Anewfrontier in biodiversity inventory: a proposal forestimators of phylogenetic and functional diversityPedro Cardoso1,2*, Francois Rigal2, PauloA. V. Borges2andJos e C. Carvalho2,31Finnish Museumof Natural History, University of Helsinki, POBox 17 (Pohjoinen Rautatiekatu 13), 00014 Helsinki, Finland;2Azorean Biodiversity Group (CITA-A) and Portuguese Platformfor Enhancing Ecological Research &Sustainability (PEERS),University of the Azores, 9700-042 Angra do Herosmo, Portugal; and3CBMA Molecular and Environmental Centre,Department of Biology, University of Minho, 4710-057 Braga, PortugalSummary1. Complete sampling of all dimensions of biodiversity is a formidable task, even for small areas. Undersamplingis the norm, and the underquantication of diversity is a common outcome. Estimators of taxon diversity (TD)are widely used to correct for undersampling. Yet, no similar strategy has been developed for phylogenetic (PD)or functional (FD) diversity.2. We propose three ways of estimating PDand FD, building on estimators originally developed for TD: (i) cor-recting PD and FD values based on the completeness of TD; (ii) tting asymptotic functions to accumulationcurves of PDand FD; and (iii) adapting nonparametric estimators to PDand FDdata.3. Using trees as a common framework for the estimation of PDand FD, we tested the approach with Europeanmammal and Azores Islands arthropod data. We demonstrated that dierent methods were able to considerablyreduce the undersampling bias and often correctly estimated true diversity using a fraction of the samples neces-sary to reach complete sampling.4. Besides the utility of knowing the true diversity of an assemblage fromincomplete samples, the use of estima-tors may present further advantages. For instance, comparisons between sites or time periods are possible only ifeither sampling is complete or sampling eort is equivalent and sucient to allowsensible comparisons. Also, asPD and FD asymptote faster than TD, comparisons between these dierent dimensions may require unbiasedvalues. The framework nowproposed combines taxon, phylogenetic and functional diversity into a single frame-work, oering a tool for future developments involving these dierent facets of biological diversity.Key-words:accumulation curves, alpha diversity, arthropods, Azores, European mammals,extrapolation, functional diversity, nonparametric estimators, phylogenetic diversity, sampling biasIntroductionFromscalesassmallasasingletree,whichcanhousethou-sandsofspecies(Erwin1982), totheentireplanet, whichishometomillions(Mora etal.2011),wearealwaysfarfrombeingabletocount everysinglespeciesfrommost groups.Even studies restricted in space, time and taxonomic scope faceintractable problems when trying to reach complete lists of spe-cies (Cardoso 2009; Coddington etal.2009). As species rich-ness increases withthe number of individuals or samples,observed richness almost invariably underestimates true rich-ness (Coddington etal. 2009).The problemof undersampling andconsequent bias indiversity descriptors has long been recognised as both ubiqui-tousandfundamentaltocorrect.Withoutcompleteinvento-ries of communities, comparisons of species richness cannot bereliablymadeifonedoesnotconsiderthesampling eortorcompleteness attained (Gotelli & Colwell 2001). Manyresearchers address this concern by estimating alpha diversityusingtechniquesthatadjustforsamplingeortorcomplete-ness (Longino, Colwell &Coddington 2002). The sameproblems have been identied for beta diversity, since underes-timationof similarityalsooccurs becauseof thefailuretoaccount for unseensharedspecies (Chaoetal. 2000, 2005;Cardoso, Borges & Veech 2009). However, the same attentiongiven to taxon diversity (TD) bias due to undersampling andhow to correct it has never been given to other facets of biodi-versity, namelyphylogenetic diversity (PD) andfunctionaldiversity (FD).Taxondiversityisthemostcommonmeasureofbiodiver-sity, usually expressed in terms of species richness where alphaor gamma diversity is concerned. However, underlying the useof TDis the simplistic assumption that the taxa are equally dis-tinct fromone another, disregarding the fact that communitiesarecomposedofspecieswithdierentevolutionaryhistoriesand a diverse array of ecological functions. Thus, the last dec-ade has seen a growing interest in complementary representa-tionsof biodiversity, includingPDandFD(Devictoretal.2010; Cardoso etal. 2014). *Correspondence author. E-mail: pedro.cardoso@helsinki.2014TheAuthors.MethodsinEcology andEvolution 2014BritishEcologicalSocietyMethods in Ecology and Evolution 2014, 5, 452461 doi: 10.1111/2041-210X.12173Phylogenetic diversity takes evolutionary relationshipsbetween taxa into account (Faith 1992) and reects how muchevolutionary history is behind the species constituting the com-munities. Assemblages with identical TD may be considerablydierent with respect to their evolutionary past, depending onhow far the species diverge from their nearest common ances-tor(Webbetal. 