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Estimating the Diets of Animals Using Stable Isotopes and a Comprehensive Bayesian Mixing Model
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Title: Estimating the Diets of Animals Using Stable Isotopes and a Comprehensive Bayesian Mixing Model
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Creator: Hopkins, John
Ferguson, Jake
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Publication Date: January 3, 2012
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Abstract: Using stable isotope mixing models (SIMMs) as a tool to investigate the foraging ecology of animals is gaining popularity among researchers. As a result, statistical methods are rapidly evolving and numerous models have been produced to estimate the diets of animals—each with their benefits and their limitations. Deciding which SIMM to use is contingent on factors such as the consumer of interest, its food sources, sample size, the familiarity a user has with a particular framework for statistical analysis, or the level of inference the researcher desires to make (e.g., population- or individual-level). In this paper, we provide a review of commonly used SIMM models and describe a comprehensive SIMM that includes all features commonly used in SIMM analysis and two new features. We used data collected in Yosemite National Park to demonstrate IsotopeR's ability to estimate dietary parameters. We then examined the importance of each feature in the model and compared our results to inferences from commonly used SIMMs. IsotopeR's user interface (in R) will provide researchers a user-friendly tool for SIMM analysis. The model is also applicable for use in paleontology, archaeology, and forensic studies as well as estimating pollution inputs.
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Funding: Publication of this article was funded in part by the University of Florida Open-Access Publishing Fund and J.M. Ponciano (Department of Biology, University of Florida). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript. No additional external funding was received for this study.
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EstimatingtheDietsofAnimalsUsingStableIsotopes andaComprehensiveBayesianMixingModel JohnB.HopkinsIII 1 ,JakeM.Ferguson 2 1 DepartmentofEcology,MontanaStateUniversity,Bozeman,Montana,UnitedStatesofAmerica, 2 DepartmentofBiology,UniversityofFlorida,Gainesville,Florida, UnitedStatesofAmerica Abstract Usingstableisotopemixingmodels(SIMMs)asatooltoinvestigatetheforagingecologyofanimalsisgainingpopularity amongresearchers.Asaresult,statisticalmethodsarerapidlyevolvingandnumerousmodelshavebeenproducedto estimatethedietsofanimals—eachwiththeirbenefitsandtheirlimitations.DecidingwhichSIMMtouseiscontingenton factorssuchastheconsumerofinterest,itsfoodsources,samplesize,thefamiliarityauserhaswithaparticularframework forstatisticalanalysis,orthelevelofinferencetheresearcherdesirestomake(e.g.,population-orindividual-level).Inthis paper,weprovideareviewofcommonlyusedSIMMmodelsanddescribeacomprehensiveSIMMthatincludesallfeatures commonlyusedinSIMManalysisandtwonewfeatures.WeuseddatacollectedinYosemiteNationalParktodemonstrate IsotopeR’sabilitytoestimatedietaryparameters.Wethenexaminedtheimportanceofeachfeatureinthemodeland comparedourresultstoinferencesfromcommonlyusedSIMMs.IsotopeR’suserinterface(inR)willprovideresearchersa user-friendlytoolforSIMManalysis.Themodelisalsoapplicableforuseinpaleontology,archaeology,andforensicstudies aswellasestimatingpollutioninputs. Citation: HopkinsJBIII,FergusonJM(2012)EstimatingtheDietsofAnimalsUsingStableIsotopesandaComprehensiveBayesianMixingModel.PLoSONE7(1): e28478.doi:10.1371/journal.pone.0028478 Editor: LyleKonigsberg,UniversityofIllinoisatChampaign-Urbana,UnitedStatesofAmerica Received January17,2011; Accepted November9,2011; Published January3,2012 Copyright: 2012Hopkins,Ferguson.Thisisanopen-accessarticledistributedunderthetermsoftheCreativeCommonsAttributionLicense,whichpermits unrestricteduse,distribution,andreproductioninanymedium,providedtheoriginalauthorandsourcearecredited. Funding: PublicationofthisarticlewasfundedinpartbytheUniversityofFloridaOpen-AccessPublishingFundandJ.M.Ponciano(DepartmentofBiology, UniversityofFlorida).Thefundershadnoroleinstudydesign,datacollectionandanalysis,decisiontopublish,orpreparationofthemanuscript. Noadditional externalfundingwasreceivedforthisstudy. CompetingInterests: Theauthorshavedeclaredthatnocompetinginterestsexist. *E-mail:jbhopkins3@gmail.com Introduction Stableisotopeswerefirstusedtoinvestigatetheforagingecology ofanimalsinthe1970s[1–5].Earlystudiesusedstableisotope analysis(SIA)todeterminetherelativeimportanceoffoodsources toanimalsbycomparingdistributionsofisotoperatios(expressed asisotopevalues;derivedbelow)foranimaltissuestothefoods theyconsumeaftercorrectedforfractionation(thesortingof isotopesduringnaturalbiochemicalprocesses)—atechnique primarilyusedwhenfoodsourceshaddistinctlydifferentisotope values(e.g.,C 3 andC 4 plants,orpreythatdifferintrophiclevel) [2,6].Isotopevalues(e.g., TM X, TM Y)areexpressedindelta( TM ) notationaspermil( % )units(orpartsperthousand): d X ~ R sample R s tan dard { 1 1000 whereRistheratioofheavytolightisotopes(e.g., 13 C/ 12 Cor 15 N/ 14 N)inthesampleandthestandard[7].Sampleswitha lowerratioofheavyisotopesrelativetothestandardwillyielda negativevalueandsampleswithhigherratioswillhaveapositive value. Forthepastfewdecades,SIAhasgainedpopularityamong ecologists(e.g.,[8–13]).Inparticular,stableisotopemixture models(oftencalledmixingmodels;hereinafterSIMMs)are commonlyusedtoestimatetherelativecontributionofassimilated dietarysourcestothetissuesofanimals(i.e.,theconversionoffood nutrientsintotissuesbytheprocessesofdigestionandabsorption), andifcertainassumptionsaremet(Table1),thedietsofanimals. Euclidiandistanceformulaswereusedinsomeearlystudies(e.g., [14–18]);however,thesemethodsdidnotprovidecorrectsolutions forobservedandsimulateddata[19].Specifically,theseEuclidean distancemodelsfailedtopreservemassbalance,anapplicationof thelawofconservationofmasswhichstatesthattheproportional assimilateddietarycontributions(mass)flowingintoanorganism orpopulationareconstrainedtosumtoone.Recently,variantsof mass-balancemodelshavedevelopedrapidly[19,20].Although theprofusionofSIMMs(manyofwhicharediscussedinthis paper)indicatestheimportanceofthisfieldtoecologists,current modelsrequireresearcherstomaketradeoffs(Table1)when choosingonemodeloveranother. Allmodelsdiscussedinthispaperusethesamebasic methodologyforestimatingproportionalsourcecontributionsto thedietsofanimals.Forexample,aduelelement(X,Y),threesource,mass-balance,linearmixingmodelisdescribedbythe followingequations[20]: d X m ~ f 1 d X 1 z f 2 d X 2 z f 3 d X 3 d Y m ~ f 1 d Y 1 z f 2 d Y 2 z f 3 d Y 3 1 ~ f 1 z f 2 z f 3 1 Thissystemofthreeequationsyieldsthreeunknownproportional sourcecontributions( e 1 e 2 e 3 )foramixture(m)when TM Xand TM Yvaluesareknownformixturesandsources(thelatteradjusted toaccountforisotopicdiscrimination;describedbelow). PLoSONE|www.plosone.org1January2012|Volume7|Issue1|e28478

