp (not shown).

From Fig. 1, it is apparent how the mostpopular incidence-based metrics of dissimilarity(dissimilarity ¼ 1 � similarity) that depend onSSobs (e.g., Jaccard’s and Sørenson’s dissimilarityindex, or other related metrics; e.g., Koleff et al.2003) are strongly contingent on variation in a-diversity. When a is variable, for example whencomparing among habitats or experimentaltreatments, or even when comparing amongstudies with different sample size, comparisonsof incidence-based dissimilarity metrics cannotdiscern whether differences in dissimilarity aredue to changes in the underlying structuring ofcommunity composition across sites, or insteadsimply due to changes in a-diversity. A nullmodel approach can provide a straightforwardway to discern whether species compositionaldifferences among sites result from changes in a-diversity, or from forces causing communities tobe more, or less, dissimilar than expected byrandom chance.

Estimating deviations from the null expectationGiven the three components of a diversity

partition, a, b, and c-diversity, a null modelgenerally holds the values of two of thosecomponents constant (e.g., a and c-diversity),and can be used to ask what the value of the thirdcomponent (e.g., b-diversity) would be expectedby random chance. A randomization test canthen be used to compare the observed valuesrelative to the expected values to detect devia-tions that would indicate changes in b-diversitythat are not due to changes in a-diversity. Thesedeviations can be in either direction, whereby b-diversity is either higher or lower than expectedby chance given a-diversity and a regionalspecies pool.

Although the exact form of the null model andassociated tests will depend, to some degree, onthe nature of the data and of the question beingasked, the general principles are the same. Forexample, one could observe that average dissim-ilarity among pairs of local communities isgreater for one group of communities thananother, but if a-diversity was smaller amongthe more dissimilar communities, this result isexpected by random chance alone (see Fig. 1). Anull model is needed to discern whether the

Fig. 1. The expected relationship between the local

richness (a-diversity) in a pair of communities and the

Jaccard’s index of similarity (dissimilarity ¼ 1-similar-

ity), for the case where richness is equal in each local

community (a1 ¼ a2). Line traces the average of 999

replicates per a value, generated by sampling from a

pool of 50 species with equal abundances.


CHASE ET AL.

difference in dissimilarity deviates from randomexpectation given the changes in a-diversity, andthe results may indicate possible underlyingmechanisms of community assembly. That is,communities that either are more, or less, similarthan expected by chance can indicate somedegree of determinism in the community assem-bly process (e.g., Chase 2007, 2010, Chase et al.2009).

Here, we overview one method, known as theRaup-Crick metric (Raup and Crick 1979), thatcan provide some information on the degree towhich pairwise communities are more different(or more similar) than expected by chance. Assuch, this metric allows one to develop hypoth-esis tests about the relative magnitude of thedifferences between observed and expectedcommunities. Although the Raup-Crick metricwas initially expressed as a similarity, we focuson it as a dissimilarity; to be consistent with theconcept of b-diversity—we refer to this metric asbRC (Vellend et al. 2007) (the R code we providein the online supplement also allows the calcu-lation of similarity if desired). In this case, ratherthan representing dissimilarity among pairwisecommunities per se, as in most metrics of b-diversity, the bRC metric expresses dissimilarityamong two communities relative to the nullexpectation. Specifically, if SS1,2 is the observednumber of shared species between localities 1and 2, containing a1 and a2 species, respectively,bRC uses a randomization approach to estimatethe probability of observing .SS1,2 given repeat-ed random draws of a1 and a2 species from aknown species pool (Fig. 2). For this null modelto be most useful, knowledge of the species poolfrom which potential species are drawn isimportant.