2002; Graham&Fine2008). PDhasbeenmeasured based on phylogenetic trees or cladograms, reectingthe amount of phylogenetic information conveyed by theassemblage (Faith 1992). Related measures reecting thedegree of (un)relatedness of taxahave alsobeenproposed(Webbetal.2002;butseeHelmusetal.2007foralternativemeasures).Functionaldiversityquantiescomponentsofbiodiversitythatinuencehowanecosystemoperatesorfunctions(Til-man etal. 2001). We refer here to FD as the amount of bio-logical functionsortraitsdisplayedbythespeciesoccurringingivenassemblages. This takes intoaccount that speciesoftenoverlapintheirtraits,andassuch,theirrelationsmaybedepictedmuchinthesamewayasspeciesarerelatedinaphylogeneticframework. Communities withcompletelydif-ferentspeciescompositionmaybecharacterisedbylowvari-ationinfunctional traits, withunrelatedspecies replacingotherswithsimilarrolesinthenetwork. FDhasalsobeenquantied inmanydierent ways, suchasquadraticentropy(Rao1982),dendrogram-basedmeasures(Petchey&Gaston2002,2006)orthefunctionalhypervolumeoccupiedbytaxa(Cornwell, Schwilk &Ackerly 2006; Villeger, Mason &Mouillot 2008).Despitethewiderecognitionthat TDis oftenunderesti-mated and that estimators are frequently needed to correct forbias,fewstudieshavebeenmadetoverifywhetherthesameproblems are present in PD or FD measures (Walker, Poos &Jackson 2008; Ricotta etal. 2012) and, more importantly, noestimators have been proposed to date. In this study, we focuson the estimation of the alpha component of TD, PDand FD,although beta diversity will probably require a similarapproach. Three main strategies have been followed in the pasttoestimatespecies richness (TD) fromincompletesamples(Sober on &Llorente 1993; Colwell &Coddington 1994; Long-ino, Colwell & Coddington 2002; Gotelli & Colwell 2011): (i)ttingthelog-normaldistributiontospeciesabundancedataand estimating the part of the distribution to the left of the veilline; (ii) tting asymptotic curves to randomised accumulationcurves; and (iii) using nonparametric estimators based on theabundance or incidence of rare species. After hundreds of pub-lished tests with many dierent data sets, some of these alterna-tives were found to perform generally better (although still farfromperfect), namely the latter two(Longino, Colwell &Coddington 2002; Walther & Moore 2005). In this study, weadapt someof themostlyusedandbest-performingspeciesrichness (i.e. TD) estimators to PD and FD measures and testtheiraccuracyandabilitytocorrectforundersamplingbiaswith a variety of theoretical and empirical data sets. We showthat dierent methods can be successfully used for estimatingPD and FD, with a wide application to all kinds of organismsand sampling schemes.Materials andmethodsTwo kinds of data may be used for estimating diversity: (i) abundancedata, which may or may not be divided by sample, and (ii) incidencedata,thatis,thepresenceorabsenceofeachspeciesineachsample.From such data, a number of approaches for estimating true diversityare possible.ESTI MATI ONOFTAXONDI VERSI TYHere, we will focus on two of the mostly used and promising methods.If indeed PD and FDaccumulate with eort (often a measure of spaceor time) in the same fashion as TD, then the approaches used to esti-mate TD willbeuseful,albeit withpossibleadaptations,for PDandFD.Fitting asymptotic curvesUsually sampling is not instantaneous, that is, features (species TD,clades PDand traits FD) accumulate with the increasing number ofindividuals orsamples(Fig.1).Ifatrsttheaccumulationisfast,asmost features are yet to be sampled, the accumulation rate constantlydecreases, as an increasing number of features are already sampled andit becomes harder to nd novelty. Because the accumulation is not ran-dom, but made in a stepwise way, after randomisation of the accumula-tionprocess,itispossibletotasymptoticcurvestothedatapointsand calculate the asymptote, which should approximate the true totaldiversity (observed plus non-observed).One of the mostly used equations is the Clench or MichaelisMentencurve:Sobs aQ1 bQwhere Sobs=observed richness, Q=number of samples (x-axis) and aand b are tting parameters. After nding these parameters, the asymp-tote, that is, estimated richness (S*), is:Fig.1. Randomisedaccumulationofdiversitywithincreasingeort(dots), a tted asymptotic equation (continuous line) and its asymptoterepresenting true diversity (dashed line).2014TheAuthors.MethodsinEcologyandEvolution 2014BritishEcologicalSociety,MethodsinEcologyandEvolution,5,452461Estimatingphylogeneticandfunctionaldiversity 453S abOther equations have been tested with variable success (e.g. negativeexponential, Weibull; Sober on &Llorente 1993; Flather 1996).Nonparametric estimatorsNonparametricestimators, basedontheabundanceorincidenceofrare species, are often considered the best option for estimating speciesrichness with most data sets (Gotelli & Colwell 2011). Among the rstto be suggested for this purpose and still the mostly used are the jack-knife(Heltshe&Forrester1983)andAnneChaosformulas(Chao1984, 1987). The jackknife 1 based on abundance data is:S Sobs S1where S1=singletons, that is, the number of species known froma sin-gle individual in the samples.Its incidence data equivalent further includes a correction factor andis:S Sobs Q1Q 1Q where Q1=uniques, that is, the number of species known froma singlesample.TheChao1(Chao1984)isalsobasedontheabundanceofrarespecies:S Sobs S1S1 12S2 1whereS2=doubletons, that is, thenumberof speciesknownfromexactly two individuals in the samples. A corresponding incidence esti-mator is known as Chao 2 (Chao 1987) and is based on the presence ofrare species in samples:S Sobs Q1Q1 12Q2 1whereQ2=duplicates, that is, thenumber of species knownfromexactly two samples.Recently, Lopez etal. (2012) have proposedacorrectionfactorbased on the percentageofsingletons or uniques in the sampleduni-verse, with the reasoning that this proportion (P) is directly correlatedwithundersampling. Irrespectiveof thefunction, its correctionforabundance estimators is:SP S1 S1Sobs 2 !and for incidence estimators:SP S1 Q1Sobs 2 !Many other nonparametric estimators have been proposed to date,but hereweadapt theseeight (four formulas, withandwithout Pcorrection)withthesamereasoningbeingeasilytranslatedtoothermethods.MEASURESOFPHYLOGENETI CANDFUNCTI ONALDI VERSI TYPhylogeneticdiversity andFDsensulatocanbeexpressedinalargevariety of ways (Mouchet etal. 2010; Schleuter etal. 2010). Allmeasures may be divided into two main groups. First, those that reectoverall diversity or the amount of evolutionary history for PD and theamountoffunctionsforFDcontainedinagivencommunity,whichmay be called phylogenetic and functional richness, respectively. Thesetendtoincreasewith increasing number oftaxa,as new taxa tendtoprovide new branches in the phylogenetic or functional tree or occupynew space in the functional hypervolume. Second, there are measuresthat reect average distance between taxa, which may be called phylo-geneticorfunctionaldispersionordierentiation(Webbetal.2002;Laliberte &Legendre 2010). These are often not inuenced by the num-ber of taxa, being largely insensitive to sampling eort (Weiher, Clarke&Keddy 1998).Giventhat estimatesofPDandFDareespeciallyuseful fortherstgroupofmeasuresandthatourstudyaimstoestimatePDandFDincomparisonwithTDasmeasuredbyspeciesrichness,wewillfocusonmeasuresthatcapturethenotionofrichness.Therefore,inthe following sections, we will measure PD and FD as the sum of theedgelengthofaphylogeneticorfunctionaltree(Faith1992;Petchey&Gaston2002),althoughotherrepresentationscouldbeused,suchas functional hyperspace (see Schleuter etal. 2010; for other measuresof functional richness). Hereafter, PD refers to the standard denitionsensuFaith(1992)coveringbothaconventionalrootinthesenseofout-groupandacommonancestor.NotethatsinceTDcanalsobevisualisedusingatreewitheachtaxonlinkeddirectlytotherootbyanedgeofunitlength(startree), treediagramsprovideacommonbasis for unequivocal estimation and comparison of TD, PD and FD(Cardoso etal. 2014).ESTI MATI ONOFPHYLOGENETI CANDFUNCTI ONALDI VERSI TYHere, we propose three ways of estimating PDand FD, building on theestimatorsoriginallydevelopedforTD:rst,correctingPDandFDvalues based on the completeness of the taxon inventory; second, ttingasymptotic functions to accumulation curves of PDand FD; and third,adapting nonparametric estimators to PDand FDdata.Correcting with taxon inventory completenessInventory completeness (sensu Srensen, Coddington & Schar 2002)can be dened as:c SobsSwhere S* is computed with any of the available methods (estimators).The most straightforward way to estimate PD and FD is probablyto: (i) compute both observed TD and PD or FD; (ii) compute taxoninventory completeness as above; and (iii) correct observed PD or FDwith the completeness values:D Dobscwhere D* and Dobsare PDor FDestimated and observed, respectively.This approach should be particularly ecient if the sampling is maderandomly along the phylogenetic or functional tree. If unique taxa ineither phylogenetic or functional terms are much rarer or more abun-dant, or alternatively much harder or easier to sample than the rest ofthe assemblage, the bias may be large. Very unique parts of the tree willbe disproportionally sampled compared with that remaining, and thismakes the correctionof PDor FDvalues withTDcompletenessdicult.2014TheAuthors.MethodsinEcologyandEvolution 2014BritishEcologicalSociety,MethodsinEcologyandEvolution,5,452461454 P.Cardosoetal.Fitting asymptotic curvesThis approach follows the same strategy as for TD. The idea is to tasymptotic functions to randomised or analytical accumulationcurves of PDor FDvalues (seeNipperess &Matsen2013for ananalytical solution)andtousetheasymptotevalueastheestimate.Arandomisedcurveisgeneratedbytaking1, 