Efficient Sampling and Robust Abundance Estimation in Depletion Surveys

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Title:
Efficient Sampling and Robust Abundance Estimation in Depletion Surveys
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1 online resource (119 p.)
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english
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Bohrmann,Thomas F
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University of Florida
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Degree:
Doctorate ( Ph.D.)
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University of Florida
Degree Disciplines:
Interdisciplinary Ecology
Committee Chair:
Christman, Mary C
Committee Co-Chair:
Silliman, Brian
Committee Members:
Bolker, Benjamin M
Dorazio, Robert
Allen, Michael S

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Subjects / Keywords:
abundance -- depletion -- removal -- survey
Interdisciplinary Ecology -- Dissertations, Academic -- UF
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Interdisciplinary Ecology thesis, Ph.D.
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Abstract:
Surveys of animal abundance are central to the conservation and management of living natural resources. However, many species complicate the sampling process because of detection uncertainty. One sampling method employed to deal with this problem is depletion (or removal) surveys in which animals are sequentially removed (and not replaced) from a closed subunit of the population. The pattern of decline in catches in successive removals provides simultaneous information about the capture or detection probability (which we call catchability) and the abundance process. Our research focuses on two aspects of depletion surveys. First, we describe results and methodology useful in designing efficient surveys of abundance, i.e. learning as much about population abundance as possible for each dollar spent on sampling. Second we describe a novel method for estimation of abundance which includes both the advantages of model-based inference and design-based inference. We show that the coherent combination of these two estimation frameworks yields a useful, robust total abundance estimator. We compare the performance of this ?hybrid? estimator with a fully model-based approach using a simulation study. Throughout we refer to and provide results regarding a data set of depletion surveys of Chesapeake Bay blue crab (Callinectes sapidus) which is part of their annual stock assessment.
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In the series University of Florida Digital Collections.
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Includes bibliographical references.
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This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
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by Thomas F Bohrmann.
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Thesis (Ph.D.)--University of Florida, 2011.
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Adviser: Christman, Mary C.
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Co-adviser: Silliman, Brian.
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RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2013-08-31

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EFFICIENTSAMPLINGANDROBUSTABUNDANCEESTIMATIONINDEPLETIONSURVEYSByTHOMASF.BOHRMANNADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2011

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c2011ThomasF.Bohrmann 2

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Tomymomanddad,whostartedmakingthiswholethingpossiblealong,longtimeago 3

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ACKNOWLEDGMENTS Iwouldliketothankthemembersofmysupervisorycommitteefortheirusefulinputandconstructivecriticism.ParticularlyIwouldliketoacknowledgethechairofmycommitteeforplayinganirreplaceableroleinmygraduatetraining.Withouthernoneofthiswouldbepossible.IwouldalsoliketothanktheDepartmentofStatisticsandtheSchoolofNaturalResourcesandEnvironmentattheUniversityofFloridaforprovidingmetheopportunitiesneededtoconductthisresearch,alongwiththegeneroussupportoftheNationalScienceFoundation'sQSE3IGERTprogramwhichhasbeensoinuentialinmytraining.Finally,withoutthehelpofmymotherandlatefatherIneverwouldhavebeeninthepositiontoundertakesuchajourney,andIamforeverindebtedtothemforthisandsomanyotherthings. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 7 LISTOFFIGURES ..................................... 8 ABSTRACT ......................................... 9 CHAPTER 1INTRODUCTION ................................... 10 2EVALUATINGSAMPLINGEFFICIENCYINDEPLETIONSURVEYSUSINGHIERARCHICALBAYES .............................. 14 2.1Background ................................... 14 2.2Methods ..................................... 16 2.2.1BayesianHierarchicalDepletionModel ................ 16 2.2.2BayesianSampleSize ......................... 20 2.2.3ConnectingSampleSizetotheHierarchicalModel ......... 21 2.3Application ................................... 25 2.3.1OverviewofBlueCrabStudy ..................... 25 2.3.2ModelSpecicationandPriorDistributions .............. 26 2.4Results ..................................... 28 2.5Discussion ................................... 29 3OPTIMALALLOCATIONOFSAMPLINGEFFORTINDEPLETIONSURVEYS 33 3.1Background ................................... 33 3.2Methods ..................................... 35 3.2.1TheModel ................................ 35 3.2.2InformationDerivation ......................... 37 3.3Results ..................................... 43 3.4Discussion ................................... 43 4ROBUSTTOTALABUNDANCEESTIMATIONINDEPLETIONSURVEYS ... 49 4.1Background ................................... 49 4.2Methods ..................................... 51 4.2.1Multiple-PassSiteModel ........................ 52 4.2.2Single-PassSites ............................ 54 4.2.2.1Partialpopulationestimator ................. 55 4.2.2.2VarianceofbTp ........................ 57 4.2.2.3EstimatingthevarianceofbTp ................ 59 4.2.3TheModel ................................ 62 5

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4.2.4BayesianEstimationofModelParameters .............. 63 4.2.5AsymptoticDistributionoftheHybridAbundanceEstimator .... 65 4.3AFullyModel-BasedEstimator ........................ 67 4.4SimulationStudy ................................ 68 4.4.1SimulatingPopulationsandDepletions-DataGeneratingModel 70 4.4.2EstimationofPopulationAbundance-DataAnalysisModel .... 71 4.5SimulationResults ............................... 75 4.6Discussion ................................... 76 5CONCLUSION .................................... 83 APPENDIX APARTIALDERIVATIVESUSEDINCHAPTER3 .................. 88 BTABLEOFINFORMATIONVALUESBYCATCHABILITYANDNUMBEROFDEPLETIONPASSES ................................ 89 CRANDWINBUGSCODEUSEDTOIMPLEMENTTHEMETHODSOFCHAPTERS2,3AND4 ...................................... 94 C.1Chapter2Code ................................. 94 C.2Chapter3Code ................................. 102 C.3Chapter4Code ................................. 104 DRELATIONSHIPBETWEENABUNDANCEANDCATCHABILITYANDPOSTERIORUNCERTAINTYOFABUNDANCEESTIMATES .................. 114 ESEVERALEXAMPLESOFPOORLYBEHAVEDMARKOVCHAINS ...... 115 REFERENCES ....................................... 116 BIOGRAPHICALSKETCH ................................ 119 6