In this section, we provide a step-by-stepoverview of the bRC calculations. We do thisbecause (1) the metric was introduced in thepaleontological literature, and has rarely beenused in ecology (a search of the ISI Web ofKnowledge Science Citation Index database on12/22/2010 shows fewer than 25 citations of thepaper from ecological studies; ,1 per year); (2)some of the decisions required to implement thecalculations have not always been transparent,but can significantly alter the results; (3) theprograms available for calculating the traditionalRaup-Crick metric—PaST (Hammer et al. 2001)

and the current Vegan Package of R (Oksanen etal. 2011)—appear to have some critical limita-tions, which we address below. For any given a1

and a2, bRC compares SSobs to the distribution ofSSexp values produced by a null model. Weassume the use of presence-absence data. Al-though the bRC calculation uses SSobs and SSexp,instead of a similarity metric, precisely equiva-lent results would be generated by any similaritymetric based on SSobs, such as Jaccard’s orSørenson’s dissimilarity index. In the onlinesupplement, we provide annotated R code thatcan be used to perform these analyses.

� Step 1. Calculate the observed a1, a2, and SSobs. Forany given pair of sites, a1 and a2 are the numberof species observed in each site, and SSobs is thenumber of species that the two sites share incommon.� Step 2. Calculate the total number of species in the‘‘species pool’’ among all sites, and the proportion ofsites each species occupies (its ‘‘occupancy’’). Theseare calculated from all of the sites in the data setof interest (not just the two sites underconsideration).� Step 3. Calculate the distribution of SSexp values.Randomly draw a1 and a2 species at randomfrom the ‘‘species pool’’. The probability of aspecies being drawn is proportional to itsamong-site occupancy. Repeat this procedureat least 1000 times (ideally more).

� Step 4. Compare SSobs with the distribution of SSexp.Sum the number of random draws in whichSSexp . SSobs and one-half of the random drawsin which SSobs ¼ SSexp, and divide the sum bythe total number of random draws. This is anestimate of the probability of observing SSobs orfewer shared species given random draws fromthe species pool.� Step 5. Standardize the metric to range from�1 to 1.Subtract 0.5 from the value from step 4, andmultiply by 2.

Step 5 represents our primary modification ofthe original Raup-Crick metric (other thanexpressing it as a dissimilarity rather thansimilarity), by re-scaling it to vary from �1 to 1.A value of 0 represents no difference in theobserved (dis)similarity from the null expecta-tion; a value of 1 indicates observed dissimilarityhigher than the expected in any of the simula-tions (communities completely more different


CHASE ET AL.

Fig. 2. The relationship between the local richness in a pair of communities (a1, a2) and the expected number of

shared species between them (SSexp) for a) the case where a1¼ a2, and b) all possible combinations of a values,

which produces a three-dimensional surface, but is otherwise the same as in panel a. The null expectation is

shown as a solid black line in panel a, along with histograms showing the distribution of SSexp for five values of

pairs of a values (5, 25, 35, 40, 45) (based on 999 randomizations), demonstrating how the distribution is

approximately normal when a¼ c/2, but has reduced variance and increased skewness as a approaches 0 or c.Calculations for bRC for two hypothetical cases of the observed number of shared species are shown. In the area

below the solid line communities share fewer species than expected and therefore have high bRC, while above the

solid line more species are shared than expected, corresponding to low bRC. These bRC calculations are not shown

in the three-dimensional version of panel b simply for clarity.


CHASE ET AL.

from each other than expected by chance), andvice versa for a value of �1 (communitiescompletely less different [more similar] thanexpected by chance).

The main limitations of existing softwareimplementations of the Raup-Crick metric areas follows: In the current Vegan package of R(Oksanen et al. 2011), the distribution of SSexp iscalculated analytically by a hypergeometricdistribution. As such, every species in theregional pool is given equal weight, includingspecies that are listed with zero abundances.Disregarding species frequencies has a largeeffect on value of the bRC. It may be possible touse a Fisher noncentral hypergeometric distribu-tion to incorporate a weighted species pool,however the simulation approach we take ismore flexible. In the PaST (PAleontologicalSTatistics) program (Hammer et al. 2001), itappears that steps 1–4 are identical to ours,except that a maximum of only 200 randomiza-tions are performed in step 3. This limitednumber of randomizations results in low powerto distinguish differences in the degree by whichsets of communities that differ from the nullexpectation (e.g., when two groups of communi-ties both differ from the null expectation, valuesof bRC quickly converge on 1 [or �1] with fewerrandomizations); this limitation is especiallysevere for species rich communities.