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Table1. AcomparisonofSIMMassumptionsandfeaturesamongcommonlyusedSIMMs.ModelsIsotopeRSIAR Semmens etal.2009MixSIRIsoConcIsoErrorIsoSource SIMMassumptions: Elementalconcentration(e.g.,[C]and[N]) ofalldietaryitemsareequal N Differentsourceconcentrations fordietarysources XXX Elementsareassimilatedwith thesameefficiency N Differentassimilationefficiencies fordietarysources XYY Notissue-dietdiscrimination N Variationassociatedwithpredicted discriminationfactors XXXX N Includesafixed‘‘discriminationerror’’ term(calculated apriori ):errorassociated withtheregressionmodelusedtopredict discriminationfactors X Noisotopicrouting N Differentialallocationofisotopically distinctdietarysourcestodifferenttissues OtherSIMMfeatures: UsesaBayesiananalyticalframeworkXXXX UsesafullyBayesianapproachbX Samplingprocedureusedto estimateparameters MCMCMCMCMCMCSIRMLMLML Usesrawdata(notparameterestimates ofrawdata)tosimultaneouslyestimate parameters(randomvariables):dietary sources(includingisotopiccorrelation, variation),measurementerror, proportionalsourcecontributionsat thepopulation-andindividual-levelbX Measurementerror:variationassociated withSIA:samplepreparationerrorand errorduringmassspectrometry;applied toeachobservationinthestudy X Y Sourceprocesserror:inherent isotopicvariationofthesampledsource (i.e.,withinandbetweenindividualplants andanimalsofthesamespeciesortaxa) XXXXX Mixtureprocesserror:inherentisotopic variationinasub-sampledtissue(e.g., non-homogenizedhairs,feathers,claws fromthesameindividual)and/orsample ofmixtures(e.g.,population) XXXXXX Correlationofisotopevaluesinsources: accountsforthelinearrelationshipamong isotopevaluesfordifferentelements X X AresidualerrortermXXX Individual-levelsourceestimationusing hierarchicaldesign XX Priorinformationassociatedwithsources (e.g.,sourceproportions,distributionof isotopevalues,elementalconcentrations) andmixtures(e.g.,measurementerror) XXXX Calculatesproportionaldietarysource estimateswhen n + 1sources XXXXaX Fourmixingmodelassumptions(italics)commonlyviolatedwhenestimatingtheproportionaldietarycontributionofsourcestothedietsofanimals ,andthemodel featurethataddresseseachviolatedassumption.AlistofotherfeaturesincludedinSIMMsandtheirdefinitions.Xdenotesthemodeladdressesthea ssumptionor includesthefeatureandYindicatesthefeatureisnotexplicitlyincluded(e.g.,modelmayaccountforerrorusinganarbitrarytolerancemeasure). MCMC(Markovchain MonteCarlo),SIR(sequentialimportanceresampling),andML(maximumlikelihood)denotessamplingmethodusedwhenestimatingparameters. IsotopestoEstimateDiet:ReviewandModel PLoSONE|www.plosone.org2January2012|Volume7|Issue1|e28478

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Inthefollowingsections,wediscusstheSIMMscommonlyused toestimatedietaryparametersandfollowthisreviewwithdetails aboutourcomprehensiveSIMM,IsotopeR.FrequentistSIMMsIsoError .PhillipsandGregg[21]refinedtheapplicationof linearmass-balanceprocedures(equationset1)withIsoError. ThisSIMMcanbeappliedtosystemswherethenumberof sourcesdonotexceedn + 1(n=numberofisotopesystems); however,whensourcesdoexceedn + 1,thesystemofequationsis underdeterminedandthemodelcannotbeused.IsoError calculatesdeterministicsolutionsandallowsausertheabilityto incorporatetheprocesserrorandtheisotopiccorrelationin sourcesandmixtures(Table1). Isoerrordoesnotaddressmanyoftheassumptions(Table1) thatmaybeviolatedwhenestimatingdietsusingSIMMs [19,22,23].Inaddition,neitherIsoErrornorthemassbalance equations(equationset1)areconstrainedtoyieldproportional sourcecontributions( f variablesinequationset1)intheinterval (0,1).Therefore,whendatafalloutsidetheisotopicmixingspace (theareaorvolumecontainedinthespaceformedbylines connectingthesourcesinmultivariateisotopespace)becausean importantfoodsourcewasoverlooked,thewrongdiscrimination factorwasappliedtoasource,oramixingmodelassumptionwas violated[24],nonsensicalnegativeproportionsarecalculatedfor dietarycontributions.IsoConc .Moststableisotopemixingmodelsassumethatthe elementalconcentrationsofdietaryitemsareequal.Althoughthis assumptionisvalidformanycarnivoreandherbivores,itisoften violatedforomnivoreswhofeedonavarietyofdietarysourcesat differenttrophiclevels[24].IsoConcwasdevelopedtoestimate thecontributionofeachsourcetothedietsofanimalsbyassuming asourcecontributionisproportionaltotheassimilatedbiomassof thesourcemultipliedbytheelementalconcentration(e.g.,%C, %N)ofthesource[24].Thismodelwasthefirsttotransforma polygonalmixingspacetoacurvedmixingspace[24]. Standardlinearmixingmodelsandtheexamplespresentedin PhillipsandKoch[24]assumedthatallsourcesareequally digestible.Inresponse,Robbinsetal.[25]pointedouttheneedto considerdigestibilitywhendeterminingtheelementalconcentrationsofsources.Inreply,KochandPhillips[26]calculatedthe digestibilityofmacronutrientsinfoodsourcesandincludedthe correctedelementalcontributionsofthesesourcesintheirdiet estimation.Byincorporating‘‘concentrationdependence’’and explicitlyincludingthedigestibilityofsourcesintheircalculation, thisSIMMmadeasignificantstridetowardsestimatingmore accuratedietaryparameters[26].However,unlikeIsoError, IsoConcdoesnotallowtheusertoincorporatevarioussources oferrorinherenttoSIMManalysis.IsoSource .IsoSourcewasdevelopedtocalculatethefrequency andrangeofpotentialsourcecontributionsinsituationswherethe numberofsourcesexceedsn + 1[27].Usingthestandardlinear mixingmodel,IsoSourcesystematicallycreateseachcombinationof possiblesourcecontributions(thatsumto1.0)byacertain increment(e.g.,0.01).Next,themodelpredictsmixtureisotope valuesforeachcombinationusingsourceisotopevalues(means).If thesepredictedvaluesfallwithinacertaindesignatedmassbalance tolerance(e.g., 6 0.1 % ;whichaccountsfortheerrorassociatedwith measurementandsourcevariability)thenthecombinationis consideredafeasiblesolution;PhillipsandGregg[27]suggested reportingthedistributionoffeasiblesolutions. Thismodelcanbehelpfulatinferringpossibledietcompositions whenauniquesolutioncannotbecalculated,butithaslimits wheninvestigatingmanyecologicalquestions[28].Inparticular, eachfeasiblesolutionisnomoreprobablethananother;therefore, theresultsaredifficulttointerpret—especiallywhentherangeof certainsourceproportions(minimumandmaximumvalues selectedfromthesolutionsetforaparticularsource)iswide (e.g.,0.1–0.9).BayesianSIMMsBayesianSIMMsallowecologiststofitprobabilitymodelsto isotopicdata.Thesemodelscanincludevarioussourcesof uncertainty,greaterthann + 1sources,priorinformation,anda hierarchicalstructureinaflexibleandintuitiveestimation framework.Specifically,theseBayesianmodelsallowusersto efficientlyestimatenumerousparameterswhileavoidingcalculationofmultidimensionalderivatives,asinlikelihoodmethods. SeveralBayesianSIMMshavebeenusedtoestimateproportionaldietarycontributionsatthepopulation-[29,30]and individual-level[31].Theearliestmodel,MixSIR(v.1.0.4)[29], estimatesthejointposteriorprobabilityofsourcesusedbyanimals (reportedasmarginaldistributionsforeachdietarysource contribution)byimportancesampling(lessefficientsampling methodthanMarkovchainMonteCarlosampling)and incorporatesthefollowingisotopicinformationinthemodel:(1) sourcemeanandstandarddeviation,(2)tissue-dietdiscrimination factormeanandstandarddeviation,(3)mixturedata(single consumerorsampledpopulation),and(4)aDirichletprioronthe proportionalestimators(recommendedbyJacksonetal.