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Apparentsurvivalestimationfromcontinuousmark – recapture/resightingdata AndrewB.Barbour 1 *,Jos eM.Ponciano 2 andKaiLorenzen 1 1 SchoolofForestResourcesandConservation,ProgramofFisheriesandAquaticSciences,UniversityofFlorida,7922NW 71stStreet,Gainesville,FL,32653,USA;and 2 DepartmentofBiology,UniversityofFlorida,Gainesville,FL32611,USA Summary 1. Therecentexpansionofcontinuous-resightingtelemetrymethods(e.g.acousticreceivers,PITtagantennae) hascreatedaclassofecologicaldatanotwellsuitedfortraditionalmark – recapturestatistics.Estimatingsurvival whencontinuousrecapturedataisavailableensuesapracticalproblem,becauseclassicalcapture – recapture modelswerederivedunderadiscretesamplingschemet hatassumessamplingevent sareinstantaneouswith respecttotheintervalbetweenevents. 2. Toinvestigatetheuseofcontinuousdatainsurvivalan alysis,weconductedamodelstructureadequacysimulationthattestedtheCormack – Jolly – Seber(CJS)andBarkerjointdatasurvivalestimationmodels,which mainlydierthroughtheBarkersinclusionofsecond aryperiodinformation.Wesimulatedapopulationin whichsurvivalanddetectionoccurredasanearcontinuous (daily)processandcollapseddetectioninformation intomonthlysamplingbinsf orsurvivalestimation. 3. Whilebothmodelsperformedwellwhensurvivalwastimeindependent,theCJSwassubstantiallybiasedfor lowsurvivalvaluesandtime-dependentconditions.Add itionally,unliketheCJS,th eBarkermodelconsistently performedwellovermultiplesamplesizes(numberofmarkedindividuals).However,thehighnumberofparametersintheBarkermodelledtoconvergencedicultie s,resultinginaneedforanalternativeoptimization method(simulatedannealing). 4. WerecommendtheuseoftheBarkermodelwhenusingcontinuousdataforsurvivalanalysis,becauseitoutperformedtheCJSoverabiologicallyreasonablerangeofpotentialparametervalues.However,thepracticaldif“cultyofimplementingtheBarkermodelcombinedwithitsshortcomingsduringtwosimulationsleavesroom forthespeci“cationofnovelsta tisticalmethodstailoredspe ci“callyforcontinuousmark – resightingdata. Key-words: Barkerjointdata,Cormack – Jolly – Seber,modelstructureadequacy,telemetry Introduction Reliablebiologicalinferencesabouttheprocessesdriving survivalofindividualsinapopulationdependontheproper formulationofstochasticproce ssmodelsthatareconfronted withcapture – recapture/resightingdata.Suchmodelstranslatefundamentalbiologicalque stionsintotestablehypothesesthatfurtherourunderstandingofthesystemofinterest (Cohen2004;Gimenez etal. 2007).Whensuchmodels areinappropriatelyformulated,biascausedbystructural errorscanleadtounreliablestatisticalinferences(Pradel& Sanz-Aguilar2012). Forcapture – recapture/resightingdata,formulationofan appropriatestochasticprocessmodelrequiresconsideration ofthestructureofthedatacollected(e.g.discretevs.continuoussamplingevents),thetypeofdatacollected(e.g. recapture,resightingordead recovery)andthebiological characteristicsofthestudysystem(e.g.openvs.closed populations).Forexample,multiplemodelshavebeen developedtoestimatesurvivalfromopenpopulationswhen usingdiscrete-resightingdata(Hightower,Jackson& Pollock2001;McClintock&White2009;Johnson etal. 2010)ordiscrete-recap turedata(Lebreton etal. 1992).The recentexpansionofcontinuous-resightingtelemetrymethods (e.g.acousticreceivers,PITt agantennae;Heupel&Simpfendorfer2002;Barbour&Adams2012)hascreatedaclass ofecologicaldatanotwellsuitedforstandardstatistical methodswhenfatesareunknown(Kie etal. 2010).Without aninvestigationofpropermodelformulation,theinformationcontainedinthisdatawillnotbefullyharnessed,and statisticalinferencesmaybeweakormisleading(Strong etal. 1999). Severalprevioussurvivalstudiesusingcontinuousresightingdatacollapsedcontinuousresightingsintodiscretetimeintervalsandappliedexistingdiscrete-timemodels.For example,Heupel&Simpfendorfer(2002)appliedHightower, Jackson&Pollock(2001)sdisc rete-timemodeltocontinuousresightingdatabycollapsingres ightingsintoweeklysampling bins.Similarly,Adams etal. (2006)collapsedcontinuousresightingdataintoweeklyinte rvalsandestimatedapparent survivalwiththediscreteCormack – Jolly – Seber(CJS)model. *Correspondenceauthor.E-mail:snook@u”.edu 2013TheAuthors.MethodsinEcologyandEvolution 2013BritishEcologicalSociety MethodsinEcologyandEvolution 2013doi:10.1111/2041-210X.12059

