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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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