For our implementation of bRC, one could askwhether SSobs for a pair of communities issignificantly different from the null expectationby assessing whether jbRCj . 0.95 (two-tailedtest, alpha ¼ 0.05). Such differences wouldindicate whether a given pair of communitiesshare fewer (approaching 1) or more (approach-ing�1) species than expected by random chance.More commonly, however, values of bRC arecalculated for all pairwise combinations ofcommunities, and these can be analyzed usingthe same set of statistical methods as the manyother pairwise indices of dissimilarity (Tuomisto2010a, b, Anderson et al. 2011).

Mean bRC among habitats, experimental treat-ments, or time periods will be close to 0 whencommunity assembly is highly stochastic anddispersal is high among communities, and willapproach �1 when deterministic environmentalfilters shared across sites create highly similarcommunities (Chase et al. 2009, Chase 2010).

Alternatively, bRC will be closer to 1 if determin-istic environmental filters favor dissimilar speciescompositions, for example, if there were strongbiotic structuring forces creating very differentcommunities on adjacent sites (e.g., checkerboarddistributions determined by competitive interac-tions; Diamond 1975), or if dispersal among sitesis very low, leading to dispersal limitation.

It is worth noting that the influence ofenvironmental filtering and dispersal limitationon bRC (or any b-diversity metric) will depend onthe sampling scale. If environmental filtering isstrong, sites with similar environmental condi-tions should be more similar than expected (bRC

, 0), while sites with dissimilar environmentalconditions should be less similar than expected(bRC . 0). Likewise, when dispersal limitation isstrong, nearby pairs of sites will be more similarthan expected (bRC , 0), whereas distant pairs ofsites will be less similar than expected (bRC . 0).

Caveats and considerationsbRC is explicitly conditioned upon variation in

a-diversity, and thus provides a more appropri-ate metric than other measures of dissimilarityfor comparing the dissimilarity among commu-nities that vary in a-diversity (e.g., Jaccard’s orSørenson’s dissimilarity index). However, allmetrics of b-diversity and associated null modelanalyses have limitations, and bRC is no excep-tion.

Specifying the regional species pool.—In practice,the species pool is typically characterized as theset of species (and their frequencies acrosslocalities) observed during the sampling of agiven set of localities. However, the size of thepool will depend on the number of localitiessampled (Gotelli and Colwell 2001), and corre-spondingly, the deviation of SSobs from SSexp willdepend on the accuracy of the estimation of thespecies pool. Additionally, the choice of how todefine the regional species pool will often have tobe tailored to the question being addressed. Forexample, if within-habitat bRC is being comparedamong habitats that differ in the set of speciesthat can live there for deterministic reasons, thechoice of whether to include only the species thatlive in a particular habitat type, or whether toinclude the species that live in each habitat typewill strongly influence the value of the bRC (seeGotelli and Graves 1996, Kraft et al. 2008,


CHASE ET AL.

Cornwell and Ackerly 2009 for related discus-sions). bRC values will be closer to 0 (smalldeviation from the null-expectation) if the speciesfrom only that habitat are included in the speciespool used to calculate SSexp, relative to the casewhere species from multiple habitat types areincluded in the species pool. At the same time, itis not advisable to use a regional species poolthat is so large (e.g., all of the species of aparticular group across biogeographic zones)that all communities would have exceptionallylow bRC values. While resolution of the regionalspecies pool issue is beyond the scope of thepresent paper, a simple ‘rule of thumb’ forincluding a species in the regional species poolto generate meaningful interpretations of bRC