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LISTOFTABLES Table page 2-1Estimatedaverage95%HPDcredibleintervallengthsforsiteabundanceforthevevesselsusedtosurveybluecrabandfornumberofdepletionpassesp=2,3,...,10. .................................... 32 2-2Posteriormeansforcatchabilityofthevevesselsandthestandarderrorsofthoseestimates. ................................... 32 3-1Optimalnumberofdepletionpassespersiteforcatchabilityvalueslessthan.50. .......................................... 48 3-2Optimalnumberofdepletionpassespersiteforcatchabilityvaluesgreaterthan.50. ........................................ 48 4-1Partialresultsfromasimulationstudycomparingourhybridabundanceestimatortoafullymodel-basedestimator. .......................... 82 4-2Additionalresultsfromasimulationstudycomparingourhybridabundanceestimatortoafullymodel-basedestimator. .................... 82 B-1Fisherinformationvaluesforcombinationsofcatchabilityandnumberofdepletionpassespersiteforcatchabilityvalueslessthanorequalto.50. ......... 89 B-2Fisherinformationvaluesforcombinationsofcatchabilityandnumberofdepletionpassespersiteforcatchabilityvaluesgreaterthan.50. .............. 91 7

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LISTOFFIGURES Figure page D-1Expectedposteriorstandarddeviationofsiteabundanceestimates ....... 114 E-1Severalexamplesofpoorly-behavedMarkovChains ............... 115 8

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyEFFICIENTSAMPLINGANDROBUSTABUNDANCEESTIMATIONINDEPLETIONSURVEYSByThomasF.BohrmannAugust2011Chair:MaryC.ChristmanCochair:BrianR.SillimanMajor:InterdisciplinaryEcologySurveysofanimalabundancearecentraltotheconservationandmanagementoflivingnaturalresources.However,manyspeciescomplicatethesamplingprocessbecauseofdetectionuncertainty.Onesamplingmethodemployedtodealwiththisproblemisdepletion(orremoval)surveysinwhichanimalsaresequentiallyremoved(andnotreplaced)fromaclosedsubunitofthepopulation.Thepatternofdeclineincatchesinsuccessiveremovalsprovidessimultaneousinformationaboutthecaptureordetectionprobability(whichwecallcatchability)andtheabundanceprocess.Ourresearchfocusesontwoaspectsofdepletionsurveys.First,wedescriberesultsandmethodologyusefulindesigningefcientsurveysofabundance,i.e.learningasmuchaboutpopulationabundanceaspossibleforeachdollarspentonsampling.Secondwedescribeanovelmethodforestimationofabundancewhichincludesboththeadvantagesofmodel-basedinferenceanddesign-basedinference.Weshowthatthecoherentcombinationofthesetwoestimationframeworksyieldsauseful,robusttotalabundanceestimator.Wecomparetheperformanceofthishybridestimatorwithafullymodel-basedapproachusingasimulationstudy.ThroughoutwerefertoandprovideresultsregardingadatasetofdepletionsurveysofChesapeakeBaybluecrab(Callinectessapidus)whichispartoftheirannualstockassessment. 9

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CHAPTER1INTRODUCTIONAnimalabundanceestimationiscentraltothemanagementofimportantpopulationswhetherthemanagementismotivatedbyconservationobjectivesorbecausethepopulationisasourceoffoodforhumans.Surveysofanimalabundanceofteninvolvethecomplicatingfactorofimperfectdetectabilityoftheanimals.Insuchcases,animalsofinterestmayinfactresideatasurveysitebutgoundetectedbytheresearchersurveyingthesite.Thisresultsinobservedcountsthatareoftenlowerthantheactualnumberofanimalsatthesurveysite.Oftendetectability,denedasthelikelihoodananimalpresentatasurveysiteisincludedinthecount,isestimatedjointlywithabundanceusingsamplingschemesthatcapturebothprocesses.Anappropriatemodel,then,forthesamplingprocessaccountsforuncertaintyinbothdetectabilityandabundance.Astandardmodelforthissamplingprocessisthebinomialdistribution(Moran1951;RoyleandDorazio2006,2008)withunknownprobabilityequaltothedetectabilityrateandalsounknownsizeparameter.Theassumptionsmadeaboutthecaptureprocessrelatetothestructureofthemodelandmodelsmaybemodiedtomoreaccuratelyportraythedetectionprocess.However,acommonunderlyingassumptionisthattheeventofananimalbeingcapturedorobservedduringasurveyisindependentfromthecaptureorobservationofotheranimals.ThebinomialdatamodeloranequivalentmultinomialformulationisappliedthroughoutthisdissertationandtheaccompanyingassumptionsareoutlinedanddiscussedinChapters2,3and4.Thesamplingparadigmofinterestinthisdissertationisdepletion,alsoknownasremoval,sampling.Indepletionsamplinganimalsareremoveduponcapturebytheresearcheruntilsamplingiscompleted.Multiple,successivedepletionsatasurveysite,i.e.animalsarecaptured,countedandwithheldfromthelocalpopulationuntilsamplingterminates,provideameansofestimatingbothabundanceanddetectability. 10

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Indepletionsurveys,werefertodetectabilityascatchability,sincethesamplingtypicallyinvolvescapturinganimals.Catchabilityisdiscussedindepththroughoutthisdissertationhoweverwedeneitformallytobetheprobabilityananimalthatispresentatadepletionsiteiscapturedduringadepletion.Oftenassumptionsaboutcatchabilitymotivatethenalformofthesamplingmodel.Forexample,Manytiemi,Arjas,andRomakkaniemi(2005)describeamodelinwhichcatchabilitychangesfromoneanimaltoanother,i.e.thereisnosinglecatchabilityparametercommontoallanimals.Relaxationofassumptionsmayalsoinvolvethesetofanimalssusceptibletocapture.Forexample,inmultipledepletionsatasite,animalsmayenterorleavethedepletionareaduringthesurveyortheresearchermayinadvertentlyshiftsamplinglocationduringthedepletionsurvey.Suchoccurrencesviolatetheassumptionthatthelocalpopulationatadepletionsiteisclosedthroughoutthesurvey.However,appropriatemodicationstoadepletionmodelaccountforsuchbehavior(cf.Ragoetal.2006).Mathematicaldepletionmodelswerepresentedasearlyasthe1930s(LeslieandDavis1939).LaterMoran(1951)consideredtheproblemfromastatisticalstandpointandoutlinedthebinomialformulationforthesamplingprocess.Thecorebinomialsamplingmodelortheequivalentmultinomialformulationformultiplepasssurveyspervadesthedepletionmodelingliterature(Zippin1956;Seber1982;Pollock1991;Wyatt2002;Dorazio,Jelks,andJordan2005).AlthoughChapters2,3and4ofthisdissertationconsiderdifferentaspectsofdepletionsurveys,withineachchapterthebinomialormultinomialdistributioniscentraltothemodeling.Therstaspectofdepletionsurveysweconsideristheeffectofdifferentdepletiontechniques.InChapter2,wedeneadepletiontechniquetobeanyapparatusorsetofpersonnelthatisusedinthedepletion.Forexample,thechoiceofvesselorgeartypemayinuencesamplingefciency(i.e.howmuchtheresearcherlearnsaboutabundanceorcatchabilityfromconductingdepletionsurveys).Ifmultipledepletiontechniqueshavebeenemployedinthepastthenthesamplingefciencyofthese 11