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Duringamultiyearstudy,Cameron etal. (1999)collapsed 4months(NovemberthroughFebruary)ofcontinuousresightingsintoasingleencounteroccasionlabelledasJanuary1st eachyearandthenestimatedannualsurvivalwithadiscrete multistatemodel.Hewitt etal. (2010)tookasim ilarapproach, butusedadiscreteCJSmodel. Theuseofcontinuousdataindiscrete-timemodelsviolates theassumptionthatsamplingocc asionsareinstantaneouswith respecttotheintervalbetweenp eriods(e.g.acohortismarked inasingleday,aprolongedperiodoftimeelapses[e.g.a month],thenasubsequentcapture – recaptureeventoccurs overasingleday;Pollock etal. 1990).Somestudieshaverecognizedandaccountedforthisissue(Barbour,Boucek& Adams2012a;Bowerman&Budy2012;Ruiz-Guti errez etal. 2012;Mintzer etal. 2013),butitisunknownhowviolatingthis assumptionbiasessurvivalpr obabilitiesinstudiesthathave not.Here,weexplorethisissu ebysimulatingapopulationof markedindividualsthatareresightedonarelativelycontinuous(daily)basisandcollapsingthesecontinuousresightings intodiscrete-timebins.Weth enestimatetheknownsurvival valueswithtwosurvivalestimationmodelstodetermine whetheramodelcurrentlyexiststhatisappropriatefor estimatingsurvivalfromcontinuousresightings.MaterialsandmethodsMODELSTRUCTUREADEQUACYWeusedamodelstructureadequacy (MSA)approach(Taper,Staples &Shepard2008)totestwhethertwosurvivalestimationmodelscould beusedforunbiasedestimationofsur vivalfromcontinuous-resighting data.MSAselectsmodel sbasedontheirabilitytoanswerspeci“cscienti“cquestionsgiventhecurrentunderstandingoftherelevantaspects oftherealworld.UndertheMSAapproach,amechanisticsimulation modeliscreatedtorepresenttheunderlyingprocessofinterest,and candidatemodelsareusedtoestimat e/predicttherelevantmetricfrom simulateddata.Thisallowsinvestigationoftwotypesoferrorinthe testedmodels:structural(errors ofapproximation)andestimation (uncertaintyinparameterestimates;Taper,Staples&Shepard2008). Inadditiontotheseerrortypes,theMSAapproachitselfissubjecttoa thirdtypeoferror.Formulatione rroroccursduetodierences betweenthemechanisticsimulat ionmodelandthetrueunderlying processes. Accordingly,weformulatedamec hanisticsimulationmodelofa markedpopulationinwhichindividua lsurvivalanddetectionoccurred asanearcontinuous(daily)process. Wethengenerateddatasetsfrom thesimulationmodelusingarangeofparametervaluesthatfully encompassedbiologicallyplausibleconditions.Foreachofthescenarios,wetestedtheabilityoftwoestimationmodels(CJSandBarker jointdata)torecoverthebasicprope rtiesofthesurvivalparameter.We iteratedthisprocessforeachparameterset100times.Weevaluated structuralerrorbycalculatingrelativebiasandpercentcoverageof survivalestimatesfromeachestimat ionmodelaftersimulatingpopulationsfrommultipleknownparametervalues.Weassessedestimation errorinasecondsimulationbyvaryingthenumberofmarkedindividualsinthesimulatedpopulation.Finally,toevaluatetherobustnessof themodelinferencestounavoidable formulationerrors,weaddedan additionalbiologicalprocess,aseveredisturbanceevent,inathird simulation.SURVIVALESTIMATIONMODELSWeemployedtwosurvivalestimationmodels,theCJS(Lebreton etal. 1992)andtheBarkerjointdata(Barker1997,1999).TheCJSmodel assumessamplingperiodsthatareinstantaneouscomparedtothe intervalbetweensamplingevents(Pollock etal. 1990).Incomparison, theBarkermodeliscomposedofbothinstantaneousprimaryperiods ( i and i + x )andcontinuoussecondaryperiods( i i + x ),withsecondaryperiodsbeingtheinterval( x )betweenprimaryperiods(Fig.1). Duringprimaryperiods,individualsarecapturedandrecapturedinan identicalfashiontotheCJSapproach.However,secondaryperiods occurbetweenmarkingperiodsan dallowmarkedindividualstobe resightedaliveordeadonacontinuousbasis. TheCJSmodelestimatestwoparam eters:(1)survival,estimatedas eitherapparentsurvival( ;survivalconfoundedbyemigration)when emigrationoccursandtruesurvival( s )whenemigrationdoesnotoccur and(2)recaptureprobability( p ).TheBarkermodelestimatesseven parametersduetotheadditionalinformationfromcontinuoussecondaryperiods(Table1).TheBarkermodelestimatestruesurvival( s ) whensecondaryperiodsareconductedovertheentirerangeofa markedpopulationorwhenemigrationdoesnotoccur,and otherwise.Oursimulationmodelsdidnotincludeemigration;therefore,all survivalestimateswillhe reafterbereferredtoas s .SIMULATION1:ST RUCTURALERRORTosimulatetheuseofcontinuousdatafordiscretesurvivalestimation, wesimulatedapopulationthatsurv ived/diedandwasdetected/not detectedonadailybasisandcolla psedthesedailydetectionsinto monthlysamplingbins.Weassumedasystemclosedtoemigrationin whichallindividualsweremarkedduringthe“rstdaywithnotagging mortality.Therefore,staticpara meters(thoseheldconstantoverall iterations)includedthenumberofmarkedindividuals( n = 1000)and thenumberofdaysforthesimulation( d = 180).Thevariableparameters(thosewealteredbe tweeniterations)ofthesimulationmodelwere limitedtotruemonthlysurvival( sm)andtruemonthlyrecaptureprobability( pm).Tofullyencompassthebiologicallyplausiblerangeof parametervalues,wecreatedsimulationmodelsusing50known sm(a Fig.1. Schematicdiagramofprimaryandsecondarysamplingperiods fortheCormack – Jolly – SeberandBarkerjointdatamodels.Primary samplingperiodsaredenotedby i i + 1 etc.andendinallsimulations atmonth m = 6.TheBarkermodelincludessecondarysampling information,whichconsistsoftheopenintervalbetweenprimaryperiods.Here,weusedevenintervalsoflength x = 1;however,uneven intervals,andintervalsnotequalto1,couldbeused. 2013TheAuthors.MethodsinEcologyandEvolution 2013BritishEcologicalSociety, MethodsinEcologyandEvolution2 A.B.Barbour,J.M.Ponciano&K.Lorenzen

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sequencefrom0 5to1 0)and50known pm(asequencefrom0 02to 1 0)values.Thisresultedin2500variableparametercombinations. Weconstructedthemechanisticsimulationmodel(AppendixS1)in theprogramR(RDevelopmentCoreTeam2011).Foreachindividual, weconductedaBernoullitrial(abinomialcoin”ip)eachdaytodeterminewhethertheindividualsurvivedordiedwithadailysurvivalprobability( sd)ofeqn1: sd s 1 = 30 m: eqn1 Eachdayanindividualsurvived, asecondBernoullitrialwasconductedtodeterminewhethertheindividualwasdetected.Toconvert monthlyrecaptureprobabilityt odailyrecaptureprobability( pd),we calculatedthedailyprobabilityofn otbeingrecapturedandsubtracted thisvaluefromone,eqn2: pd 1 1 pm 1 = 30 : eqn2 Therecaptureprobabilityneedsto becomputedthiswaysincethere aremanypossiblecombinationsforanindividualtobedetectedatleast onceinagivenmonth,butthereisonlyonepossiblewaytonotbe recaptured.Subtractingtheprobabilityofnondetectionfromone accountedforallpossiblerecapturecombinations. Afterrunningthemechanisticsimulationmodel(AppendixS1)fora givenvariableparameterset,wecollapseddailydetectionsinto monthlybins( m = 6),inwhichindividualswereeitherdetectedornot, tocreatecapturehistoriesforeac hindividual.FortheCJS,these monthlybinsrepresentedprimaryperiods,butwereusedasthesecondaryperiodsintheBarker.FortheBarkermodel,wesetthecapturehistoryvaluesinallprimaryperiods,withtheexceptionofthetagging event,tozero.WecreatedcapturehistoriesfortheCJSbytwomethods.Inthe“rstmethod,whichmirroredAdams etal. (2006),wecollapseddailydetectionsinto6monthlybinsasdescribedabove, meaningthetaggingeventwasincludedinthe“rstmonthofdetections. Weleftalltimeintervalsasthedefaultlengthofone.Inthesecond method,wesetthemarkingeventasanindependentprimaryevent, therebycreatingaseventhbin(sixintervals)inthecapturehistory. Whenusingthesecondmethod,weadjustedforuneventimeintervals withintheRMARKpackage(Laake&Rexstad2008)forR.Usingthe midpointofeachresightingmonthasourreferencepoint,wesetthe “rstinterval(betweenmarkingandthe“rstresightingmonth)toequal alengthof0 5months.Thus,eachsubsequenttimeintervaloccurred betweenthemidpointsoftheresightingmonthsandwasoflength1 0. Sinceshorter-termsurvivalestima tesmaybeofinterestincertain studies,weranaseparatesimulati onusing10-daybins,insteadof30daybins.WecreatedcapturehistoriesfortheCJSbythesecond method,treatingthemarkingeventse paratelyfromresightinginformation. WeestimatedsurvivalwiththeCJSandBarkermodelsusing programMARK(White&Burnham1999)accessedbytheRMARKpackage(Laake&Rexstad2008).FortheBarkermodel,we“xed F at1and F’ at0asnoemigrationoccurred,andwe“xed