[32]and incorporatedinSemmensetal.[31]).AlthoughMixSIRmay calculatereasonabledietaryestimatesinsomecases,itscredible intervalsmaybetoonarrowbecausethemodeldoesnotaccount forvariationamongindividualsandothersourcesoferror (Table1). Currently,twootherBayesianSIMMsarecommonlyused [30,31].Semmensetal.[31]builtthefirsthierarchicalBayesian modeltoaccountforintra-populationvariabilityinresourceuse whenestimatingthedietofapopulation(hereinafterSemmenset al.model).Thismodelisveryusefulbecauseitallowsresearchers toestimatedietsatboththepopulation-andindividual-level.In general,hierarchicalmodelsareusedtomakesuchindividual-level inferencepossible;however,difficultiesmaypersistwhenestimatingindividualdiets.Specifically,thesehierarchicalmodelsuse informationfromthepopulation-leveltoestimateindividualdiets; therefore,whenthepopulationsamplesizeislarge,individual estimateswillbepulledtothepopulationmean[33].Currently,it isunknownwhattheidealsamplesizeisforindividualswhen makingindividual-levelinference.However,itiscertainthatthe populationhasamajorinfluenceonindividualdietestimatesand repeatedmeasuresforindividualswillimproveinference[33]. aXdenotesthatthemodelprovidessolutionswhensourcesexceed n + 1,butsolutionsarenotcomparabletoothermodels(i.e.,outputlistsrangesofpotential solutions,notparameterestimates).bXindicatesWardetal.(35)wasthefirststudytousethisapproach.However,thismodel(35)hasrecentlybeenintroduced;therefore,ithasnotbeenc ommonlyused. doi:10.1371/journal.pone.0028478.t001 Table1. Cont. IsotopestoEstimateDiet:ReviewandModel PLoSONE|www.plosone.org3January2012|Volume7|Issue1|e28478

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Anothermodel,theSIARmodel[30]—originallydevelopedas anRpackage[34]andfirstdescribedbyJacksonetal.[32]— allowsausertoincorporateunequalelementalconcentrationsin sourceswhenestimatingthedietsofanimalsatthepopulationlevel.AlthoughthesenewBayesianmodelsprovidereasonable estimatesforproportionaldietarycontributions,theylackthe abilityperformananalysisthatincorporatesbothconcentration dependenceandindividual-levelestimationsimultaneously. Here,weexploretheassumptionsassociatedwithSIMManalysis, combineSIMMfeatures(i.e.,componentsofthemodelexpressed inmathematicalterms),anddeveloptwonewfeaturesforour comprehensiveSIMMmodelcalledIsotopeR.Weusethe hierarchicalmodelstructureofSemmensetal.[31]andthe concentrationdependenceformulationoriginallydevelopedby Phillips&Koch[24]asthefoundationforourmodel,while incorporatingallotherSIMMfeaturestomoreaccuratelyinfer proportionaldietcompositions(Table1).Inaddition,weuseafully BayesianapproachsimilartoWardetal.(35)tojointlyestimate parameters.Jointestimationisusefulwhenestimatingmultiple dependentquantitiesbecauseitaccountsfortheinherentuncertainty associatedwiththejointestimationprocess.Notaccountingforthis uncertaintycanleadtooverlyprecisecredibleintervals. WevalidatedIsotopeRbyestimatingtherelativecontributionof sourcestothedietsofmalefood-conditioned(FC;[36])blackbears ( Ursusamericanus )sampledinYosemiteNationalPark(YNP).Our purposewastouserealdatatoestimatedietaryparametersusing IsotopeR,nottoaccuratelyestimatetherealdietsofYNPblack bears.Wealsoexaminedtheeffectofeachfeatureoninferenceby systematicallyremovingthemfromthemodelindependently. Lastly,wecomparedIsotopeRestimatestothosefromother frequentlyusedmodels.Methods SamplingMixtures.YosemiteNationalParkWildlifeManagement stafflive-capturedFCblackbearsprimarilyinYosemiteValley formanagementpurposesfromAugust2005throughSeptember 2007(TableS1).Theycapturedandimmobilizedbearsinculvert trapsaccordingtoParkServiceprotocol.Theycollectedbear tissuesinaccordancetoWildlifeManagementprotocol.Forhair, theycollectedtenormorefull-lengthguardhairsfromalongthe spinesorupperlimbsofbearsduringspringandearlysummeror fromthelowerlimbsorflanksinlatesummerandfall.We assumedhairscollectedduringspringandearlysummermonths weregrownthepreviousyear,whereashairscollectedinthefall weregrownthecurrentyear[37].Sources .Wecollectedthefollowingbearfoodsopportunisticallyin2007becausetheywereidentifiedbypreviousdiet studies(i.e.,fecalanalysis)asbeingimportantnaturalfoodsourcesfor bearsthroughoutYNP[38,39]:acorns(Quercuskelloggii,Quercus wislizenii),manzanitaberries(Arctostaphylosspp.),grass(Agrostis spp.),forbs(Trifoliumspp.,Lupinusspp.,Montiaspp.),andanimals [ants(Formicida),wasps(Vespidae),bees(Apidae),termites (Isoptera),andmuledeer(Odocoileushemionus)](TableS2).We usedtheisotopevaluesforthesefoodstoestimatetheisotopic signatureofnaturalsources(100%plantdiet,100%animaldiet). Wecollectedhumanhairsamplesin2009fromfloorclippings attwosalonsandonebarbershopinSt.Louis,MO(n=20;Table S3);collectingthesesamplesfromthegarbagedidnotrequirean ethicspermit.Wecomparedisotopicresultsfrom2009toresults froma2004nation-widesurveyofhumanhair(n=52)[40].We foundthatthetwosampleswereisotopicallyindistinguishable (2004: d13C( x x )= 2 16.9 6 0.8, d15N( x x )=8.8 6 0.5;2009:TableS3; t= 2 0.79,df=71.62,P=0.43);therefore,wepooledsamplesto formthehumanfoodaggregate(i.e.,100%humanfooddiet; Fig.1,Tables2A&C).Weassumedthatbearson100%human fooddietwouldbeisotopicallysimilartohumansbecauseboth humansandbearsaremonogastricomnivores;thus,itislikelythat theydiscriminateagainst14Nand12Cbyasimilarmagnitude. Weestimatedtheelementalconcentration([C]and[N])ofthe averagehumandietintheUnitedStatesbyanalyzingnutrientdata fromtheUSDANationalNutrientDatabase(NDB:http://www. nal.usda.gov/fnic/foodcomp/search/;TableS4).First,wedeterminedamountofdigestibleCandNinsamplesfromeachfood group(n $ 3fooditems).Thenweweighedthefoodgroupbasedon thefractionalcontributionsofthesefoodgroupstothedietsof humans[41].Lastly,weusedtheweightedvaluestoestimatethe averagedigestible[C]and[N]forhumanfoods(TableS4).Weused theseestimatestoconstructtheisotopicmixingspaceusedinour exampledietanalysis,andunliketheplantandanimalaggregate, thisaggregatewasnotestimatedusingBayesianmethods.Samplepreparation,analysis,andSuesseffectcorrectionWerinsedguardhairswitha2:1chloroform-methanolsolution toremovesurfaceoils.Weoven-driedplantsandhomogenized eachsample.Wethenweighedallsamplesintotincups (4 6 6mm).TheStableIsotopeLaboratoryatUniversityof California,SantaCruz,CAanalyzedsamplesfortheircarbon ( d13C)andnitrogen( d15N)stableisotopiccompositionby continuousflowmethodsusingaCarlo-Erbaelementalanalyzer