would be to include those species that canpossibly colonize a given site within a reasonabletime period. Furthermore, it is worth keeping inmind that each alternative designation of theregional pool essentially asks a different researchquestion, so therefore consistency or discordancein results among different pool designations (aswell as those that do and do not disentangle a-diversity) can provide insight into ecologicalquestions. For example, to examine the role ofdispersal limitation in creating patterns of com-munity dissimilarity relative to environmentalfeatures, one could compare whether communitycompositional dissimilarity deviates from thenull expectation at relatively small scales (whenthe regional pool is constrained to that observedin a given set of sites) to the deviation obtainedwhen the regional pool incorporates a largernumber of species that could potentially colonizea site, but only very rarely do so. There are someinstances where one might want to includespecies in the pool that are not observed in thesampled communities, but known to be present.The R-code we provide in the online supplementdoes not currently allow for the pool to be higherthan those species that in at least one of thesamples, but this could certainly be modified ifthe investigator feels that it is useful. However,we note that while increasing the size of thespecies pool will increase the absolute magnitudeof the deviation from the null expectation, it willnot generally influence the relative deviationsamong pairs of communities.

Species occupancies.—We suggest that the pro-portion of sites occupied by each species should

be incorporated into the null model calculationsof SSexp (though this can be bypassed in the R-code we provide) (see also Connor and Simberl-off 1978, Raup and Crick 1979). If all species areconsidered equally likely to occupy a given sitein the null model, SSobs will often be muchgreater than SSexp, and bRC will be more negative(see Gotelli and Graves 1996, Kembel and Hub-bell 2006, Hardy 2008 for related discussion). Apotentially difficult question concerns how toincorporate occupancies of species when thereare multiple habitat types (or treatments withinan experiment). Species will likely vary in theirdegree of occupancy of patches across differenthabitat types, if, for example, a species occupies alarge proportion of the available sites in habitatsin which it is favored, and many fewer sites inhabitats that it is less favored. Here, if one wantsto include occupancies into the calculation ofSSexp, and include both habitat types, it would beessential to ensure a relatively equal samplingeffort on both of the habitat types, so thatoccupancies would be representative of the entireregion (Vellend et al. 2007).

Power to detect differences at low and high a-diversity.—Despite its advantages, bRC, and infact all pairwise dissimilarity metrics, suffer fromsome undesirable (and unavoidable) propertieswhen a-diversity is either low or very highrelative to the species pool. The problems stemfrom the fact that SSexp must take an integervalue, and the number of theoretically possiblevalues of SSexp can be quite small in some cases(Fig. 2a). For example, if min(a1, a2)¼ 5, there areonly six possible values of SSexp: 0, 1, 2, 3, 4 and5. For the hypothetical case where the number ofspecies in the species pool is 50, a1¼ a2¼ 5, andspecies are equal in their occupancies of sites, theminimum possible SSexp (0) is in fact a highlyprobable outcome of the null model (probability. 0.55). Thus, there is low power to detect anydeviation from the null expectation in such cases.This is an inherent property of metrics that relyon species presence-absence information, andcannot be remedied by any kind of quantitativecorrection or analysis, but must be kept in mindwhen making such calculations. One solution isto down-weight the influence of any such valuesin downstream analyses based on the full matrixof pairwise dissimilarities.


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An illustration of the utility of null models in b-diversity studies

To illustrate the importance of null models inaddressing hypotheses regarding b-diversity andthe possible mechanisms by which it is created,we show data from two disparate studies thatboth examined the effects of disturbance onpatterns of community structure (Chase 2007,Anderson et al. 2011). In the first case, data werecollected from experimental freshwater ponds,ten of which were controls, and ten of whichwere subjected to drought in the middle of theexperiment (Chase 2007). Although droughtdecreased a-diversity somewhat (data notshown), it had a marked influence on b-diversityas measured using Jaccard’s dissimilarity index,with ponds exposed to drought being muchmore similar to one another compositionally thancontrol ponds (Fig. 3a). bRC values on this datasetshow that communities in the drought pondswere more similar than expected by chancerelative to the control ponds (Fig. 3b), suggestingthat changes in b-diversity were not simply dueto a random influence of drought on species ineach locality, and instead likely resulted becausedrought imposed a systematic ecological filterthat removed a subset of species in each of thecommunities exposed to drought (Chase 2007).