2,. . ., nrandomsam-ples, nbeingthetotal numberofsamples. Foreachsamplinglevel,theobservedPDorFDiscalculatedasthebranchlengthalreadysampledfromtheglobal tree(withnsamples). Therandomisationprocedureis repeatedalargenumber of times (werecommendatleast 1000 to guarantee smoothness of the curves) allowing obtainingameanPDorFDforeachsamplinglevel. Infact, theRpackagepicante (Kembel etal. 2010)includes a function, specaccum.psr, thatbuilds randomised accumulation curves for phylogenetic species rich-ness, andsoftware is alsoavailable tocompute analytical curves(pplacer suite matsen.fhcrc.org/pplacer/ or DavidNipperesss Rfunctions davidnipperess.blogspot.com.au). We propose to usesuchcurvestotdierentasymptoticfunctions,suchastheClench,negative exponential or Weibull.Adapting nonparametric estimatorsHere, we propose an adaptation based on phylogenetic or functionaltrees. Givenatree(Fig.2)thatisprogressivelycompletedwiththeaccumulation ofsamples,singleton or doubletonedge length may becalculated as the sum of length of edges represented by a single or twoindividuals, respectively. Conversely, unique or duplicate edges may becalculated as the sum of length of edges represented in a single or twosamples, respectively. Other abundance or incidence classes may be cal-culated similarly. This way, many nonparametric estimators, includingthe eight tested here, may be used to estimate PDor FD.The adaptation of the jackknife estimators from TD to PD and FDisstraightforwardandrequiresnofurtherchanges.AsfortheChaofunctions, because PD and FD are measured in real numbers, the useoftheoriginal functionsisnotpossible. IfS1issmallerthan1, thenumerator S1(S11) will be negative and the estimated diversity will besmaller than the observed. The same is true for Q1. We propose to sub-stitute these unit correction constants by the minimum branch lengthofterminal branchesfoundinthetree. Thisisinfacttheminimumvalue that S1 or Q1 may take. This minimumvalue should exclude nulldistances, as these do not bring any new information. Because in TDtheminimumdistanceis1(ifTDismeasuredinastartreeofunitlength), it is possible to apply these generalisations of the original ChaoformulastoTD,PDorFD.Infact,ifthisminimumvaluechangesfrom1 to close to 0, the formula seamlessly changes from the bias-cor-rectedtotheclassicformoftheChaoestimators(Gotelli&Colwell2011). The Chao 1 adaptation is:S Sobs S1S1 min2S2 min;where min=minimumtaxon, phylogenetic or functional distancebetween any two species, excluding null distances. The correspondingChao 2 adaptation is:S Sobs Q1Q1 min2Q2 minFinally, itshouldbenotedthatusuallynonparametricestimatorsunderestimate diversity when very few samples are used. For that rea-son, it is common to calculate and plot accumulation curves for the esti-mated values in addition to the usual observed diversity curves(Srensen, Coddington & Schar 2002; Lopez etal. 2012). If the esti-mator curve asymptotes, it may be a good indicator of estimator reli-ability. In the case of the estimation of PD or FD, using real numbersinstead of integers (as in TD) means that depending on the particularsamples randomly chosen for building the accumulation curve, particu-larly the rst few samples, some of the estimates may reach unrealisticvaluesandthesewill disproportionatelyinuenceaverageestimates.This is critical for the Chao estimators, as both use the number of dou-bletons or duplicates in the denominator. Instead of using averages asusualforTDaccumulationcurves,weproposetousemedians whenplotting estimated PD or FD accumulation curves calculated with theChaoestimators. Thisoptionshouldprovideverysimilarvaluestoaveraging inmostcases,butavoids spurious inationofestimates inparticular cases.PERFORMANCETESTI NGOFTHENEWESTI MATORSTwo methods may be used to test the performance of the new estima-tors. The rst is to create articial communities with a known numberofspecies,speciesabundancedistributionandspatialdistributionofeach species in a computer-simulated landscape. Articial trees depict-ing theoretical phylogenetic or functional relationships between speciesmust also be created for PD or FD analyses. Then individuals or plotsarerandomlysampled,andthesedataareusedtoestimatediversity.Estimators are compared based on their ability to recover true, known,diversity (Brose, Martinez &Williams 2003).The second method is to use empirical data from areas exhaustivelysampled, so that we may assume that the observed diversity is near thereal one. Randomfractions of the data are extracted fromthe data set,and these are used to estimate total diversity. Estimates are then com-paredwithknowndiversity(Colwell&Coddington1994).Here,weused both methods in performance testing.All analyses wereperformedintheRstatistical environment (RDevelopment Core Team 2013) using the vegan (Oksanen etal. 2011)and spatstat (Baddeley & Turner 2005) packages. An R script to esti-mate TD, PDand FDusing the three proposed methods is given