p and r to0sincenorecaptureoccurredduringprimaryperiods, andwedidnotsimulatedeadrecoveries(parameterde“nitionsin Table1).Thesimulationsdidnotincludetimevariabilityinsurvivalorrecaptureprobability;ther efore,weusedti me-independent estimatesfor s R and R intheBarker,and s and p intheCJS estimationmodel. Weranthemechanisticsimulatio nmodel100timesforeachofthe 2500variableparametercombinations,resultingin250000totaliterations.Foreachiteration,wecomputedtherelativedepartureoftheestimatedsurvivalfromthetruesurvivalas,eqn3: Relativedeparture ^ sm sm s 1 m: eqn3 Therelativebiaswasthenestimatedastheaverageoveralliterationsof theserelativedeparturesforagivenvariableparametercombination. Additionally,wequanti“edpercentcoveragebycountingthenumber ofsuccessfuliterationspervariableparametersetinwhichthetrue valuefor smwasincludedinanestimationmodels95%con“dence intervalof ^ sm.Forthefullsimulationrun(all250000iterations),we “rstusedthedefaultNewton – RaphsonoptimizationmethodinprogramMARK,andthenreranthefullsimulationwithanalternative optimizationmethod(simulatedan nealing)fortheBarkermodelsince thismodelfailedtoconvergeinmultipleinstances.SIMULATION2:ESTIMATIONERRORWerepeatedsimulationonewithtwoalterationstothemechanistic simulationmodel.First,we“xedthevariableparameters( smand pm) toavalueof0 9,asthesevaluesapproximatedinitialestimatesfrom aknownstudysystem(Barbour,Boucek&Adams2012a;Barbour etal. 2012b).Second,wemadethenumberofmarkedindividuals ( n )avariableparameter,withvaluesrangingfrom n = 50to n = 1000byincrementsof50.Wethenreranthesimulationaspreviouslydescribedanditeratedthesimulation1000timesforeach n valuewhileusingsimulatedanne alingforoptimizationwiththe Barkerandthesecondmethodofca pturehistorycreationforthe CJS.Wedeterminedpercentcoverageandestimatedrelativebiasof survivalateach n valueforeachestimationmodelinidenticalfashiontosimulationone.Besidesco mputingtherelativebiasasthe average,relativedeparturefromthetruesurvivalprobability,we kepttrackofthe2 5and97 5percentilesofthedistributionofthese relativedepartures. ToaddresscoverageissueswiththeCJS,weconductedaparallel simulationthatusedparametricbo otstrappingtocreatecon“dence intervals.For n = 200and500,werepeatedthepriorsimulationforthe CJSfor500iterations,butusedthere sultsofeachiterationtorun1000 bootstrappedsimulations.Weusedthe2 5and97 5percentilesof Table1. Barkerjointdatamodelparameterde“nitionsinprogramMARKParameterDe“nition siTheprobabilitythatananimalaliveat i isaliveat i + 1 piTheprobabilitythatananimalatriskofrecaptureat i isrecapturedat i riTheprobabilitythatananimaldiesin i i + 1isfounddead RiTheprobabilityananima lthatsurvivesfrom i to i + 1isresighted(alive)sometimebetween i and i + 1 R0 iTheprobabilityananimalthatdiesin i i + 1withoutbeingfounddeadisresightedalivein i i + 1beforeitdied FiTheprobabilitythatananimalatriskofrecaptureat i isatriskofrecaptureat i + 1 F0 iTheprobabilitythatananimalnotatriskofrecaptureat i isatriskofrecaptureat i + 1(thisde“nitiondiersfromBarker(1997)in ordertoforceprobabilitydriveninternalconstraints;White&Burnham1999) 2013TheAuthors.MethodsinEcologyandEvolution 2013BritishEcologicalSociety, MethodsinEcologyandEvolutionSurvivalestimationfromcontinuousdata 3

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maximumlikelihood(ML)survivalestimatesfromthese1000bootstrappediterationstoconstructcon“denceintervalsforeachofthe500 iterationspersamplesize( n ).Weusedtheseparametricbootstrapcon“denceintervalstotestcoverageofthetruesurvivalvalue.SIMULATION3:FORMULATIONERRORTodeterminetheabilityoftheestimationmodelstoaccountforadditionalbiologicalcomplexityintheformofadisturbanceevent,we alteredthemechanisticsimulationmodeltoincludeamonthoflowsurvival.First,we“xed smto0 90and pmto0 90andmaintained n = 1000.Then,forthethirdmonthofthesimulation(days61 – 90),we lowered smto0 30torepresentaseveredisturbanceevent. Wecreatedtwomodelstructurestoaccountforthedisturbance event,andweusedthemtoestimatesurvivalusingtheCJSandBarker models.The“rstmodelstructureallowedforfulltimedependencewith respecttosurvival, sm( t ).Oursecondmodelstructure, sm( d ),representedthetruth-generatingprocess, themechanisticsimulationmodel. Inthismodel, smforthedisturbancemonthw asestimatedseparately fromtheother,time-independent smperiods.Forallestimationmodels, theotherparameterswerecalculatedinidenticalfashiontosimulation one. Weiteratedthissimulation1000 timesandusedsimulatedannealingforoptimizationwiththeBarkerandthesecondmethodofcapturehistorycreationfortheCJS.Weselectedthemostparsimonious modelstructureforeachestimationmodelaftereachiterationby identifyingthemodelwiththeminimumAkaikesInformationCriterion(AIC;Akaike1973)score.Generally,modelswith D AICvalues < 2havesubstantialsupport,andmodelswith D AIC > 10have nosupport(Burnham&Anderson2004).Wethensummarizedand plottedthesimulateddistributionoftheMLsurvivalestimates. Finally,wedeterminedthepercentcoverageasthenumberof successfuliterationsduringwhichthe95%con“denceintervalof ^ smforthegiveniterationincludedthetruevalueof sm.ResultsTheBarkermodelestimatedsurvivalfromcontinuousresightingdatawithminimals tructuralerror,whiletheCJS modelonlyperformedwellundertime-independentconditions withhighsurvival.UnliketheCJS,theBarkermodelperformedwellacrossmultiplesamp lesizesofmarkedindividuals ( n ).Additionally,theBarkermode lreliablyestimatedsurvival whenweaddedbiologicalcompl exitytothemechanisticsimulationmodel.However,theBarke rmodelsoptimizationfailed toconvergeforsomecombinationsofparametervaluesusing Newton – Raphsonsmethod,necessitatingtheuseofsimulated annealing.Wesummarizethesi mulationresultsinaseriesof contourplots(Figs2 – 4)inwhichweplottedtherelativebias (sub“guresaandb)andpercentcoverage(sub“guresc andd)ateachofthe2500parametercombinationforagiven simulationrun.SIMULATION1:ST RUCTURALERRORWhenconstructingcapturehistoriesundermethodone,the CJSmodelmoderatelyunderestimated sm(Fig.2a)andrarely demonstratedanacceptablelevel ofcoverage(Fig.2c).Creatingcapturehistoriesundermethodtwo,whichseparated markingfromresightinginformation,resultedinrelatively unbiasedestimates(Fig.2b)withpropercoverageexceptwhen survivalwaslowandespeciallywhencombinedwithhigh recaptureprobability(Fig.2d).Movingfrommonthlyto10daybinswhileusingmethodtwoofcapturehistorycreation didnotsubstantiallyaectresultsfortheCJS(Fig.3a,c). Incomparison,theBarkermodelestimated smwithaconsistent,minorpositivebias,(Fig.4a),butfailedtoconvergemultipletimeswhenusingNewton – Raphsonoptimization (Figs4cand5).Thiswaslikelyduetothehighnumberof estimatedparametersleadingtolocalminimaduringnumericaloptimization.However, 100%ofmodelrunsconverged whenusingsimulatedannealingandcoverageestimates consistentlyranged90 – 98%(Fig.4b,d).Whenusing10day insteadofmonthlybins,theperformanceoftheBarker modelwasreducedatlowrecapt ureprobabilities,withcoverageapproaching0%andrelativebiasexceeding 10 0% (Fig.3b,d).SIMULATION2:ESTIMATIONERRORWhenalteringthenumberofmarkedindividuals( n ),the Barkermodelreliabilitycoveredthetruevalueof smin c .95% oftheiterationsforevery n tested(Fig.6).Relativebiasforthe Barkermodelwasnear0 0%,withvariabilityinthedeparture fromthetruthdecreasingwithincreasing n .TheCJSmodel coveredthetruevalueof smin95%oftheiterationswhen n waslow( n = 50,100),butas n increasedto1000,coveragefell below86%(Fig.6).Thisoccurre dbecausecon“denceintervalsbecamenarroweras n increased,whilerelativebiaswas maintainedat0 005%(Fig.6).Thus,theprobabilityofcovering smwiththeCJSdecreasedwithincreasing n .Whenusing parametricbootstrappingtoaddresstheCJSpoorcoverage, coveragedecreasedfrom92 4%to88 4%at n = 200andfrom 88 8%to81 4%at n = 500. 05 06 07 08 09 10 020610 05 06 07 08 09 10 05 06 07 08 09 10 05 06 07 08 09 10 –010 –008 –006 –004 –002 000 002 004 00 02 04 06 08 10 Monthly survivalMonthly recapture probabilityRelative bias % Coverage020610 020610020610(a) (b) (c) (d) Fig.2. Estimatedrelativebias(a,b)andpercentcoverage(c,d)of monthlysurvivalestimatesfortheCormack – Jolly – Seber(CJS)estimationmodelwhenusing30-daysamplingbins.Capturehistorieswere createdbytwoseparateapproaches,asdetailedinthemethods.Plots include:(a)methodonerelativebias,(b)methodtworelativebias,(c) methodonepercentcoverageand(d)methodtwopercentcoverage. Forvisualclarity,relativebiasvalueslessthan 5 0%wereplotted as 5 0%:thisisapparentinthelowerright-handcornerof(b). 2013TheAuthors.MethodsinEcologyandEvolution 2013BritishEcologicalSociety, MethodsinEcologyandEvolution4 A.B.Barbour,J.M.Ponciano&K.Lorenzen