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techniquesmaybeevaluatedandcomparedusingthemethodswepresentinChapter2.ThesemethodsinvolvethecouplingofaBayesianhierarchicalmodelingparadigmwithmethodsfromtheBayesiansample-sizeliterature.Applicationofourmethodprovidesestimatedexpectedcredibleintervallengthsforlocalabundancebasedondatafromdepletionsurveysconductedbyeachofthetechniques.Estimatingcredibleintervallengthsisusefulforcomparisonamongtechniques.Thatis,techniqueswhichareexpectedtohaveshortercrediblelengthsforabundancearelikelytoyieldmoreinformationperuniteffortorcost.Suchinformationmaybeusefulwhenconsideringwhichtechniquestouseforfuturedepletionsurveys.Chapter3alsoinvolvesthedesignofefcientdepletionsurveys.Whenasubsetofsitesischosenatwhichtoconductmultipledepletionpasses,oftenthefocusofthosesitesistolearnaboutthecatchabilityparameter.Highprecisionofestimatesofcatchabilitymaytranslatetohighprecisionofabundanceestimatesatothersiteswherethesamecatchabilityprocessisinvolvedinthesampling.Fisherinformationmeasurestheamountofinformationaboutaspeciedparameterinasample(CasellaandBerger2002).Technically,maximizingFisherinformationaboutagivenparameterisequivalenttominimizingthevarianceofthebestunbiasedestimatorofthatparameter(CasellaandBerger2002).InChapter3,wederivetheFisherinformationforthecatchabilityparameterandmaximizeitwithrespecttoaspectsofthesamplingdesign.Inparticular,weobtainanoptimalallocationofeffort(howmanymultiple-passsitesversushowmanypassespersite)withrespecttotheFisherinformationaboutcatchability.Suchresultsarefurtherusefulindesigningfuturedepletionsurveysinwhichasubsetofsitesarespeciedasmultiple-passsitesandanothersetisspeciedassingle-passsites.Insuchadesign,theinformationaboutcatchabilityobtainedfromthemultiple-passsitescombineswiththesingle-passcountstoyieldestimatesofabundance.Whentotalabundanceestimationistheobjectivetypicallyadesignwillinvolvesamplingasubsetofthehabitatofthepopulationofinterest.Eithermodel-basedor 12

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design-basedorsomecombinationofthesemethodsprovideoptionsforestimatingabundance.Little(2004)reviewedthesestatisticalframeworksandalsodiscussedtheconceptofhybriddesign-based,model-assistedestimators.Fullymodel-basedestimatorsfordepletiondatasuchasthoseofWyatt(2002)andDorazioetal.(2005)requirethespecicationofamodelfortheanimalabundanceprocess,andhenceparticularknowledgeofthepopulation.Alternately,design-basedmethodssufferwhendetectabilityisunknown(ThompsonandSeber1994).InChapter4weintroduceahybridestimatorusefulfortotalabundanceestimationunderadepletionsamplingschemethatdealscoherentlywithbothoftheseissues.Ourhybridestimatorhastheadvantagethatnomodelneedbespeciedfortheabundanceprocess(buthastheexibilitytoincorporateanabundancemodel)whileamodeldescribingthecatchabilityprocesscanbespecied.Thisallowsaresearchertoaccountforsite-to-sitevariationincatchabilitywithanappropriatemodelofcatchabilitywithoutrequiringanabundanceprocessmodeltobespecied.Wedemonstrateinasimulationstudytheilleffectsofmis-specifyingtheabundanceprocessmodelonafullymodel-basedestimatoraswellasgoodperformanceofourestimatorunderanalogousmis-specication.Theproblemofabundanceestimationispervasiveintheecology,conservationandsheriesscience.Therefore,inthisdissertationweaimtoimproveuponexistingmethodologyusefulforsomesubsetofsuchstudies.Theconceptofdepletionsurveys,whicharecommonlyusedforanimalabundanceestimation,hasprovideduswithameanstohopefullyimprovetheprocessofestimatingtotalabundanceforsometypesofanimals.WereferthroughoutthedissertationtothepopulationofChesapeakeBaybluecrab(Callinectessapidus)whichisthebasisforourmotivatingdataset,butwebelievethemethodspresentedinChapters2,3and4andperhapstheideasforfutureresearchpresentedinChapter5willproveusefulformanyotherpopulationsofinterest. 13

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CHAPTER2EVALUATINGSAMPLINGEFFICIENCYINDEPLETIONSURVEYSUSINGHIERARCHICALBAYES 2.1BackgroundBoththemanagementandconservationoflivingnaturalresourcesrequiretheestimationofpopulationabundance.However,abundanceestimationcanbecomplicatedforvarioustaxa;complicationssuchasless-than-perfectdetectabilityprofoundlyimpactthesamplingprocessandsubsequentestimationofabundance.Less-than-perfectdetectabilityaddsalayerofuncertaintybecausecountsmadeatasitedonotreecttheabundancethere,onlysomeunknownproportionofit.Onemethodofestimatingtheproportionbeingcollectedistoconductdepletion(orremoval)sampling.Indepletionsampling,personnelsuccessivelyremoveindividualsfromsomedesignatedareaofthepopulation,withholdingthoseindividualsfromthepopulationuntilthesurveyiscomplete.Inthismanner,diminishingcatchesfromthedepletedpopulationprovideinsightintobothlocalabundanceandcatchability,withcatchabilitydenedtobetheprobabilityananimalatthesurveysiteiscaughtduringadepletionpass.Thisinformation,coupledwithanappropriatestatisticalmodel,allowstheresearchertoestimateabundancebothatthesitesofthesurveysandelsewhere.Unfortunately,accurateestimationofpopulationabundancesismademoredifcultbydecreasingnaturalresourcebudgetsandthereforemaximizingsamplingefciencyisvital.Inthismanuscriptwepresentmethodsforevaluatingsamplingefciencyindepletionsurveysbycouplingthestatisticalmethodologiesofhierarchicalmodelingandsamplesizeevaluationinanovelway.Hierarchicalmodelinginvolvesdescribingprocessesofinterestusingindividualmodelswhicharethencoupledinalargermodelingframework(RoyleandDorazio2008).Samplesizeevaluationinvolvesconnectingtheinuencesofsamplesizeonmeasuresofuncertaintyaboutspeciedparameters.Samplingefciencyrelatestotheamountofinformationobtainedperuniteffortorcost.Weaimtoevaluateandcomparethesamplingefcienciesofdatacollection 14