in theSupporting Information (Estimate.r, Data S1).Fig.2. Hypothetical phylogenetic or functional tree (with total phylo-genetic or functional diversity=25) with three species and two sam-ples with abundance of species per sample. S1=singleton edges,S2=doubleton edges, S = edges represented by more than two indi-viduals, Q1=unique edges, Q2=duplicate edges.2014TheAuthors.MethodsinEcologyandEvolution 2014BritishEcologicalSociety,MethodsinEcologyandEvolution,5,452461Estimatingphylogeneticandfunctionaldiversity 455Theoretical data setsWecreatedonearticialcommunitywith10000individualsdistrib-uted in 100 species following a log-normal species abundance distribu-tion. Tosimulate aphylogenetic or functional tree, we randomlyattributed10binaryalleles/traitvaluestoeachspeciesandfromthismatrix built a dendrogramusing UPGMA(although any other methodcould have been used, as this tree was purely random). The individualsof each virtual species were then distributed in three simulated squaresurfaces with varying levels of intraspecic interaction. This way speciesfollowed aggregated, random or uniform (overdispersed) distributionsinspacedependingonwhetherinteractionswerepositive,neutralornegative, respectively. Next, we simulated the sampling of each of thethree communities by superimposing a 10910 grid over each square,each of the 100 cells being a sample. Finally, by randomising the orderofthesamples1000times, webuiltrandomisedtaxonandphyloge-netic/functional diversity accumulation curves based on the sumof thebranch length and the respective estimated values for all levels of under-sampling (from1 to 99 samples).Empirical data setsWe used two empirical data sets previously used by our team for betadiversity partition (see Cardoso etal. 2014 for details). These are bothexhaustive,withknowntotaldiversityvaluesandadequatephyloge-netic and functional data, and represent very dierent organisms andspatial scales.The Atlas of European Mammals (Mitchell-Jones etal. 1999) pro-videdthedistributionofthe160nativemammalsinEurope,eastofroughly 30E, in a 50950km resolution. We built the phylogenetictree for these species by extracting the phylogenetic relationship fromthe world-wide mammal supertree provided by Bininda-Emonds etal.(2007). We tested the ability of the estimators to calculate the PDof allnative European mammals with increasing number of cells, how manysamples/cellswouldbeneededtocorrectlyestimatetotal Europeanmammal PD and how much of the bias of observed PD is eliminated.Because no abundance data are available (and in fact would probablybe uninformative at this scale), we did not calculate abundance-basedestimators.The North-Atlantic Azorean archipelago, with nine islands, presentsamosaicoflanduses, whichreplacetheoncealmosthomogeneouscover of laurel forest (Cardoso etal. 2013). Atotal of 36 sites in the nat-ural forestsofTerceiraIslandweresampledforepigeanarthropodsusing 30 pitfall traps per site (Gaspar, Borges & Gaston 2008). Func-tionalcharacteristicsrelatedtoresourceusewerecollatedforallthearthropod species, and a functional tree was built (see Cardoso etal.2014 for details). Based on the distribution ofthe 28 endemic speciessampled and the respective functional relationships in the global func-tional dendrogram, wethencalculatedtheoverall FDthat canbefound in natural forests. We tested the ability of the estimators to calcu-late the true FDif instead of using 30 pitfall traps we had used anythingfrom 1 to 29 traps per site. We also calculated how many traps per sitewould be needed to correctly estimate total FD and how much of thebias inherent to observed diversity was eliminated.Accuracy measurementThe behaviour of the estimators and their ability to correct undersam-plingbiascanbetestedinanumberof dierent ways(Walther&Moore 2005). To test the new estimators with the empirical data sets,we used the scaled mean square error:SMSE 1A2QXQj1Sj A2where A=real total diversity and Sj=estimated diversity for the jthsample. We chose this measure among the many possible options as itis:(i)scaledtotruediversity,sothatsimilarabsolutedierencesareweighted according to how much they represent of the real value; (ii)scaled to the number of samples, so that values are independent of sam-plesize; (iii)squared, sothatsmall, mostlymeaninglessuctuationsaround the true value are down-weighted; and (iv) independent of posi-tive or negative deviation from the real value, as such dierentiation isusuallynotnecessary. Thisvaluewascalculatedforbothdatasets,using the sample numbers n/29to n/21(nine values) for the Europeandata set, so that the rst, highly biased, part of the curve was heavilyweightedcompared withthe last,mostlyunbiased,partofthecurve,and all 29 values for the Azorean data set.ResultsTHEORETI CALDATASETSFitting asymptotic curvesForillustrativepurposes,weshowtheresultsofcurvettingwith only ve samples (Fig.3). Even with only 5%of the theo-reticalassemblagessampled,thepercentageofobservedspe-cies