interfacedwithanOptimagassourcemassspectrometer. WecorrectedalltissuesfortheSuesseffect,whichisdefinedas theglobaldecreaseof13CinEarth’satmosphericCO2,primarily duetofossilfuelburningoverthepast150years[42–44].Based onicecorerecords[45],weappliedatime-dependentcorrection of 2 0.022 % peryear[46](to2009)toallsampleisotopevalues, except2009humanhair.IsotopeR’smodelfeaturesUnlikeotherSIMMmodelsweincorporateallfeatures currentlyusedinSIMManalysisaswellasotherimportant features(Table1).AppendixS1describesIsotopeRfeatures, illustrateshowfeaturesinterrelate,anddefinespriordistributions. Forthoseinterested,wealsoprovidethemodellikelihood (AppendixS2).IsotopeR’sstructureishierarchical(similartothe Semmensetal.model),suchthatanindividualestimateis conditionalonthegrouporpopulation’sdistribution.The hierarchicalstructureofthemodelallowsustomakestatistical inferenceoneachindividualinthepopulation,eventhoughwe onlyhaveoneobservationforeachindividual.Althoughwe calculateindividualestimatesusingonlyoneobservation,the structureofourmodelallowsforrepeatedobservationsofthe sameindividual.Includingrepeatedmeasuresforeachindividual consumerwouldresultinlessinfluencefromthepopulation-level andmoreaccurateindividual-levelestimates. WhereascurrentSIMMsconsiderinputparametersasknown quantities,IsotopeRconsidersthemrandomvariables.Similarto Wardetal.(35),thesevariablesareestimatedusingafully Bayesianapproach,whichincorporatesalltheuncertainty associatedwiththejointestimationprocess.Inouranalysis,we jointlyestimated75parametersusingthefullIsotopeRmodel. Incorporatingtheuncertaintyassociatedwithestimatingmultiple parametersleadstomoreaccurateintervals[47]forsourcesand theirconcentrations.Wereported95%credibleintervals,aswellas meansandstandarddeviationstoillustrate(Fig.1)andstatistically summarize(Table2B)ourisotopicmixingspace.Inadditionto definingourmixingspace,wesimultaneouslyestimatedthejointIsotopestoEstimateDiet:ReviewandModel PLoSONE|www.plosone.org4January2012|Volume7|Issue1|e28478

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posteriorprobabilitydistributionofthesampledpopulation’s dietarysourcecontributions.Intheend,wereportedmarginal posteriordistributionsforeachdietarysourceatthepopulation(Fig.2,TableS5)andindividual-level(Fig.3,TableS6). WefollowthetransformationalproceduredescribedbySemmens etal.[31]toestimateproportionaldietcontributionsusingMarkov chainMonteCarlo(MCMC).Thisapproachassumesthatthe observedisotopicdistributionofanindividualiandelementeisa mixturedistribution(Mi,e)wheretheisotopicdistributionofeach sources(Xs,e)isweightedbytheindividual’sassimilateddiet proportion(fs,e,i)ofeachelement.Forastudywithnfoodsources, theindividual’sobservedisotopicdistributionisgivenby Mi e~ Xn s ~ 1fs e iXs e, 2 wherethevectorofdietproportionsforeachelementsumsto1, suchthat Xn s ~ 1fs e ijj ~ 1 : 3 Specifically,weassumethatthevectorof fs’sinequation2are randomvariablesdistributedusingthecenteredlog-ratio(CLR) transformationdescribedbySemmensetal.[31].ThistransformationallowsustouseMCMContheproportionsinequation3onthe continuousrealline,andthentransformresultstotheinterval[0,1], resultinginestimatorsofproportions.Duetolowacceptancerates, approachessuchasimportancesamplingaredifficulttoapplywhen estimatingnumerousparameters.Therefore,weusedaGibbs sampler(aMCMCalgorithm).Isotopiccorrelation.Isotoperatiosfordifferentelementsare oftenassumedtobeindependentbecauseindependentbiochemical andecologicalprocessesareultimatelyresponsiblefortheir fractionation[24].Althoughtheprocessesexplainingmostofthe variationindifferentelementsmaybedifferent(e.g.,photosynthetic pathwayforcarbonvs.trophicenrichmentfornitrogen),secondary factorscanleadtocouplingbetweenisotopicratiosofdifferent elements[27,40,48,49].Forexample,severalbear(Ursidae)studies thatusedSIAprovidedevidencethatthenutritionalpathwaysof carbonandnitrogenmaybelinkedandthestrengthofcorrelation mayincreasewithtrophiclevel[13,50,51]. Ignoringcorrelationsinamodel’scovariancestructurecanhave effectsonbothpointestimates[52]andtheirintervals[53]. BesidesIsotopeR,IsoErroristheonlymodelthatconsiders isotopiccorrelationinmixingmodelcalculations[21];however, weuseadifferentapproachtoincludethisinformationinour estimationprocess.IsoErrorcalculatesthecorrelationcoefficient (r)ofthesourcesandthemixtureandappliesthesevaluesto Figure1.IsotopicmixingspaceforFCblackbearssampledinYosemiteNationalPark. Isotopevalues( d13Cand d15N)formalebears (opencircles)capturedinYNPandtheirestimatedfoodsources.Estimatedmeansforsourceaggregates(100%plantdiet[greencircle],100%animal diet[orangecircle],100%humanfooddiet[bluecircle])andprocesserror(1SD;dashedovals)wereestimatedbyIsotopeRanddefinedthevertices ofthedietarymixingtriangle;theshapeofeachsourceaggregateillustratesthedegreeofestimatedisotopiccorrelationofobservationsusedto defineeachsource(seeFig.4).Variationsindietarycontributions(%)ofplants(P),animals(A),andhumanfood(HF)areshownalongtheedgeofthe mixingtriangle(solidgrayline)thatconnectsestimatedsourcemeans;labelsdenotethecontributionofdietwhenconsumerslieattheintersectio n ofthemixingtriangleedgeandgraydashediso-dietlines(withinthetriangle).Theblackdashedtriangleillustratestheapproximatetotalmixing spaceat1SD.Measurementerror(notshown)wasalsoestimatedbyIsotopeRandappliedtoeachsourceobservationwhenestimatingsource aggregatesandtoeachbearinthemixingspace.Theinsetillustratestheisotopicmixingspaceifconcentrationdependencewasnotincludedinthe analysis. doi:10.1371/journal.pone.0028478.g001 IsotopestoEstimateDiet:ReviewandModel PLoSONE|www.plosone.org5January2012|Volume7|Issue1|e28478

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correctthevarianceestimates.Incontrast,weestimatedrforall sourcesusingBayesianmethodsandincludedtheseestimatesas termsinthecovariancematrix(AppendixS1, # 9).MeasurementError.Weestimatedmeasurementerrorand appliedittoeachobservation.Specifically,wemeasuredthiserror fromcalibrationrunsusedtoensurethemassspectrometer’s accuracy.Becausethesecalibrationsarerunonstandards,we jointlyestimatedthemeasurementerror(AppendixS1, # 1,2,3) oftheinstrumentalongwiththeremainingmodelparameters.ResidualError.Weincludedaresidualerrorterminour modeltoaccountfortheerrorotherwiseunaccountedforinthe mixture.Ouruseofanerrorterm(AppendixS1, # 15, # 25, # 26)is consistentwithstandardlinearregressionmodelsandissimilarto otherSIMMs(e.g.,[30,31]).Thistermtakesintoaccount unexplainedvariation,thatis,variationnotincludedinsources, discriminationprocesses,sub-samplingerror,ormeasurementerror.Processanddiscriminationerror.Differencesbetweenthe isotoperatiosintissuesofconsumersandtheirdietarysourcesresult fromfractionationandstoichiometriceffects(i.e.,isotopicrouting) [54].Ingeneral,animaltissuesare15N-and13C-enrichedrelativeto theirdietsbecauselighterisotopes(14N,12C)arepreferentially eliminatedviawaste[6]andrespiration[2],respectively,allowing heavierisotopes(15N,13C)tobeassimilatedintoanimaltissues. Thesedifferencesarecommonlycalled‘‘discriminationfactors’’and