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SIMULATION3:FORMULATIONERRORWhena30-daydisturbancewasincorporatedintothesimulationmodel,AICsupporteddierentmodelstructuresforthe CJSandBarkerestimationmodels(Table2).FortheCJS, AICselectedthefullytime-dependentmodelforsurvival [ sm( t )],everyiterationwithnosupportgiventothetruth-generatingmodel, sm( d )(Table2).FortheBarkerestimationmodel, thestructuralmodelrepresenti ngthetruth-generatingprocess, time-independentestimatesforallperiodsexceptforthedisturbancemonth, sm( d ),wastheminimumAICmodelin89 9% ofthe1000iterations.Howeve r,thetime-dependentmodel receivedconsiderableAICsupport(Table2). Forthetruth-generatingandtime-dependentmodelstructures,wecomparedtheestimatedrelativebiasinsurvival obtainedbytheCJSandtheBarkermodel.Whenthestructuralmodelwasthe sm( d )model,theCJSmodelcoveredthe truevalueof smforthedisturbancein0 1%oftheiterations withanestimatedrelativebiasof33 3%.Coveragefornondisturbance smwas0 0%,witharelativebiasof 6 3%.Incomparison,theBarkermodelcoveredthetruevalueof smduring thedisturbancein93 8%oftheiterations,witharelativebias of2 5%.Coveragefornondisturbance smwas83 7%,witha relativebiasof1 0%.Whenwecomparedresultsfromthe sm( t )model,theBarkeroutperformedtheCJSmodel(Fig.7). TheCJSmodelprovidedrelativ elyunbiasedestimatesforall monthsexcepttwoandthree,attributingasubstantialproportionofthe smdeclineinmonththreetomonthtwo(Fig.7). Forthedisturbancemonth,theCJSestimationmodelcovered thetruevalueof smin0%oftheiterationswithameanrelative biasof44 2%.TheBarkermodelcoveredthetruevalueof smduringthedisturbancein76 4%oftheiterationswithamean relativebiasof5 6%.DiscussionWepresentedthe“rstassessmentofthestatisticalpropertiesof survivalestimatorswhencontinuous-timedataareavailable yetadiscrete-timesamplingmodelisusedforestimation. Usingamodelstructuraladequacyapproach(Taper,Staples &Shepard2008),wedemonstratedthatsubstantialbiasexists whencontinuouscapture – recaptureinformationisdiscretized forsurvivalestimation.Theextentofthebiasdependsupon theestimationmodelused,withtheBarkerjointdatamodel outperformingtheCJS. The“rstmethodofcapturehistorycreationfortheCJS introducedsubstantialbiasbeca usewecodedallindividualsas aliveinmonthonedespitetherebeing29daystosuccumbto mortality.Usingthesecondmethodofcapturehistorycreation,theCJSfailedtoreliablyestimatesurvivalinmostsimulationsandresultedinbiasatlow s withbiasworseningathigh p .Thecomplexitiesofhowthesurvivalanddetectionprocesses 05 06 07 08 09 10 020610 05 06 07 08 09 10 05 06 07 08 09 10 05 06 07 08 09 10 –010 –005 000 005 010 00 02 04 06 08 10 Monthly survivalMonthly recapture probability(a)(b) (c)(d)Relative bias % Coverage020610 020610 020610 Fig.3. Estimatedrelativebias(a,b)andpercentcoverage(c,d)for survivalestimatesfromtheCormack – Jolly – Seber(CJS)(a,c)andBarkerjointdata(b,d)estimationmodelswhenusing10-daybins.CJS capturehistorieswerecreatedundermethod2,andBarkeroptimizationBarkerwasconductedbysimulatedannealing.Plotsinclude:(a) CJSrelativebias,(b)Barkerrelativ ebias,(c)CJSpercentcoverageand (d)Barkerpercentcoverage.Forvisualclarity,relativebiasvaluesless than 10 0%wereplottedas 10 0%:thisisapparentinthelower left-handcornerof(b). 05 06 07 08 09 10 020610 020610 020610 020610 05 06 07 08 09 10 05 06 07 08 09 10 05 06 07 08 09 10 –010 –008 –006 –004 –002 000 002 004 00 02 04 06 08 10 Monthly survivalMonthly recapture probability(a)(b) (c)(d)Relative bias % Coverage Fig.4. Estimatedrelativebias(a,b)andpercentcoverage(c,d)for monthlysurvivalestimatesfromtheBarkerjointdataestimation modelwhenusing30-daybins.Optimizationwasconductedbytwo separatemethods:Newton – Raphson(NR)(a,c)andsimulatedannealing(SA)(b,d).Plotsinclude:(a)NRrelativebias,(b)SArelativebias, (c)NRpercentcoverageand(d)SApercentcoverage. 0204060810 05 06 07 08 09 10 000 005 010 015 020 025 030 Monthly survivalMonthly recapture probabilityConvergence failure % Fig.5. Percentageof100iterationsateachknownparametercombinationthatfailedtoconvergewhenusingtheNewton – RaphsonoptimizationmethodwiththeBarkerestimationmodeland30-day samplingbins. 2013TheAuthors.MethodsinEcologyandEvolution 2013BritishEcologicalSociety, MethodsinEcologyandEvolutionSurvivalestimationfromcontinuousdata 5