is higher than 80%in all cases. However, most of the real-worldassemblagesareaggregated,andthisis,probably,theworst-case scenario for sampling and estimation of diversity assuchassemblagesrequiremuchmoreeort tobethoroughlysampled than assemblages where species are randomly or uni-formlydistributedinspaceandhencemuchmoreaccessiblewhensamplingeortconcentratesinfewsitesormicrohabi-tats. But even in aggregated assemblages, the asymptote of theClench function reaches a value very close to the real.Nonparametric estimatorsIn general, nonparametric estimators are capable of correctingmuchofthebias of observeddiversity,with asimilar behav-iourfortaxon, phylogeneticorfunctional diversity(Fig.4).Their behaviour does, however, dier, with the jackknife for-mulas for incidencedataoftenovershootingwithveryfewsamples and both the Jack and Chao formulas for abundancedata underestimating with aggregated communities. The adap-tations here proposed for PD and FD are, however, at least asecient as the original formulations for TD.EMPI RI CALDATASETSPhylogenetic diversity of European mammalsFor the nonparametric estimators, between 100 and 500 of the2000cellsareneededtoreachthetruespeciesrichness(TD)value (Fig.5), with both jackknifes being particularly ecient.For PD, the correction based on taxon inventory completenessis remarkablyecient, withbetween5and100cells beingneeded to reach the true values, with the P-corrected versions2014TheAuthors.MethodsinEcologyandEvolution 2014BritishEcologicalSociety,MethodsinEcologyandEvolution,5,452461456 P.Cardosoetal.of estimators performing particularly well. The direct nonpara-metricestimationapproachrequires about 100cells for allfunctions. The Clench curve clearly underperforms in all casescomparedwiththenonparametricestimators.Regardingtheaccuracy of the dierent methods, all estimators largelyreducedthebiasoftheobservedvalues(AppendixS1, TableS1, Supporting information).Functional diversity of Azorean arthropodsInallcasestheestimatorspredicthigherdiversitythaneec-tively observed in the 36 sampled sites, which may indicate thatinfactoursampleduniverseisstillincomplete(Fig.6).TheFD estimates based on TD completeness reach the asymptotevery quickly, at about ve traps, while in all other cases about1520 samples (traps) are needed (Fig.6). Regarding the accu-racy, all estimators had rather similar abilities to correct bias,although the Clench method seems to perform slightly better(Appendix S1, Table S2, Supporting information). As theSMSEvalues are based on the total observed diversity, and thismay be underestimated, these values should, however, be inter-preted with caution.DiscussionThis study sought to explore the impact of undersampling onobservedTD,PDandFDandtoadaptsomeofthemostlyusedandbest-performingspeciesrichnessestimatorstoPDand FDmeasures.In general, the simple correction of observed PDor FDwithtaxon inventory completeness values seems to performsurpris-ingly well. This approach even outperforms TD estimates, forwhich the estimators were specically built. This may be due totwo factors in combination. First, TDestimators are known toconsistently undercorrect with a lownumber of samples(OHara 2005; Cardoso2009; Lopez etal. 2012). Second,observed PDand FDare intrinsically less biased than TDwitha lownumber of samples, as the rst taxa sampled add on aver-agemorediversitythanthelattertaxa, whichshouldsharepartsofthetreewiththeonespreviouslysampled. Thetwoeects combined mean that with a lownumber of samples, it ispossible toveryeectivelycorrect PDor FDvalues. Thiswould not be the case if species richness estimators were moreecient with low sampling, in which case estimated PD or FDwould initially overshoot the true value.Bothasymptoticfunctions andnonparametricestimatorsusedforPDorFDseemtoperformaswellasforTD, forwhichtheywererstcreated. Althoughresultsarevariable,thismeansthatinmanyempiricaldatasetsatleast7080%completenessisneededtohavereliableestimates(Srensen,Coddington&Schar2002; Cardoso2009). Reachingsuchlevels is, however, not guaranteed in many studies, particularlywhen dealing with hyperdiverse taxa, such as most arthropods(Coddington etal. 2009). Overall, better estimators are neededFig.3. Randomised accumulation curves for observed diversity up to ve samples (full black line) and the respective tted Clench functions extrapo-lated up to 100 samples (dotted black line). Theoretical assemblages following aggregated, random and uniform spatial distributions were built, aswas a random phylogenetic/functional tree connecting all species (see text for details). The true diversity of the assemblages is also shown (full greyline).2014TheAuthors.MethodsinEcologyandEvolution 2014BritishEcologicalSociety,MethodsinEcologyandEvolution,5,452461Estimatingphylogeneticandfunctionaldiversity 457forTD, PDandFD, but inall cases, thebiasinherent toobserved diversity is in fact reduced, so their utility is