willvarydependingonfactorssuchasthetaxonandtissueanalyzed [55],aconsumer’snutritionalstatus(e.g.,[56,57]),sex[58],andthe macromolecularcompositionofdiet(e.g.,[12,23,59–61]). Discriminationfactorsareoftenestimated(meanandSD)from resultsfromcontrolleddietstudies,andareusedtoshiftfoodsources toconsumersinanisotopicmixingspace.Thesecorrectionsare criticaltoaccuratelyestimatingproportionaldietarycontributions usingSIMMs[23]. Discriminationfactorsextractedfromtheliteratureareassumed tobetrueandpredictedcorrectlyfromregressionmodelsfittedto controlleddietdata[55].Usingthesefixedvaluescanresultin erroneousresultswhenestimatingmixeddietsoffree-ranging animalsusingSIMMs[62].Recentresearchsuggeststhatsome controlledstudieshaveusedinvalidprocedurestopredict discriminationfactors[58,61,63].Forexample,studiesthatfed captivebearscontrolleddiets[50,64]regressedtissueisotope valuesonfoodisotopevalues.Thepredicteddiscriminationfactor foreachnaturalfoodsourcewasthedifferencebetweenthe isotopevalueofthefoodsourceandthepredictedisotopevaluefor thetissue;thelattercalculatedfromenteringthefoodisotopevalue intotheregressionmodel.Robbinsetal.[61]notethatregression coefficientscalculatedbysuchmethodsarebiasedatestimating discriminationfactorsbecausetissueisotopevalues(dietisotope value + discriminationfactor)anddietisotopevalues(tissueisotope value–discriminationfactor)areautocorrelated.Predicting discriminationfactorsusingthesecovariates(inregression equations)yieldspuriousresults;therefore,discriminationfactors obtainedbysuchmethodsshouldnotbeusedtoestimatethediets ofanimalsusingSIMManalysis.Furthermore,resultsfromrecent controlleddietstudiesusingSprague-Dawleyratssuggestthat correlationsbetweendiscriminationfactorsanddietaryisotope valuesareartifactsoftheassociationbetweendiscriminationand biologicallysignificantcharacteristicsofdiet(e.g.,%N,%protein) thatcorrelatewithdietaryisotopevalues.Therefore,ifaregression approachisused,discriminationfactorsshouldberegressedon biologicallysignificantcharacteristicsoffood,ratherthanfood isotopevalues. WeusedregressionmodelsdevelopedbyKurle[58]topredict thetissue-dietdiscriminationfactorsofeachsampledbearfood.In thisstudy,wedefineddiscriminationfactorsasthedifferences betweenisotopevalues( d13Cand d15N)forbearhairandsampled bearfoods(expressedusing D notation: D Xtissue-diet= d Xtissue2 d Xdiet). Kurle[58]fittedregressionmodelstodatacollectedfroma controlleddietstudywhereomnivorousratswerefedvariousdiets thatequilibratedtotheirtissues.Becauseratsareoftenusedas proxiesforwildomnivores,weusedtheregressionequations developedinKurle[58]topredictdiscriminationfactorsforthehair ofmalebearsondifferent%proteindiets.Specifically,weentered theestimated%protein(x)ofplantandanimalfoods—determined bymultiplying%Nofsampledfoodsby6.25,orcalculatedfromthe NDB # (acornsonly)—intotheregressionequations( D13C = 2 0.14x + 7.43 ; D15N = 0.14x 2 2.10 )providedbyKurle[58]to Table2. Bearfoodsources.Aggregate d13C( % ) d15N( % )rD13C( % ) D15N( % )%C%NDigest[C]Digest[N] A.FrequentistmodelsDiscriminationincluded: Plants 2 21.47(2.83) 2 1.48(1.61) 2 0.2945.41(3.92)1.57(1.03)47.29(3.43)3.51(3.09) Animal 2 27.44(1.82)11.71(1.74) 2 0.8348.26(3.81)12.17(1.69)51.50(0)12.17(1.69) Human 2 16.94(0.79)8.78(0.47)0.5852.83(2.54)6.88(1.10) Bear 2 21.60(0.88)4.37(0.68)0.17 B.IsotopeRestimates Plants 2 21.72(2.66) 2 1.42(1.61) 2 0.2845.45(3.94)1.57(1.03)47.28(3.91)3.42(2.28) Animal 2 27.43(1.61)11.69(0.29) 2 0.9148.28(3.86)12.14(1.70)51.50(0.06)12.18(1.63) Human 2 16.95(0.29)8.78(0.27)0.69Fixedestimates(sameasA) C.OtherBayesian models Discriminationseparate: Plants 2 27.53(2.25) 2 0.75(1.19)6.06(0.90) 2 0.73(0.90)45.41(3.92)1.57(1.03)47.29(3.43)3.51(3.09) Animal 2 24.23(0.71)3.16(1.00) 2 3.22(1.48)8.55(1.48)48.26(3.81)12.17(1.69)51.50(0)12.17(1.69) Human 2 16.94(0.79)8.78(0.47)DiscriminationincludedFixedestimates(sameasA) A)Discrimination-correctedplant(n=48),animal(n=29),andhumanfood(n=72)sources(aggregates)calculatedfromthesampledata.(B)Plantand animalsources estimatedbyIsotopeR.HumanfoodconcentrationsarefixedasinAandC(seeTableS4).(C)RawisotopevaluesanddiscriminationfactorsusedinIsoSo urceandother Bayesianmodels.Meanand(1SD)reported. doi:10.1371/journal.pone.0028478.t002 IsotopestoEstimateDiet:ReviewandModel PLoSONE|www.plosone.org6January2012|Volume7|Issue1|e28478

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Figure2.Modelcomparisons. Meansand95%credibleintervals(denotedbyerrorbars)calculatedbyIsotopeR(bluecircles)andotherBayesian (orangecircles)models.Thebluedashedlineandgraybarindicatestheestimatedmeanand95%credibleintervalforthefullIsotopeRmodel, respectively.Frequentist(openblackcircleswithconfidenceintervals)anddatacloningestimates(opengreencircles)arealsoillustrated. doi:10.1371/journal.pone.0028478.g002 IsotopestoEstimateDiet:ReviewandModel PLoSONE|www.plosone.org7January2012|Volume7|Issue1|e28478

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predict D13Cand D15Nvaluesforeachsampledbearfood.Wethen addedeachsample’s D valuetoeachsample’smeasuredisotope value.Ultimately,theprocesserroroftheaggregateincludesthe inherenterrorassociatedwiththeisotopicvariationofthesamples intheaggregateandthevariationofdiscriminationfactors associatedwitheachsampleintheaggregate. CurrentBayesianmodelsandsomefrequentistmodelsallow userstoapplyfixeddiscriminationfactors(predictedfrom regressionequations,orextractedorinferredfromtheliterature) andtheassociateduncertaintyofeachsourcetoestimatedietary parameters.Itiscommonforresearcherstousediscrimination factorsfromtheliteratureinsteadofperformingacomplementarycontrolledexperimentontheirspeciesofinterest.Often researcherseitherusediscriminationfactorsfromasingle controlledstudythatinvestigateddiscriminationinthesame taxonorresearchersuseanaveragediscriminationfactor calculatedfrommultiplestudies(e.g.,awaterfowlstudycalculated themeandiscriminationfactorfromvariouscontrolledstudieson birds).Inadditiontocalculatingthepredicteddiscrimination factorforeachplantandanimalsample,wecalculatedtheerror (i.e.,appliedasadiscriminationerrorterminthemodel;Appendix S1, # 4)associatedwiththeregressionmodelsusedtopredictthese discriminationfactors.Therefore,allknownerrorassociatedwith thediscriminationprocessisaccountedforinourmodelstructure.Concentrationdependence.SIMMsthatfailtoaccountfor stoichiometryindietarysourcesmaydistortdietaryestimates[26]. Includingunequalelementalconcentrationsofsourceswhen calculatingdietaryestimatesusingSIMMswillalterthepolygonal Figure3.DietaryestimatesgeneratedbyIsotopeRandtheSemmensetal.model. Proportionaldietaryestimates(marginalposterior probabilitydistributions)forindividualbears( n =11)estimatedbyIsotopeR(bluelines)andtheSemmensetal.model(orangelines).Dottedlines denotepopulation-leveldietaryestimates. doi:10.1371/journal.pone.0028478.g003 IsotopestoEstimateDiet:ReviewandModel PLoSONE|www.plosone.org8January2012|Volume7|Issue1|e28478