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operatejointlymakeitdiculttounequivocallyascertainwhy biasincreasesatlow s .Onepossibilityisthatthesampleinformationatsuchvaluesislowenoughtogenerateparameter identi“ablyproblemsassociatedwithproblematicjointpro“le likelihoods(Ponciano etal. 2012).Findinganapproximation ofthebiasinaverysimplecase forwhichthelikelihoodfunctionallowsananalyticaltreatmentoftheproblemmayshed lightonthisissue. Despiteanexpectationthatincreasingsamplesize( n )would improveCJSmodelperformance,increasing n resultedinan unchangedbiasanddecreasingcoverageduetooverlynarrow con“denceintervals.Thus,wecreatedparametricbootstrap con“denceintervalssincetheyhavebettercoverageproperties whentheMLestimateisunbiased(Efron&Tibshirani1993). However,ourimplementationofparametriccon“denceintervalsexacerbatedthecoverageproblem.Theconstantbiasin parameterestimatesacrosssa mplesizessuggeststhataparametricbootstrapconstantbia scorrection(constantacross dierentvaluesofthetrueparameters)oftheestimatemay improvecoveragepropertiesandthuswarrantsadetailed simulationstudyexploringthisissue. Duringthedisturbanceeventsimulation,weusedanarbitrarymethodofcapturehistorycreationfortheCJS,inwhich wehadtheabilitytoperfectlybracketthedisturbanceevent withinasinglesamplingperiod.Evenwiththisprescient knowledge,theCJSreturnedbiasedestimates,makingitunlikelytoperformwellunder“eldconditionswheresuchknowledgedoesnotexist.TheCJSspoorcoverageanddicultlyin dealingwithbiologicalcomplexityseemtomakethismodela poorchoiceforusewithcontinuousdatainreal-lifeapplicationsthatrequiretime-depen dentestimates.However,while estimateswerebiasedinthetime-dependentsimulation,the Percent coverage Relative bias0850 0875 0900 0925 0950 –0050 –0025 0000 0025 0050 02505007501000Number markedValue Model Barker CJS Fig.6. PercentcoverageandrelativebiasfortheBarkerandCormack – Jolly – Seber(CJS)undervariousnumbersofmarkedindividuals( n ).CJS capturehistorieswerecreatedundermethodtwo,andoptimizationfortheBarkerwasconductedbysimulatedannealing.Relativebiascalculatedas theaverage,relativedeparturefromthetru esurvivalprobabilityover1000iterationsper n andwasplottedwiththe2 5 – 97 5percentilesofthedistributionoftheserelativedepartures.Onthe x -axis, n wasreducedby10fortheCJSandincreasedby10fortheBarkermodelforvisualclarity. Table2. AkaikesInformationCriterion(AIC)tableresultsfrom simulationthree,whichincludeda1-monthdisturbanceevent Model k MeanAICcMean AICc%AICcselected (a) sm( t )74620 10 0100 sm( d )34812 0191 90 0 (b) sm( d )44618 10 089 9 sm( t )84622 24 110 1 Monthlysurvival( sm)estimateswereeithertime-dependent sm( t )or timeindependentexceptforthedisturbanceperiod sm( d ).Thenumber ofestimatedparameters( k ),themeanAICscoreover1000simulated iterationsandthepercentofitera tionsgivingAICsupporttoamodel aregiven.Thesimulationranforthe:(a)Cormack – Jolly – Seberand(b) Barkerjointdatamodels. 04 06 08 10 12345 Time periodSurvival estimateModel Barker Truth CJS Fig.7. Time-dependentestimatesofmonthlysurvival, sm( t ),forthe Cormack – Jolly – Seber(CJS)andBarkerestimationmodelsplotted againstthetruevaluefor sm.CJScapturehistorieswerecreatedunder methodtwo,andoptimizationfortheBarkerwasconductedbysimulatedannealing.Meansurvival estimatesplottedwiththe2 5 – 97 5percentilesofthedistributionofmaximumlikelihoodestimatesfromthe 1000iterations. 2013TheAuthors.MethodsinEcologyandEvolution 2013BritishEcologicalSociety, MethodsinEcologyandEvolution6 A.B.Barbour,J.M.Ponciano&K.Lorenzen