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techniques(e.g.gear,vesselorevenpersonnel)usedindepletionsurveys.Todoso,werstrequireadepletionmodel.SuchmodelsdatebacktotheworkofLeslieandDavis(1939),andwerefurtherdevelopedbyMoran(1951)andZippin(1956).Morerecently,researchershaveimplementedBayesianorlikelihood-basedhierarchicalmodelsfortheanalysisofdepletionsurveys(Wyatt2002;Dorazioetal.2005;Mantyniemietal.2005;andRivotetal.2008).InBayesianstatistics,dataarecombinedwithpriorknowledgeandamodeltoyieldinformation(theposteriordistribution)thatisapplicableforconductinginferencesabouttheparametersofinterest(Gelmanetal.2004).Further,hierarchicalmodelingallowscoupledphysicalorbiologicalprocessestobemodeledindividuallyandthencombinedcoherently(RoyleandDorazio2008;Cressieetal.2009).Forexample,indepletionsurveyswecompartmentalizetheabundance,catchabilityanddataprocessesandcombinethemtoformthehierarchicalmodel,therststatisticalmethodologyemployedhere.Thesecondstatisticalmethodologyinvolvesevaluationofsamplesize,whichweconsiderfromtheBayesianperspectiveinordertocomplementourdepletionmodel.Often,Bayesiansamplesizeevaluationstargetposteriorvarianceasameasureofinformationandconnectthattotheamountofdatacollected(e.g.Pham-GiaandTurkkan1992).However,weapplythemethodsofJoseph,Wolfson,andDuBerger(1995)andconnectsamplesizetothelengthofthehighestposteriordensity(HPD)credibleintervalforsomespeciedposteriorprobabilitylevel.HPDcredibleintervalsaredenedtobethesmallestsetofparametervaluescomprisingsomespeciedproportionoftheposteriordistribution(inthiscase,(1)]TJ /F10 11.955 Tf 12.69 0 Td[())andarenotnecessarilycenteredaroundtheposteriormean.ThelengthoftheHPDcredibleintervaliscloselytiedtothevarianceoftheposteriordistribution.HPDcredibleintervalsarealsoeasilycalculatedandareinterpretedonthescaleofthevariableofinterest,inthiscaseabundance.ByconnectingthesetwoBayesianmethodologies,theresearchercanevaluatetheconnectionbetweensamplesize(numberofdepletionpasses)and 15

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informationintheposteriordistributionofpopulationabundanceatthestudysite.Inparticular,foragivendepletiontechnique,wedescribehowtoestimatetheexpected(1)]TJ /F10 11.955 Tf 12.77 0 Td[()HPDcredibleintervallengthforanynumberofdepletionpasses,whichcanbeusedtocomparedepletiontechniquesorspecifysamplesizesforfuturesurveys.Forillustration,weapplythemethodologytoadatasetofdepletionsurveysforbluecrab(Callinectessapidus)intheChesapeakeBay.Weestimateexpected(1)]TJ /F10 11.955 Tf 12.26 0 Td[()HPDcredibleintervallengthsforvevesselspreviouslyusedindepletionsurveysofthebluecrabandforvaryingnumbersofdepletionpasses.Wediscussthediscrepanciesinsamplingefciencyrevealedbyapplyingthismethodologyandtheimplicationsforfuturesampling.Finally,weprovideinsightintowhydepletiontechniquesvaryintermsoftheirsamplingefciency. 2.2Methods 2.2.1BayesianHierarchicalDepletionModelWebeginbypresentingageneralhierarchicalBayesianmodelusefulforanalyzingdepletionsurveydata.Thehierarchicalframeworkprovidesseveraladvantages.Hierarchicalmodelingindepletionsamplingpermitsustodeneunobservableprocesses(e.g.theunderlyingaveragepopulationdensity)andtocoherentlyconnecttheseprocessestoobservabledatawhichariseduringthedepletionsurveys(countsofanimals).Throughoutweassumethatcatchabilitycanvaryacrosssitesbutisconstantforallpassesatasite.Usingthisframework,wenowdenethegeneraldepletionmodel,thatis,withoutdistributionalassumptions,andpresentappropriatenotation.Wesupposethatdepletiontechniquet=1,2,...,ThasbeenusedtosurveyItsitesyieldingatotalofI=TXt=1Itdepletionsites.Ingeneral,wewilluseasuperscriptofttodenoteaquantity'srelationshipwithdepletiontechniquet;forexample,catchabilitiesandcatchdata 16

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dependonwhichtechniquewasused.AlsodeneSttobethesetofsitessurveyedbytechniquet.Allowingitoindexthesites,deneqtitobethecatchabilityoftechniquetatsitei,andletNibetheabundance.LetYtikrepresentthecountobservedusingtechniquetduringthekthdepletionpassatsitei,wherek=1,2,...,piwithpitotaldepletionpassesatsitei.Assumingforconveniencepi=patallsites,wesummarizethedataatasiteasthevectorYti=fYti1,Yti2,...,Ytipg.Finally,denetheItpmatrixYttosummarizethedatafromthetthtechniqueandhavingjthrowYtj.Weassumeabundanceatasite,Ni,isarandomvariabledependentontheknownareaofthesite,ai,andanimaldensityatthatsite,i.Fordistributionsofrandomvariables,wedenotetheprobabilitymassfunction(p.m.f.)orprobabilitydensityfunction(p.d.f.)ofarandomvariableXby[X]indicatingProb[X=x]andtheconditionaldistributionofXgivenZandWtobe[XjZ,W]indicatingProb[X=xjZ,W].ThusNihasp.m.f.[Niji,ai].Further,thedensityiisitselfrandomanddistributed[ij]withparametersthatdescribetheabundanceprocessatthepopulationlevel.Finally,intheBayesiancontextisrandomandhaspriordistribution[].Unliketheabundanceprocess,thecatchabilityprocessvariesbydepletiontechnique.Letcatchabilityatsiteiwhensurveyedbytechniquet,qti,dependonparameterstsothatqtihasp.d.f.[qtijt]withparameterst.Similartothehigherlevelabundanceparameters,weplacepriordistributionsonthet,[t].Weassumecatchabilitiesareindependentacrossdepletionsites,andwefurtherassumeindependencebetweencatchabilityandabundanceprocesses(i.e.thatcatchabilityisnotinuencedbyabundance).Havingdenedtheabundanceandcatchabilitydensities,let[YtijjNij,qti]bethep.m.f.forthecounts,conditionalonabundanceNijandcatchabilityqti.Nijisthenumberofanimalsavailabletobecaughtforthejthdepletionpassatsiteianddependsonthe 17