certainlyconrmed.Weshouldalsomentionthatwedidnottryadaptingtheanalytical estimators of unconditional variance existingforaccumulationcurves of nonparametricformulas (Gotelli &Fig.4. Randomised accumulation curves for observed and estimated diversity. Theoretical assemblages following aggregated, randomand uniformspatial distributions were built, as was a random phylogenetic/functional tree connecting all species (see text for details). The true diversity of theassemblages is also shown (target). Note that the x-axis is in log10 scale to facilitate comparisons with very low sampling levels. Sobs=observeddiversity, Jack 1 (P) abund=jackknife 1 estimator with abundance data, Jack 1(P) inc=Jackknife 1 estimator with incidence data, Chao 1(P) =Chao estimator with abundance data, Chao 2(P)=Chao estimator with incidence data (P=P-corrected version for all Jack and Chao formulas).Fig.5. Observed and estimated species richness (TD) and phylogenetic diversity (PD) of European mammals with the randomised accumulation ofcells in Europe. The curves in the middle panel were calculated correcting the observed PDvalues using the completeness of TD(see text for details).The curves in the right panel were calculated either tting the Clench asymptotic function to each point in the observed PDcurve or using nonpara-metricestimators. Notethat thex-axis is inlog10scaletofacilitatecomparisons withverylowsamplinglevels. Sobs=observeddiversity,Clench=ttedasymptoticcurve,Jack1(P)inc=Jackknife1estimatorwithincidencedata,Chao2(P)=Chaoestimatorwithincidencedata(P=P-corrected version for both Jack and Chao formulas).2014TheAuthors.MethodsinEcologyandEvolution 2014BritishEcologicalSociety,MethodsinEcologyandEvolution,5,452461458 P.Cardosoetal.Colwell 2011). Only conditional variances are possible to cal-culate, and these approach 0 when all samples are included.ALTERNATI VEMETHODSAlthough we have focused on asymptotic functions and non-parametric estimators, taxa richness (TD) has been estimatedthroughothermeans. Asymptoticfunctionshavebeencriti-cisedfor not relyingonastrongtheoretical basis, beingastrictly phenomenological method. Dierent functions may ta curve equally well, yet results may vary dramatically(Sober on&Llorente1993), andresidualsoftenreveal thatmanyfunctions donot correctlyt curve shapes (OHara2005). For these reasons, Gotelli & Colwell (2011) do not rec-ommend the method. Possible better alternatives exist (Colwelletal. 2012), but these are probably much harder to adapt forPDand FD.Another often used approach for species richness estimationis to t a species abundance distribution to a truncated para-metric distribution and estimate the portion to the left of theveil line. This unseen part of the distribution corresponds totheundetectedspecies.However,PDandFDdonotfollowsuch theoretical abundance distributions. In fact, it would beunclear to dene what abundance is in this context.GOI NGBEYONDDI VERSI TYESTI MATESBesidestheobviousappeal andutilityof knowingthetruediversity of an assemblage fromincomplete samples, the use ofestimators may present further advantages. Comparisonsbetweensites, regionsandtimeperiodsareonlypossibleifeither the sampling is complete, which often is not the case, orthe sampling eort is equivalent and sucient enough to allowsensible comparisons. The way this problemhas been typicallyFig.6. Observed and estimated taxon (TD) and functional (FD) diversity of endemic Azorean arthropods with the randomised accumulation oftraps. The curves in the middle panels were calculated correcting the observed FD values using the completeness of TD (see text for details). Thecurves in the right panels were calculated either tting the Clench asymptotic function to each point in the observed FDcurve or using nonparametricestimators. Sobs=observed diversity, Clench=tted asymptotic curve, Jack 1(P) abund=jackknife 1 estimator with abundance data, Jack 1(P)inc=jackknife 1 estimator with incidence data, Chao 1(P) =Chao estimator with abundance data, Chao 2(P)=Chao estimator with incidencedata (P= P-corrected version for all Jack and Chao formulas).2014TheAuthors.MethodsinEcologyandEvolution 2014BritishEcologicalSociety,MethodsinEcologyandEvolution,5,452461Estimatingphylogeneticandfunctionaldiversity 459corrected is through rarefaction. Rarefaction allows the com-parisonof diversityamongassemblages onanequal-eortbasis (Gotelli & Colwell 2001; Chao &Jost 2012). Rarefactionof PDor FD, either through randomisation procedures or ana-lytically(Walker, Poos&Jackson2008; Ricottaetal. 