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isotopicmixingspace,andinsomecases,includemixturesthatmay havebeenpreviouslyoutsidethemixingspace[24].Similarto IsoConc[24]andSIAR[30],westrayedfromtheassumptionthat concentrationsareequalamongsources.Specifically,IsotopeR jointlyestimatedtheconcentrations(CandN)foreachsource (Table2B)andincorporatedtheassimilationefficiency(i.e., digestibility)ofdifferentfoods(AppendixS1, # 10,11,12,13,14). Weincludedthedigestibilityofeachfoodsourcebecauseprevious studies[25,26]suggestitisimportanttoconsiderwhenincorporating concentrationdependenceinmixingmodelcalculations.In particular,weestimateddigestible[C]and[N]ofhumanandbear foodsbyanalyzingnutritionaldatafromtheNDB(TableS4)and sampledbearfoods(TableS2),respectively.Calculationsare describedinKochandPhillips[26]andinTableS2andS4.Aggregatingplantsandanimals.Weaggregatedsampled bearfoodsinto3sources:100%plantdiet,100%animaldiet,and 100%humanfooddiet.Wegroupedacorns(n=15),berries (n=9),grass(n=9),andforbs(n=15)intoaplantaggregate (n=48),anddeer(n=5)andinsects(n=24)intoananimal aggregate(n=29)(Tables2&S3).Weaggregatedthesenatural foodsources[38,39]becausetheywerebiologicallysimilar[65] andisotopicallydifferent(TableS2). Weusedthethreeaggregatedsourcestoestimateajoint probabilitydistributionofproportionaldietarysourcecontributionsforthesampledpopulationandeachindividualbear.These distributionsonlyprovideinferencetothefoodsweincludedinthe modelandwilllikelybebiased,consideringtheomnivorousdiets (i.e.,theyeatotherplantandanimalfoodsbesidesthespecies includedintheanalysis)ofYNPblackbears.Priordistributions.Thepriordistributioncanhaveaneffect oninferencesinBayesiananalysis.Inparticular,thepriorcanbe especiallyinfluentialwhensamplesizesarelow;insuchcases,using priordistributionsderivedfrompastresultscanimproveinference [66].Noninformativepriordistributions(distributionsthatplaya minimalroleintheposteriordistribution),alsoreferredtoasvague, flat,diffuse,oruninformative,areusedinBayesiananalysis‘‘tolet thedataspeakforthemselves,sothatinferencesareunaffectedby informationexternaltothecurrentdata’’[66,61]. WhenconductingBayesiananalysesitisimportanttoascertain theinfluenceofthepriorontheposteriordistribution;evenwhen usingnoninformativepriors.Likelihoodmethodssuchasdata cloningmaybeusedtoexaminesuchinfluence[67].Foreachofthe multivariatenormaldistributionsinthisstudy,weusedanormal distributionpriortoestimatemeanparametersandgamma distributionsforvarianceparameters.Weassessedtheeffectpriors hadoninferencebyconductingadatacloningproceduredescribed byLeleetal.[67].Forthisprocedure,wereplicatedthedataset (n=10)andusedthesecopiestoswamptheposteriordistribution, effectivelyminimizingtheinfluenceofthepriordistribution[67]. Datacloningproceduresyieldestimatoroutputthatareasymptoticallyequivalenttomaximumlikelihoodestimators.Weevaluated theinfluenceofpriordistributionsonouranalysisbycomparingthe datacloningestimatestoIsotopeR’sestimates.ModelcomparisonsWecalculatedsummarystatisticsforsourceaggregatesandused themasinputparametersinallmodelsexceptIsotopeR(Tables2A &2C).Weestimatedtheproportionalsourcecontributions(means and95%credibleintervals)forthesampledpopulationusingthe fullIsotopeRmodelandcomparedtheseestimatestothosewhen eachIsotopeRmodelfeaturewasindependentlyremovedfromthe model(Fig.2).Inaddition,wecomparedestimatesbyIsotopeRto estimatescalculatedbycommonlyusedSIMMs(Fig.2,TableS5). Lastly,wecomparedindividualdietaryestimatesforbears calculatedbyIsotopeRtothosecalculatedbytheSemmenset al.model(Fig.3). Bayesianmodelshavedifferentconvergenceproperties;therefore,weraneachmodelusingadifferentnumberofiterations.We ranaburninof5 6 105drawsforallIsotopeRmodels,followedby 15 6 105iterationsofMCMC.Wethinnedourresultingchainby every1,000drawsduetostrongautocorrelationinsome parameters.TheSemmensetal.modelusedaburninof 15 6 103draws,followedby15 6 104iterationsofMCMCthat werethinnedbyevery100draws,whereasSIARwasrunata burninof4 6 105draws,followedby1 6 106iterationsthatwere thinnedbyevery300draws.MixSIRwasrunataburninof5 6 103draws,followedby3 6 104iterations.Results SIAanddietanalysisWeanalyzedtheisotopiccomposition( d13C, d15N)ofhairfor 11maleFCblackbears(TableS1)andestimatedtheirdietsusing IsotopeR(Fig.2,TableS5;AppendixS3).Theproteincontentof sampledplantsandanimalswereoutsidetheboundsoftheprotein contentinratdiets[58];therefore,weextrapolatedthe discriminationfactorsusedinthisstudy.Specifically,theestimated proteincontentofsampledplants(range=2.5–23.1%)wasless thanratdiets(range=30–40%)andtheestimatedproteincontent ofsampledanimals(60.5–98.1%)wasgreaterthanratdiets(Table S2).Eachpredicteddiscriminationfactorforeachsamplewas addedtotheisotopevalueofeachsample(TableS2).Weused theseadjustedvaluestoestimateplantandanimalsource aggregates(TableS2).IsotopeRestimatedallthreesources (Table2B)andtheisotopicmixingspace(Fig.1).Wenotethat sourcedata(Tables2A&2C)andIsotopeRestimatesforsources (Table2B)wereessentiallyequivalent. IsotopeRestimatedmeasurementerror( d13C: x x =0.34; d15N: x x =0.12)andappliedthiserrortoeachobservation.IsotopeRalso includeddiscriminationerror( D13C=1.96; D15N=0.37)inits estimationprocess.Wecalculatedisotopiccorrelationforusein IsoError(Table2A)andIsotopeRestimatedthisrelationship (Fig.4,Table2B).Animalandhuman d13Cand d15Nvalueswere highlycorrelated(Figs.4,Table2)andallsourcecorrelationswere similartoestimatescalculatedfromthedata(Tables2Avs.2B). Wefoundthatestimatingcorrelationintheresidualerrorterm wasunnecessarybecausethecorrelationinthebearpopulation (r=0.17)wasaccountedforbythecorrelationinthesources. Estimatedelementalconcentrationsamongfoodsourceswere non-constant,causingthelinesthatconnectthesourcesinthe isotopicmixingspacetobecurvilinear(Fig.1).Specifically,the isotopicdataforanimalmatterhadahigher[N]thansampled plants(t=47.40,df=47.12,P= 0.001;TableS2),regardlessof whetherdigestibilitycorrectionswereincludedintheestimation (non-digest:t=6.98,df=47,P= 0.001;digestt=9.96, df=47.12,P= 0.001).Asexpected,ignoringtheeffectof concentrationdependenceamongsourceshadaconsiderable effectoninference(Fig.2,TableS5).IsotopeRfeaturesWeremovedeachfeaturefromthemodelindependentlyand comparedinferencetoresultsfromthefullIsotopeRmodel(Fig.2, TableS5).Removingcorrelationandmeasurementerrorindependentlyhadaneffectonsourceestimates(especiallyforhumanfood); althoughwenotedifferencesaresimilartoMonteCarloerror ( 3%).Removingtheresidualerrortermanddiscriminationerror term(thelatterindependentlyhavingalargereffect)alsohadan effectondietaryestimatesandincreasedthesizeofthecredibleIsotopestoEstimateDiet:ReviewandModel PLoSONE|www.plosone.org9January2012|Volume7|Issue1|e28478