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initialabundanceNi.ThatisNij=Ni)]TJ /F6 7.97 Tf 13.63 15.21 Td[(j)]TJ /F3 7.97 Tf 6.59 0 Td[(1Xk=1Ytikfork=2,3,...,pandwhereNi1=Ni.Withmultiplesitesbeingsurveyed,itisconvenienttodenethevectorsNt=fNi:i2Stg,t=fi:i2Stgandqt=fqti:i2Stgforeachtechniquet.Byourindependenceassumptions,therespectivejointdensitiesoftheabundancesandcatchabilitiesare,foreachtechniquet:[Ntjt,a]=Yi2St[Niji,ai][tj]=Yi2St[ij][qtjt]=Yi2St[qtijt] (2)wherea=fa1,a2,...,aIgisthevectorofallsiteareas.Undertheassumptionofindependenceamongdepletionpasses,thisyieldsthefollowingjointp.m.f.forthecountsobservedbytechniquet:[YtjNt,qt]=Yi2StpYj=1[YtijjNij,qti]. (2)Havingdenedtheappropriatedensitiesandnotation,wenowdiscussestimationandinferenceviathemodel.InBayesianstatistics,estimationandinferenceareconductedbyconsideringaspectsoftheposteriordistributionofparametersofinterest,conditionalonthedata(Gelmanetal.2004).Foragivendepletiontechniquet,weobtainthejointposteriordensity,uptoanormalizingconstant,as [Nt,t,qt,,tjYt,a]/[YtjNt,qt][Ntjt,a][tj][][qtjt][t]. (2) 18

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TheproportionalityresultsfromBayes'Theorem.TheequalityholdsrstbecausethedataYtareindependentofthehigherlevelparameterswhenconditionedonNtandqtandsecondbecauseoftheassumptionofindependenceoftheabundanceandcatchabilityprocesses.Finally,toobtain( 2 )wehavealsousedthepropertythatjointdistributionscanberewrittenasproductsofappropriateconditionaldistributionsandmarginaldistributions.Wealsoconsiderthejointdistributioninvolvingalltechniquestogether.Doingsoisusefulbecausealthoughthecatchabilityprocessesvaryfromonetechniquetoanother,theabundanceprocessdoesnot.Inotherwords,weobtaininformationaboutthepopulationdensityparametersusingalldepletiontechniquestogether.ThejointdistributioninvolvingtheTtotaldepletiontechniquesconditionalonthesetofalldataY=fY1,Y2,...,YTgis[N1,N2,...,NT,q1,q2,...,qT,1,2,...,T,1,2,...,T,jY,a]/TYt=1[YtjNt,qt][Ntjt,a][qtjt][tj][][t]=[]TYt=1[YtjNt,qt][Ntjt,a][qtjt][tj][t]. (2)Weuse( 2 )toconductinferenceontheabundanceprocessdescribedbywhenmultipledepletiontechniqueshavebeenused;specically( 2 )isusedtoobtainthemarginalposteriordistribution[jY,a]whichweobtainbelow.Fromtheappropriatejointposteriordistribution( 2 )or( 2 )themarginalposteriordistributionoftheparametersofinterestisobtainedbyintegratingovertheotherunknownparameters.Thisallowsustoaccountfortheuncertaintyofallparametersevenwhenconductinginferenceonastrictsubsetoftheparameters.Forthepurposesofevaluatingsamplingefciencyofdepletiontechniques,weareinterestedinthefollowingmarginalposteriordistributions:NtjYt,aandtjYt,afort=1,...,TandjY,a.Thersttwoposteriordistributionsaredepletiontechnique-specic.Thethird 19

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aggregatesallobservationsacrossdepletiontechniquestoinformtheabundanceprocess.Thersttwomarginalposteriordistributionsareobtainedasfollows:[NtjYt,a]/ZZZZ[YtjNt,qt][Ntjt,a][tj][][qtjt][t]dqtdtddt (2)and[tjYt,a]/ZZZXNt[YtjNt,qt][Ntjt,a][tj][][qtjt][t]dtdqtd. (2)Themarginalposteriordistributionforissimplytheintegralof( 2 )overallotherparameters.Thisis[jY,a]/ZZZXN1XNT[]TYt=1[YtjNt,qt][Ntjt,a][qtjt][tj][t]dN1dNtd1dtd1dtdq1qt. (2)Expression( 2 )allowsustoestimateabundancesatthesiteswheredepletionsamplingwasconducted.Expression( 2 )isusefulasameansofunderstandingimportantfeaturesofcatchabilityforeachoftheTtechniques.Laterwewillrequiresimulationtoestimatecertainquantities,and( 2 )willbeusedtogeneratelikelypopulationdensityvaluesfromwhichsimulatedabundancescaninturnbegenerated.Although( 2 )canbeusedtopredictabundancesatnon-sampledsites(andthereforeestimatetotalpopulationsize),thatisthesubjectofanothermanuscript. 2.2.2BayesianSampleSizeHavingintroducedthegeneralhierarchicalBayesiandepletionmodel,weconsidersamplesizeanalysisasawaytoinformdecisionsregardingdepletionsurveys.Thesetwomethodologiescombinetomeasuretheamountofinformationadepletiontechniqueisexpectedtoobtainduringsampling.Wemeasureinformationintermsofposterioruncertainty(variability)ofthesingle-siteabundanceestimateversusthenumberofdepletionpasses.Forexample,arelativelyefcientdepletiontechniquewillyieldless 20