In the second case, data were collected from 10transects collected before and after an El Ninoinduced coral bleaching event from Indonesia(Warwick et al. 1990) and re-analyzed using avariety of b-diversity metrics in Anderson et al.(2011). Following the bleaching event, there wasa marked decrease in a-diversity (data notshown), and a large increase in b-diversity, asmeasured using Jaccard’s dissimilarity index(Fig. 3b). However, there were no differences inbRC values before and after the bleaching event(Fig. 3d) (see also Anderson et al. 2011). Thecontrast between the results for Jaccard’s dissim-ilarity index, which suggest a large increase in b-diversity after the disturbance, and the bRC

results, which suggest no change in b-diversityrelative to the null expectation once the effect ofdisturbance on a-diversity was considered, em-phasize the importance of this approach.

Comparing the effects disturbance in thesedisparate systems simultaneously using metricsthat are a-dependent (e.g., Jaccard’s dissimilarityindex) and a-corrected (e.g., bRC) allows one to

delve into the possible classes of mechanisms bywhich disturbance acts on these communities. Inthe coral reefs, disturbance appears to have actedprimarily through random sampling effects,where there was no difference bRC before andafter the coral bleaching event. Alternatively, inthe ponds, drought disturbance acted moredeterministically, such that bRC was considerablylower in the ponds that experienced droughtrelative to the control ponds. Similar inferencescan be made when comparing patterns of b-diversity among communities that differ in otherfactors that may influence a-diversity, such aspredators (Chase et al. 2009), pathogens (Smith etal. 2009), or productivity (Chase 2010).

CONCLUSIONS

Ecologists have become increasingly interestedin the patterns of, and processes leading to, thesite-to-site (or time to time) compositional differ-entiation among localities—b-diversity—buthave also increasingly recognized problems withits definition and analysis (e.g., reviewed inTuomisto 2010a, b). The bRC approach allowsone to disentangle variation in communitycompositional dissimilarity across sites fromvariation in the a-diversity of those sites as longas those sites are embedded in the same regionalspecies pool. As such, the bRC metric can be usedto address the question ‘does the compositionalvariation among communities differ from a null-expectation?’ (Condit et al. 2002, Dornelas et al.2006), and the related ‘to what degree docommunities deviate from the null expectation,and how do abiotic or biotic factors influence thisdeviation?’ (e.g., Chase 2007, 2010, Chase et al.2009, Smith et al. 2009).

Although we espouse the bRC approach as auseful metric, there are several different ways toimplement null models in order to make infer-ences about patterns of b-diversity. The specificform of the null model will depend critically onthe question being asked, as well as the scope ofthe data being analyzed. The bRC metric isappropriate when comparisons are made amongcommunities that can reasonably be consideredto be a part of the same regional species pool.However, when comparisons of b-diversityamong biogeographic regions that vary in thesize of the regional species pool (e.g., along


CHASE ET AL.

latitudinal gradients) are of interest, a related, but

distinct null-modeling approach (e.g., Crist et al.

2003, Crist and Veech 2006) will be needed to

disentangle the relative contributions of a-diver-sity and c-diversity to variation in b-diversity.

ACKNOWLEDGMENTS

Discussions with T. Knight were instrumental in the

early development of these ideas, and comments fromN. Sanders, N. Swenson, H. Tuomisto, H. Cornell, T.

Crist, R. Ptacnik, and three anonymous reviewers

improved this manuscript. This work was conductedas part of the ‘Gradients of b-diversity’Working Group

supported by the National Center for EcologicalAnalysis and Synthesis, a Center funded by NSF(Grant #EF-0553768), the University of California,Santa Barbara, and the State of California. JMC andKGS’s research was also supported by WashingtonUniversity and the National Science Foundation (DEB0816113). NJBK was supported by the NSERC CRE-ATE Training Program in Biodiversity Research.

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Using null models to disentangle variation in community dissimilarity