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intervals(Fig.2,TableS5).Removingdigestibility,concentration dependence,andallfeaturesseparatelyfromthefullmodelhad considerableinfluencesondietaryestimates(Fig.2,TableS5).BayesianandfrequentistSIMMsPopulationestimatesgeneratedbyIsotopeR,SIAR,and IsoConcweredifferentthanotherestimatesbecausethesemodels includedconcentrationdependence.Inaddition,thedigestibility andnon-digestibilitypopulationestimatesforthesemodelswere differentwithinandamongmodels(TableS5).Resultsfromthe Semmensetal.model,MixSIR,IsoError,andIsotopeRwithout features(i.e.,Nocomponents;Fig.2)wereallsimilar(Fig.2,Table S5).Also,populationestimatesgeneratedbytheSemmensetal. modelandMixSIR’swerenearlyidentical(Fig.2,TableS5);small differencesinresultswerelikelyduetoerrorassociatedwith MCMCsamplingandbecausetheSemmensetal.modelincludes individual-levelestimation. EstimatesbySIARandIsotopeRweresimilar,yetslightly different.ThisdifferencewaslikelyduetoIsotopeRestimating dietaryproportionsattheindividual-level;includingisotopic correlationwhenestimatingthemixingspace;andaccountingfor themeasurementerrorappliedtoeachobservationinthestudy andtheerrorassociatedwithafullyBayesianapproach.Including theseimportantfeatureswillincreasetheaccuracyofestimating dietaryparameters. IsotopeR’scredibleintervalsforindividualswerewiderthan estimatescalculatedbytheSemmensetal.model(Fig.3,Table S6).Meanestimatesforhumanfoodweresimilarbetweenmodels, butplantandanimalproportionsweredifferent(Fig.3,TableS6). ThisdiscrepancywaslikelyduetothefactthatSemmenset.al. modeldidnotincludeconcentrationdependence,measurement error,orafullyBayesianapproach.Furthermore,theirmodel estimateswereessentiallythesameforeachindividual(Fig.3, TableS6),whereasIsotopeRprovidedavarietyofdietary informationforindividuals(TableS6). PointestimatesbyIsoErrorandIsoSource(toleranceof0.05) wereessentiallyidentical;however,wenote,IsoErrorprovided confidenceintervalsandIsoSourcedidnot.Itisalsoimportantto notethatmeanestimatesforthesemodelsweresimilartoallother modelsthatdidnotincludeconcentrationdependenceintheir calculations.InfluenceofpriordistributionsDatacloningandIsotopeRyieldedsimilardietaryestimates ( 3%)(Fig.2,TableS5);therefore,weconcludethatpriorshad littleinfluenceontheposteriordistribution.Wefurthertestedthe influenceofthepriorsbychangingallpriordistributionstouniform distributions,whichledtoessentiallynochange( 3%)inour estimatedpopulation-orindividual-levelestimators(Fig.2,Table S5).GiventheMonteCarloerrorpresent( 3%)theseresults suggestthatinferencesarerobustwhenusinguninformativepriors.DiscussionIsotopeRgeneratedcredibleintervalsthatweregenerallywider thanothermodels(Fig.2,TablesS5&S6);however,IsotopeR calculatedmoreaccurateparameterestimatesbecausethemodel includesallrecognizedandquantifiableSIMMfeatures,including measurementerror,concentrationdependence(withdigestibility), isotopiccorrelation,individual-levelestimation,andafully Bayesiancalculation.Collectively,thesemodelfeaturescanhave aconsiderableeffectondietaryestimateswhencomparedto commonlyusedmodels(Fig.2,TablesS5&S6). Basedontheanalysisofourdataset,theSemmensetal.model, MixSIR,andIsoError,allgeneratedverysimilarsolutions(Fig.2, TableS5).However,thesemodelsprovideinvalidestimateswhen elementalconcentrationsarenonconstant.AlthoughIsoConc incorporatesconcentrationdependenceandhadmeanestimates similartoSIAR,likeIsoSource,itdoesnotcalculateinterval estimates.SIARprovidesreasonableparameterestimates,but doesnotincorporatethesourcesoferrorandotherimportant featuresIsotopeRincludesinitsmodeldesign.Measurementerror,isotopiccorrelation,andresidual errorWesuggestSIMMusersincludemeasurementerrorintheir estimationprocedurebecauseitexists,itcanbeestimated,andits absenceintheestimationprocesscanbiasresults(Fig.2,Table S5).Previousstudieshaveshownthatnotincludingmeasurement errormayleadtobiasedparameterestimatesandcanalsoleadto alossofstatisticalpower[68].Wealsofoundthataccountingfor measurementerrorincreasedthemagnitudeofcorrelationin sources.Notaccountingforthiserrorinmeasurementsmay Figure4.Isotopiccorrelationof d13Cand d15Nineachaggregatedsource. OrangecirclesindicateaccepteddrawsfromIsotopeR’sMCMC chains;thesevaluesareusedtoestimateisotopiccorrelationandothersourceparameters.Blackcirclesdenoteobservedvalues. doi:10.1371/journal.pone.0028478.g004 IsotopestoEstimateDiet:ReviewandModel PLoSONE|www.plosone.org10January2012|Volume7|Issue1|e28478

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effectively‘washout’dependenciesbetweenvariablesandreduce estimatesofisotopiccorrelationinsources. Itisimportanttoaccountforisotopiccorrelationinsources becausethisrelationshipcanaffecttheshapeoftheisotopicmixing spaceandtheposteriorprobabilitydistributions.Determiningthe propershapeofthemixingspaceiscrucialwhenestimatingthe dietsofanimalsusingisotopicdata.Althoughtheremaynot alwaysbeenoughmeasurementsforsourceisotopevaluesto accuratelyestimatecorrelationcoefficients,ourresultssuggestthat includingtheseestimatesmaybeimportantwhenestimatingthe credibleintervalsofdietaryproportions.Inparticular,evidence fromouranalysissuggeststhattheisotopiccorrelationofbearswas explainedbyisotopiccorrelationinsources;however,future studiesshoulddetermineifaccountingforisotopiccorrelationin sourcesfullyexplainsisotopiccorrelationinmixturedata.DiscriminationerrorIsotopicdiscriminationisacomplicatedprocessandisdifficult toaccuratelymeasure[23].Asaresult,manyresearchersuse discriminationfactorsfromthepublishedliteratureandassume theywereestimatedcorrectly.Wecorrectedtheisotopevaluefor eachfoodusingapredicteddiscriminationfactorandincludedthe variabilityofthesepredictionsintheestimationofsource aggregates.Inaddition,weestimatedsourcesusingadiscriminationerrorterm,whichrepresentstheuncertaintyassociatedwith theregressionmodelsusedtopredictdiscriminationfactors. Althoughourpredicteddiscriminationfactorsareoutsidethe regressionrangeprovidedbyKurle[58],andaretherefore unreliable,weassumeinterpolatedpredictionsarevalidand suggestresearchersadjusteachsampleintheirstudyinsuch amanneriffeasible.Werecommendsamplingpreyitemsto determinetheirnutrientcompositionsbeforedecidingtherangeof biologicallysignificantdiets(e.g.,ranginginproteinquantityor quality[61])tofeedanimalsinacomplementarycontrolledstudy. Thiswillensureregressionmodelsareusefulinpredicting discriminationfactorsforconsumer’sdietarysources. Weassumedrats,bears,andhumanshavesimilardiscriminationfactorssinceomnivorousspecieshavesimilardigestive physiologies.Althoughthisassumptionisreasonable(i.e.,rats arecommonlyusedasaproxyforhumansincontrolled experiments),morecontrolledstudiesneedtobeconductedto determineifdiscriminationvariationisnegligibleamongomnivoresondifferentproteinquantityandqualitydiets.ConcentrationdependenceTheassumptionthatelementalconcentrationsamongsources areconstantwasviolatedandaddressedinouranalysis. Specifically,IsotopeRcorrectedtheisotopicmixingspace(Fig.1) byaccountingfordigestible[C]and[N]valuesforeachfood source.Whenexcludingthisfeaturefromthemodel,dietary estimateschanged(Fig.2,TableS5);alinearrelationshipbetween sources(inlayinFig.1)ledtooverestimatedsourceswithgreaterN concentrations.Similartoothermodelsthatincorporateconcentrationdependence(i.e.,SIARandIsoConc),ourfullmodel