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uncertaintyintheposteriordistributionofabundanceforagivennumberofdepletionpassesthananothertechnique.Weadaptthesample-sizemethodologyofJosephetal.(1995)inordertoevaluatedepletiontechniqueefciency.Ingeneral,samplesizeanalysisaimstoconnectsamplesizewithsomemeasureoftheinformationobtained.Researchersoftendesireaprespeciedamountofinformationandsamplesizemethodsprovidelikelysamplesizesrequiredtoyieldthatlevelofinformation.InBayesianstatistics,informationaboutaparameterorsetofparametersismeasuredbyvariabilityoftheposteriordistribution.Thehigherthevariability,themoreuncertaintythereisaboutaparameterandthereforethelessinformationtheresearcherhas.Althoughtherearemultiplewaystocharacterizetheuncertaintyintheposteriordistribution,Josephetal.(1995)measureuncertaintybythehighestposteriordensity(HPD)credibleintervallength.Foraspeciedcondencelevel,the(1)]TJ /F10 11.955 Tf 12.13 0 Td[()100%HPDcredibleintervalisdenedasthesmallestsetofpointsintheparameterspacehavinghighestposteriordensityvaluessuchthattheprobabilitytheparameterfallsinthatsetisequalto(1)]TJ /F10 11.955 Tf 12.52 0 Td[().Thatis,theminimumdensityvalue(orprobability)ofpointsintheHPDcredibleintervalisatleastaslargeasthedensityvalue(orprobability)ofanypointoutsidetheHPDcredibleinterval.MinimizingHPDcredibleintervallengthisanalogoustominimizingvariabilityoftheposterior,andthereforeprovidesausefulandconvenientmeasureofposteriorinformation. 2.2.3ConnectingSampleSizetotheHierarchicalModelWenowevaluatetheefciencyofadepletiontechniquewithrespecttoabundanceatasingledepletionsite.TheHPDcredibleintervalforNi,theabundanceatthesitei,sampledusingtechniquet,isadiscretesetofabundancevalues.WecallthisNHPDi:NHPDi=fn:Prob(Ni=njYti)Prob(Ni=kjYti)forallk=2NHPDig (2)undertheconstraintthatProb(Ni2NHPDijYti)(1)]TJ /F10 11.955 Tf 11.95 0 Td[()and 21

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Prob(Ni2NjYti)<(1)]TJ /F10 11.955 Tf 11.95 0 Td[() (2)foranystrictsubsetNNHPDi.Inotherwords,NHPDiisthesmallestpossiblesetofabundancevaluessummingtohaveposteriorprobabilityofatleast(1)]TJ /F10 11.955 Tf 11.95 0 Td[().ThecredibleintervalNHPDiforabundanceisbasedonthevectorofobservedcatchesatsitei,Ytianditslengthreectstheuncertaintyintheposteriordistributionofsingle-siteabundance.Therefore,byestimating(orcalculating,ifpossible)theexpectedHPDcredibleintervallengthforagiventechnique,wehaveameasureusefulforcomparingtheefciencyofmultipledepletiontechniques.However,estimatingexpectedcredibleintervallengthforagivendepletiontechniquetrequiresaveragingoverallpotentialcatchdatavectors.AdaptationoftheworkofJosephetal.(1995)allowsustodoso.First,letl(~y,t,)bethelengthofthe(1)]TJ /F10 11.955 Tf 12.54 0 Td[()100%HPDcredibleintervalfordepletiontechniquetandsinglesitecatchvector~y.The(1)]TJ /F10 11.955 Tf 12.68 0 Td[()100%HPDcredibleintervalisobtainedfromthemarginalposteriordistributionforabundance(thesingle-siteversionofexpression( 2 ))andl(~y,t,)isthelengthofornumberofvaluesinthatinterval.Wethenwishtoestimatetheexpectedvalueofl(~y,t,)withrespecttothedistributionofcatchvectorsforagivendepletiontechnique.Indoingso,wespecifythelengthsofthecatchvectors,i.e.,thenumberofdepletionpasses,tobe~p.WecallthisexpectationLt,~p()anditisdenedasLt,~p()=E[~ytjY,a](l(~y,t,)). (2)Thedistributionofpotentialcatchvectors,conditionalontheobserveddata,[~ytjY,a],isequivalenttotheposteriorpredictivedistributionofcatchvectorsfortechniquet(Gelmanetal.2004).Weobtainthisdistributionas[~ytjY,a]=ZZZXN[~yjN,q,~p][Nj,a][j][qjt][,tjY,a]ddqdtd 22

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(2)whereN,,qandarepresentsinglesiteabundance,density,catchabilityandarea,respectively.Noteinexpression( 2 )weconsiderexplicitlythelikelyposteriorcorrelationbetweentheparametersandtbyconsideringthemjointlyin[,tjY,a].Finally,toobtaintheexpectedHPDcredibleintervallength(expression( 2 ))fortechniquet(atthelevelandforppasses),wehaveLt,~p()=ZYpl(~y,t,)[~ytjY,a]d~y. (2)HereYpisthespaceofallpossiblep-lengthcatchvectorsonwhich[~ytjY,a]isthedistributionofcatchvectorsfortechniquet.Expression( 2 )averagesHPDcredibleintervallengthoverallpossiblecatchvectorsoflengthp.Thelikelihood,orweight,ofthesecatchvectorsisbasedontheabundanceprocessandthecatchabilityprocessfortechniquet.Thisinformationisimplementedviathemarginalposteriordistributions( 2 )and( 2 ).Asmentioned,obtainingthequantityl(~y,t,)involvescalculatingthe(1)]TJ /F10 11.955 Tf 12.09 0 Td[()100%HPDcredibleintervalusingexpression( 2 ),themarginalposteriordistributionofabundance.Inmanycases,( 2 )aswellas( 2 )and( 2 )willbeintractableintegralsthatwemustapproximateusingnumericalmethods.However,anappropriatelychosenapproximationcangreatlyincreasecomputationalefciencyforthismethod.Thisinvolvesthecomputationofl(~y,t,)usingadistributionthatapproximates[~Nj~y,Y,a](here~Nistheabundanceatthesitewithdepletiondata~y).Anapproximation,forcomputationalefciency,to[~Nj~y,Y,a]isthedensity[~Nj~y,bt,b,a]wherebtandbaretheestimates(e.g.estimatedmeans)of( 2 )and( 2 ),respectively.TheinformationfromYentersthedistributionthroughbtandb.Undercertain 23