2012;Nipperess &Matsen2013), is analternative toestimationwhen total diversity values are not needed, but the objective isto compare dierent assemblages with similar eort. Rarefac-tiondoes,however,preventonefromknowingtruediversitynumbers;estimatingthese,ifwithoutsignicantbias,shouldbe a preferable option.Another advantage of using estimated in lieu of observed orrareed diversity is clear in studies relating TD with either PDor FD. Many studies try to measure the level of redundancy inassemblagesbycomparingdierencesinTDbetweenassem-blages with respective dierences in PDor FD(Mayeld etal.2010; Gerisch etal. 2012; Whittaker etal. in press). However,asseen(AppendixS1,Supportinginformation),PDandFDare less biased than TDin the presence of undersampling. Thismeans that, due to pure sampling eects, phylogenetic or func-tionalredundancymaybeunderestimatedinmanycases,asPD and FD may reach an asymptote much before TD. Obvi-ously, this use of the estimators does prevent the application ofour rst approach, the reliance on TD completeness values, asall dimensions are corrected by the same proportion.LI MI TSANDFUTUREPERSPECTI VESIn this study, we choose to focus on trees or dendrograms, notonly because trees allow a straightforward adaptation of exist-ing estimators but also because they allow the comparison ofTD, PDand FDunder the same representation (Cardoso etal.2014). Although there is no other obvious alternative for PD,the use of dendrograms to compute FDhas been the subject ofstrongdebate(Schleuteretal.2010).Weemphasisethattheuser needs to consider that the choice of the distance and theclustering method may greatly aect the FD values obtained;some publications and associated scripts are available to guidetheresearcherinthisprocess(Mouchetetal.2008;Merigot,Durbec & Gaertner 2010). The framework proposed here canalso be adapted to other representations such as multidimen-sional hypervolumes, acommonrepresentationfor FD. Itshould be noted, however, that the adaptation is not asstraightforward as with dendrograms. Due to the nature of thecalculation of convex-hull volumes, each sample may occupy arelatively small portion of the functional space, yet when con-sidering several samples simultaneously, the space in betweensamples may be occupied, even when none of the samples indi-viduallyoccupiesit.Towhatextentthisshortcomingcausesbiases in practice remains to be studied. Additionally, the cal-culationoftheconvex-hull volumeforeachsamplerequiresthat the number of species always exceeds the number of traits(Laliberte &Legendre 2010), which, in turn, constrains the usereither to reduce the dimensionality of the functional space or toremovespecies-poorsamples. Inanycase, FDasmeasuredwith hypervolumes can always be estimated using the correc-tion with TDcompleteness or tting asymptotic functions.ConclusionsUsing a range of both theoretical and empirical examples, weshowthat current approachestoestimatingtaxondiversity(TD, usuallyspeciesrichness) areasecient orevenmoreecient whenusedtoestimatephylogenetic(PD) or func-tional (FD) diversity. The frameworkusedhere combinesTD,PDandFDintoasinglerepresentation,oeringatoolfor future developments involving these dierent facets ofbiological diversity. A number of topics are in need of futuredevelopment, namely (i) the comparison of dierentapproaches to estimating PD and FD, not only the ones pre-sentedherebutalsootherasymptoticfunctionsornonpara-metric estimators; (ii) verication of the circumstances inwhich each approach is preferable, if this is predictable at all;and(iii) the development of estimators specicallyfor PDandFDdata,potentiallysuperiortothecurrent adaptationsof TD estimators.AcknowledgementsWethankNickGotelli,JackLongino,JonCoddington,DavidNipperessandDan Faith for comments on previous versions of the manuscript. Clara Gasparcompiled all the functional data on Azorean arthropods. The Societas EuropaeaMammalogicaandparticularlyTonyMitchell-Jones providedthe EuropeanMammals Atlas data.Data accessibilityRscripts: uploaded as online supporting information (Estimate.r).FundingAzorean data derived from the DRCT projects M.2.1.2/I/003/2008 and M.1.1.2/FRCT. 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(inpress) Functional biogeography of oceanic islands andthe scal-ing of functional diversity in the Azores. Proceedings of the National Academyof Sciences USA.Received 3 December 2013; accepted 26 February 2014Handling Editor: Daniel FaithSupporting InformationAdditional Supporting Information may be found in the online versionof this article.Appendix S1. Scaled mean square error (SMSE) of each descriptor ofspecies richness (TD)orphylogenetic diversity(PD)ofaccumulationcurves for European mammals (only incidence estimators are shown)and Azorean arthropods.Table S1. Scaled mean square error (SMSE) of each descriptor of spe-cies richness (TD) or phylogenetic diversity(PD) of accumulationcurves for European mammals (only incidence estimators are shown).Table S2. Scaled mean square error (SMSE) of each descriptor of spe-cies richness (TD) or functional diversity (FD) of accumulation curvesfor Azorean arthropods.Data S1. An Rscript to estimate TD, PDand FD.2014TheAuthors.MethodsinEcologyandEvolution 2014BritishEcologicalSociety,MethodsinEcologyandEvolution,5,452461Estimatingphylogeneticandfunctionaldiversity 461