estimatesforplantsincreasedconsiderablywhileanimalsand humanfooddecreased.ThisoccurredbecauseestimatedN concentrationswerehigherforanimalsandhumanfoodwhen comparedtoplants(Table2).Correctingfordifferencesin digestibleCandNforsourceconcentrationscurvedthelinesthat connectedtheisotopicendpointsandpinchedthebottomofthe mixingspace.Thisdecreaseinareaproximatetotheplant aggregateincreasedtheestimatedproportionofplantstothediets ofbears(Fig.1).Althoughdietaryestimatesforomnivoresarenot reliablewithouttakingconcentrationdependence(withdigestibilitycorrections)intoconsideration,theeffectsofconcentration dependenceonSIMMinferenceshavenotbeenevaluatedusing captiveanimals.Therefore,inadditiontoincludingconcentration dependenceinSIMMcalculationsitmayalsobeimportantto excludeitfromanalysisandreportallresults.Greaterthann + 1sourcesEstimatorcoveragewilldecreaseasthenumberofsources increase.Thisisduetotheinabilityofthemodeltoalways estimateuniquesolutionswhenthenumberofsourcesisgreater thanthenumberofdegreesoffreedom( n + 1).Therefore,we recommendreducingtheamountofbiasinSIMManalysisby having # n + 1sources.Thiscanbeaccomplishedbyaggregating sourceswhentheyexceedn + 1,addingdimensionalitytothe mixingspacebyincludingadditionalisotopesintheanalysis,or eliminatingsourcesthatdonotsignificantlycontributetothediets ofanimalsassuggestedbypreviousdietstudies.Withouttaking oneoftheseappropriatesteps,auserwilloftencalculate confoundingresults(i.e.,inconsistentorbimodalposterior probabilitydistributions).Forexample,awolfpopulationwas partitionedintothreegroupsandaBayesianSIMMwasusedto makeinferencesaboutthedietsofgroupsandindividuals[31].For themainlandgroup,theisotopicdistributionofthesampled salmonpopulationfellinthemiddleofthewolfdistributionand directlybetweenthedeerandmarinemammaldistributions;this isotopicarrangementofsourcesconfoundedtheestimation process.Addinganotherisotope(e.g., d34S)oreliminatingmarine mammalsfromtheanalysis—onlyiftheywereshowninother studiestonotcontributetothedietsofwolvesonthemainland— wouldhavelikelyremediedthisproblem. Foromnivores,plantandanimalsmaybeaggregatedintomore groups(i.e.,moredietarysourcestoestimate)ifauserincreasesthe numberofisotopesusedtomakeinference(e.g.,including d34Sto estimatethecontributionofsalmonindietsofbearsinAlaska). Thiswouldpotentiallyincreasethepredictivepowerofthemodel [30],especiallyifsourceswere # n + 1.Itisimportanttoput sufficienteffortinusingpriordata(e.g.,resultsfromscatorgut contentanalysis)todeterminethecompletelistoffoodsourcesand toaggregatethemappropriately(e.g.,[65];assuggestedinthis study)toconstructanisotopicmixingspacethatwillproduce uniqueandbiologicallysignificantsolutions.Inaddition,such studiesarealsoimportantwhendefiningpriordistributionsin BayesianSIMManalysis.InfluenceofpriordistributionsEstimatingallparameterssimultaneously(i.e.,fullyBayesian approach)ismostusefulwhenconsumersamplesizeislow.When samplesizeincreases,estimationerrordecreases,andparameter estimateswilleffectivelybecomeconstants.Despiteoursmall samplesize(n=11),datacloningpointestimatesweresimilar ( 3%;Fig.2,TableS5)toourmodelestimates;thus,suggesting thepriorhadlittleinfluencedonIsotopeR’sparameterestimates.ConclusionsHere,weprovideareviewofcommonlyusedSIMMsandoffer anewcomprehensivemodel.Ourpurposewasnottoaccurately estimatetherealdietsofYNPblackbears.Weusedanincomplete collectionoftheplantfoodsandextrapolateddiscrimination factors;therefore,ourdietaryinferencesarelikelyincorrect. However,wedobelieveourestimatesarereasonablegivenwhat weknowaboutthedietsofFCbearsinYNPandthenutrient requirementsofbears.Inparticular,webelieveitisreasonablefor bearsthatregularlyconsumehumanfood(18–43%),whichishigh inprotein[41],toeatlessanimalmatter(0–19%)thanbearsthatIsotopestoEstimateDiet:ReviewandModel PLoSONE|www.plosone.org11January2012|Volume7|Issue1|e28478

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donotconsumehumanfood.ThisisespeciallythecaseforYNP blackbearssincemostoftheanimalmatterintheirdietsis composedofinsects[38,39].Inaddition,vegetationisclearlythe largestcontributortothedietsofbears(Fig.2,TablesS5&S6)as suggestedbypastdietstudiesconductedinYNP[38,39]. SIMMsareevolvingrapidly.Webelievethisexpeditiousprocess willresultintheabandonmentofmanymodelscurrentlyusedto estimatethedietsofanimalsandthecreationofmanynewmodels (e.g.,time-seriesmodels).BecauseIsotopeRincludesallfeatures usedincurrentmodelsaswellasothernewfeatures,webelieveit willbethemodelofchoiceformanyecologistsinterested estimatingthedietsofanimalsusingisotopicdata.Inaddition, themodelcouldbeusedasafoundationforfutureSIMM developmentbecauseofitscomprehensivestructure;wenotethat IsotopeR,likeotherSIMMs,isalsoapplicableforusein paleontology,archaeology,andforensicstudiesaswellas estimatingpollutioninputs.TheRpackage‘‘IsotopeR’’(with GUI)isavailableonCRAN(seeRvignetteandhelpfilesfor directionsonmodeluse).SupportingInformationTableS1Suess-correctedisotopevaluesformalefoodconditionedblackbears. Yeardenotestheyearthehair represents.HairwasSuess-correctedasdescribedinMethods. (DOC)TableS2Adjustedisotopicdataanddigestibilitycalculationsforsampledplantsandanimals. Discrimination factorscalculatedforplantandanimalaggregatesderivedfrom regressionmodelsinKurle[58].Digest[N]and[C]arecalculated usingthelistedequations.Concentrationsfor Quercus spp.(acorns) werecalculatedusingtheUSGSNutrientDatabase(NDB)and othernutritiondata[69–70]. (DOC)TableS3Isotopevaluesforhumanhairsampledfrom2 salonsand1barbershopinSt.Louis,MO,2009. (DOC)TableS4Humanfooddigestibilitycalculationsfor humansintheUnitedStates. Threefoods(minimum)were selectedfromeachdietarysourcecategoryprovidedbyNakamaru etal.[41].Stoichiometricmeasurementswererecordedforeach food[locatedbyenteringeachfood’sNDB # (Nutrientdatabank identifier)intotheNDBsearchfield(http://www.nal.usda.gov/ fnic/foodcomp/search/)].Digest[C]andDigest[N]arecalculatedusingthelistedformulas.Mean[C]and[N]arecalculated foreachsourcecategoryandweighedaccordingtotheweighting factors(inparenthesesnexttoeachdietarysourcecategory;[41]). Weight[C]and[N]representtheweighted[C]and[N]for humanfoodintheUnitedStates.Theseparametersarefixedand usedtoestimateproportionalsourcecontributionsinallmodels thatusesuchparameters. (DOC)TableS5Population-leveldietaryestimatesgenerated byIsotopeRandcommonlyusedSIMMs. (DOC)TableS6Individual-leveldietaryestimatesgenerated byIsotopeRandtheSemmensetal.(2009)model. (DOC)AppendixS1IsotopeR(fullmodel)operationalschematic. Indentedformulasontheleftsidedenotetermsandprior distributionsassociatedwithrandomvariables.Therightside providesadescriptionofeachformula.Subsectiontitlesfollowed bythenumberofparametersestimated.Arrowsdenotehierarchicaldependenciesamongrandomvariables. (DOC)AppendixS2IsotopeRlikelihoodequation. (DOC)AppendixS3Filesanddirectionsforrunningthefull IsotopeRexampleusedinthispaper. 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