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distributionalassumptionsthisgreatlysimpliesthedistributionusedtocalculatel(~y,t,).Intheapplicationwedemonstratetheutilityofthisapproximationwhenabetadistributionmodelsthecatchabilityprocess,apoissondistributionmodelsthecountsateachsite,agammadistributionmodelstheanimaldensityprocessandthedepletioncountdataaremodeledusingabinomialdistribution.Ingeneral,thesimplied,approximatemarginalposteriordistributionis[~Nj~y,a,bt,b]/ZZ[~yj~N,q][~Nj,a][jb][qjbt]dqd (2)whereqandarethecatchabilityandpopulationdensity.Aswewillseeintheapplication,( 2 )maybeavailableinclosedform,inparticularunderthedistributionalassumptionsdescribedaboveinwhichcaseobtainingtheHPDcredibleintervalisstraightforward.Withtheframeworkinplace,wenowdiscussfullimplementation.Typically,oneappliesMarkovChainMonteCarlo(MCMC)inordertosimulatedrawsfromintractableprobabilitydistributions(Gelmanetal.2004).Approximatingthemarginalposteriordistributions( 2 )and( 2 )willlikelyrequirefullMCMCmethods.Doingsoproducessimulateddrawsfromtheposteriordistribution.However,usingtheMCMCsimulateddrawsfrom( 2 )and( 2 ),weestimate( 2 )usingonlyMonteCarlointegrationbyemployingtheapproximation( 2 ).BycouplingfullMCMCwithMonteCarlointegration,wecanestimate( 2 )efciently.InaMonteCarloframework,equation( 2 )isapproximatedasfollows:apredictedvector~ytpforaspeciedtechniquetandlengthpissimulatedfrom~yjY,a.Toobtainsuchavector,rstalocalpopulationabundanceparametervector0issimulatedfromjYusing( 2 ).Nextapopulationdensityvalue0isdrawnfromj0.Then,anabundancevalueN0isdrawnfromNj,awherearepresentstheareaofthesimulatedsite.Similarlyforcatchabilityacatchabilityparametervectort0isdrawnfromt0jYtandthenacatchabilityq0isdrawnfromq0jt0.Usingthesimulatedabundanceand 24

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catchability,ap-lengthvectorofcatches~ytisdrawnfromytjN0,q0andthelengthoftheHPDcredibleintervall(~y,t,)iscalculatedusingtheapproximatingdistribution( 2 ).Repeatingthisprocessmanytimesandcalculatingtheaveragelengthofl(~y,t,)isanapproximationtointegrationoverYp,thespaceofallpossiblecatchvectorsoflengthp.TheresultofapplyingthismethodisanexpectedHPDcredibleintervallengthforeachdepletiontechniqueandforanynumberofdepletionpassesdesired.Theseresultsprovideabasisforcomparisonofdepletiontechniquesandalsocanbeusedforconsiderationofhowmanypassesarerequired,basedondesiredlevelsofinformationaboutthepopulation.Weillustratetheutilityofsuchresultsaswellastheprocessofapplyingthemethodologyinthefollowingapplication. 2.3Application 2.3.1OverviewofBlueCrabStudyAspartofanongoingmonitoringprogramofChesapeakeBaybluecrab(Call-inectessapidus)populationabundancebytheMarylandDepartmentofNaturalResources,depletionsurveyswereconductedat172sitesoveraseveralyearperiod.Fivedifferentvessels(T=5)eachcompleteddepletionsurveysofatleast10sites.Adepletionsurveyatasiteconsistedofp=6consecutivepassesofaboattowingadredgealongthebayoor.Samplingtookplaceduringthewintermonthswhenthebluecrabaredormantandsettledinthesubstrate.Thishelpedensurethattheassumptionthatasinglesiterepresentsaclosedpopulationduringthesixpasseswasreasonable.Thestudyareaswereuniformlya=550m2andeachpasswascomprisedofthreeconsecutivetowsofa1.83mwidedredgealongadjacent100mstrips.SeeVolstad,etal.(2000)formoreinformationonthedredging.Inordertocompleteafullstockassessment,alargesetofsingle-passsitesweresampledaswell,butwefocushereonthemultiple-passsurveys. 25

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2.3.2ModelSpecicationandPriorDistributionsSpecicationofthehierarchicalmodelinvolvesassigningdistributionstothecomponentsdescribedabove.Thisinvolvesspecicationofthecomponentsoftheabundanceprocess,thecatchabilityprocess,andthedataprocess.Therefore,werstdescribethedistributionalassumptions.Wethenprovidereasonablepriordistributionsfortheparametersinthehighestlevelofthehierarchy.Finally,wecalculatetheposteriordistributionsneededtoevaluatethesamplingefciencyofthevevesselsusedtosurveybluecrababundance.Beginningwiththeabundanceprocess,wesupposethetotalnumberofcrabsatsitei,depletedbythetthvessel,isaPoissonrandomvariablewithmean550i.ThatisNiPoisson(550i) (2)whereforeacht,i2St.Thedensityiisitselfarandomvariable,distributedasii.i.d.Gamma(1,2) (2)wheretheparameters1and2describethecrabpopulationdensityovertheentirebay.Notethatiftheobjectiveofthestudyweretoobtainanestimateofthepopulationtotal,theparameters1and2wouldlikelychangeeachyeartoreectthechangingpopulationsize.However,becauseourobjectiveistoevaluatethevessels'samplingefciency,wedene1and2toreectaveragecrabdensitythroughoutthestudy.Inthiscase,=f1,2g.Thecatchabilityprocess,whichisthefocusofthisstudy,differsfromvesseltovesselanddeterminesthesamplingefciencyofeachvessel.Wesupposetheqti,thecatchabilityatsitei,isindependentofcatchabilityatothersitesandhasdistributionqtii.i.d.Beta(t,t) (2)wheretheparameterst=ft,tgdescribethecatchabilityofthetthvessel. 26