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CJSsuccessfullyapproximatedth eoverallsurvivalduringthe simulation.Thus,ifastudyis designedtomeasureoverall survivaltheCJSmaybeanappropriatechoice. ThesuccessoftheBarkerjoint datamodelisnotsurprising, sincethemodelwasformulatedforasituationinwhichcontinuousresightingsoccurredbetweendiscretesamplingintervals (Barker1997).AlthoughtheBar kermodelperformedwell,it didnotreach95%coverageofthetruesurvivalvalueinall simulations.Additionally,the practicalitiesofimplementing theBarkermodelwithcontinuousdatawerenotwithoutdiculties.Inoursimulation,wedi dnotconductdiscretesampling events,whichallowedustocomputethelikelihoodby“xing thevaluesoffourparameters.Inrealsituations,however,all sevenparametersinthemodelmayneedtobeestimated.Since theparametersmayvarybytime,group,orbeassociatedwith covariates,theBarkermodelrequiressubstantialexperienceto properlyformulatean apriori modelset.Ifthelikelihoodsurfaceswereproblematicwithon lythreeparameters,wewould expectnontrivialmaximizationproblemswhenthefullmodel isimplemented,particular lywhenthenumberofunknown parametersislargerelativetothedatasetathand.Thepracticaldicultyofimplementingt heBarkermodelcombinedwith itsshortcomingsduringtwosimulationsleavesroomforthe speci“cationofnovelstatistica lmethodstailoredspeci“cally forcontinuousmark – resightingdata.Astartingpointto achievesuchagoalcouldbeworkingwithcontinuous-time survivalstochasticprocessmod elswhosetransitionprobability matrixcorrespondexactlytothetransitionmatrixofafamily ofdiscrete-timestochasticprocesses(Allen2010). Oursimulationrepresentst he“rststeptowardsunderstandinghowtobestusecontinuousdatainsurvivalestimation.Withtheexceptionofthedisturbanceevent,weonly simulatedtime-independents urvivalanddetectionprobabilitywhileexplicitlyignoringem igration,whichisnotlikelyto bere”ectiveofbiologicalreality.AlthoughtheBarkermodel isdesignedtoaccountforrandomemigration(Barker1997) andhasbeenshowntoeectivelyhandlesuchmovement (Horton&Letcher2008),themodelsrobustnesstoemigrationwhentheparametersdesignedtodealwithemigration (FandF )are“xedisunknown.Whileourstudyignored theissue,wearecurrentlyusingempiricaldatafromour “eldresearchsitetodeterminerealisticratesofemigration, whichwillbeusedtoextendthiswork(A.B.Barbour, unpublisheddata). Althoughwefocusedsolelyontheuseofcontinuousdatain theestimationofdiscretesurvival,thisproblemmayhaveparallelsinothercontexts.Forinstance,datamaybegrouped alongspaceinsteadofthetimeaxis,despitetheacknowledged importanceofspatialheterogeneity(VanKirk&Lewis1997; Neubert&Caswell2000).Itisu nknown,forexample,ifthe discretizationofmodern,large -scaleGIS(GeographicInformationSystems)dataofspatialabundancedistributionsmay leadtobiasedabundanceestimators(Kleiber&Hampton 1994;Sibert etal. 1999;Adam&Sibert2002).Themathematicalintricaciesof“ndingthecorrecttime-scalerepresentation formodelling,estimationandtes tingofthebiolo gicalprocess ofinterestineachcasearenot trivial.Inthecontextofmark – recapturemodels,itisnecessarytoinvestigatewhentheunderlyingdiscrete-timeMarkovia nstructureintheBarkermodel canbeapproximatedwithacontinuous-timeMarkovprocess (e.g.seeKarlin&Taylor1981,chap.15,section2.F). Reliableunderstandingandpredictionofcomplexecologicaldatahingesontheformulatio nofproperstatisticalmodels toquantifybiologicalprocesse swhileaccountingforthesamplingschemeused.However,theecologicalliteratureis“lled withexampleswhereo-the-shelfstatisticalmodelshaveproventobeaninsucienttooltogenerateunderstandingofthe biologicalprocessesofinterestsimplybecausetheyarenottailoredtotheapplicationathandandassuch,areunabletoharnesstheinformationinthedataeectively(e.g.Strong etal. 1999).Here,wefocusedoninformingtheoreticiansandpractitionersalikeabouttheinferentialproblemsassociatedwith temporalgroupingpracticesinsurvivalestimation.Thiswork shouldbetakenasapositive “rststeptowardsseekinga model-centredsolutiontosuchdiculties.AcknowledgementsWethankJ.Nichols,J.HinesandM.Connerforcommentshelpfulinthedesign ofthesimulationandM.Allen,A.Adams,D.Behringer,DavidKoonsandone anonymousreviewerfortheirvaluableinsights.ABBwassupportedbya NationalScienceFoundationGraduate ResearchFellowshipunderGrantNo. DGE-0802270.K.L.acknowledgesfundingfromtheFloridaFishandWildlife ConservationCommission,ProjectNo.11409.ReferencesAdam,M.S.&Sibert,J.R.(2002)Populationdynamicsandmovementsofskipjacktuna( Katsuwonuspelamis )intheMaldivian“shery:analysisoftagging datafromanadvection-diusion-reactionmodel. 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