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ThevectorofcountsYtiobtainedatasiteareconditionalontheabundanceandcatchabilityatthatsite.Therefore,wesupposethecountsfromthedepletionsurveyareasequenceofbinomialrandomvariables.Wesupposetheresultofthekthtowatsitei,surveyedbyvesselthasdistributionYtikBinomial(Nik,qti) (2)whereagainNi1=Ni (2)andNik=Ni1)]TJ /F6 7.97 Tf 12.66 14.94 Td[(k)]TJ /F3 7.97 Tf 6.58 0 Td[(1Xj=1Ytij (2)fork=2,3,...,6.Inwords,thenumberofcrabscaughtduringthekthpassatasiteisaBinomialrandomvariableconditionalonthecrabsavailable(initialnumberlessthecrabscaughtduringpreviouspasses)andthecatchabilityatthatsite.Finally,inordertoconductinferenceusingthemodelinaBayesianframework,thehighestlevelparametersinthemodelmusthavepriordistributionsassignedtothem.Thechoiceofapriordistributioncanbeaninvolvedprocess.Inthecasewherethepriordistributionismeanttoconveynoinformation,thesensitivityoftheposteriordistributiontothespeciedpriorsshouldbechecked.However,forthepurposesofthismanuscriptwedonotdiscussindepththespecicationofpriordistributions.Insteadweplacediffusepriorsontheparameters1and2andontandtfort=1,2,...,T.WeassigntoeachoftheseparametersthepriordistributionGamma(.01,.01)whichhasalargevariance.Therefore( 2 )and( 2 )arefullyspecied.Forthisapplication,weapproximate( 2 )using( 2 )asdescribedinthemethodssection.Becauseofourdistributionalassumptions,wecancalculatetheapproximatingdistribution( 2 )usingbt=fbt,btgandb=fb1,b2gfromapplying( 2 )and( 2 ). 27

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Forthisstudy,weusethemeansoftheappropriateposteriordistributionasestimates.Thisresultsinaclosedformposteriordistributionwhichweusetocalculatel(~y,t,)forarealizedcatchvector~y.Recallingtheroleofsimulationmethodsinthisstudy,hereweeliminatetheneedtousefullMCMCforeachsimulatedcatchvector,whichwouldbecomputationallyintensive.Theclosedformposteriordistributionuptothestandardgammafunction)]TJ /F1 11.955 Tf 6.78 0 Td[(()obtainedbysubstitutinginto( 2 )theappropriatedistributionsaswellasourestimatesbt,bt,b1andb2is[~Nj~y,a,bt,b]=[Nj~y,a,bt,bt,b1,b2]/a a+b2~N\(~N+b1) \(~N)]TJ /F5 11.955 Tf 11.96 0 Td[(C+1)\(C+bt)\(p~N)]TJ /F5 11.955 Tf 11.96 0 Td[(p~Y1)]TJ /F7 11.955 Tf 11.96 0 Td[((p)]TJ /F7 11.955 Tf 11.95 0 Td[(1)~Y2)-222()]TJ /F7 11.955 Tf 42.45 2.65 Td[(~Yp+bt) \(C+bt+bt+pN)]TJ /F5 11.955 Tf 11.96 0 Td[(p~Y1)]TJ /F7 11.955 Tf 11.95 0 Td[((p)]TJ /F7 11.955 Tf 11.95 0 Td[(1)~Y2)-222()]TJ /F7 11.955 Tf 42.45 2.66 Td[(~Yp) (2)wherenow~Yjrepresentsthesimulatedcatchduringthejthpassfortechniquetatasinglesite.Notewehavealsosuppressedsubscriptsforsite.Also,pstillrepresentsthenumberofsimulatedpassesandC=pXj=1~Yj (2)isthesumofthesimulatedvector.Touse( 2 ),whichisknownonlyuptoproportionality,wesimplyevaluatetheexpressionfor~N=C,C+1,C+2,...,bwherebissomelargeupperbound.Then,useof( 2 )allowsustoavoidusingMCMCforeachsimulatedcatchvector,amajorreductionincomputationtime. 2.4ResultsForthedepletionsurveysofChesapeakeBaybluecrabwecomparethevetechniques,inthiscasethevessels.Thatis,foreachofthevevessels(t=1,2,...,5)weapproximatedtheexpectedlengthofthe95%HPDcredibleinterval(=.05)usingexpression( 2 ).Foreachvessel,weestimatedtheexpected95%HPDcredibleintervalforp=2,3,...,10depletionpasses. 28

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Wealsoestimatedthemeancatchabilityandstandarderrorofthemeanforeachvesselusingexpression( 2 ).Thatis,weestimatedE(g(t)jYt,a)whereg(t)=t t+t,themeanofthebetadistributiondescribingthecatchabilityofthetthvessel.ResultsforeachofthevevesselsarefoundinTable2-1andthemeansandtheirstandarderrorscanbefoundinTable2-2.AllcomputationswereconductedusingthestatisticalprogramminglanguageRv.2.10.1(RCoreDevelopmentTeam2009)andstatisticalsoftwareWinBUGSv.1.4.3(Lunnetal.2000). 2.5DiscussionByconnectingtwoexistingstatisticalmethodologies,wepresentameansforevaluatingtheefciencyofdepletionsamplingtechniques.Therstofthesemethodologies,hierarchicalmodeling,allowscomponentsofthemodeltoreectspecicprocessesandallowstheresearchertocoherentlyconnectthesecomponentsinordertomodelthesystem.EstimationofparametersinthemodelusingBayesianmethods,includingthespecicationofapriordistributionforeachparameter,allowstheresearchertofullyaccountforparameteruncertainty(RoyleandDorazio,2006)andfurtherallowsinferencestobeexpressedprobabilistically.Forexample,apossibleconclusiondrawnfromapplyingtheBayesianmodelmightbethataparameterisinsomeintervalIwithprobability(1)]TJ /F10 11.955 Tf 12.94 0 Td[().However,bothBayesian-estimatedandlikelihood-estimatedhierarchicalmodelshavebecomeusefulandpopulartoolsinecology(RoyleandDorazio,2008;Cressieetal.2009).Thesecondmethodology,Bayesiansamplesizemethodology,appliedalongsidetheBayesianhierarchicaldepletionmodel,allowsestimationofinformationgainedbydepletionsurveys.Thehierarchicalmodelinformstheresearcherabouttheprocessesofinterest(e.g.abundanceingeneralorcatchabilityofadepletiontechnique)andusesthatinformationtoproducelikelycatchvectorsforagivendepletiontechnique. 29

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Thosecatchvectorsarerandomlygeneratedconditionaloninformationaboutthetechnique'scatchabilityandtheabundanceprocess;thisistheposteriorpredictionofdataforagivendepletiontechnique.Thelengthofthecatchvectoristhenumberofdepletionpasses.Foreachgeneratedcatchvector,theHPDcredibleintervallengthforabundance(whichdescribestheinformationobtainedinasample)iscalculated.OuradaptationofthesamplesizemethodologydescribedbyJosephetal.(1995)includesapproximating( 2 )usingMonteCarlointegration.BydoingsoweapproximatetheexpectedlengthoftheHPDposteriordensitycredibleforsingle-siteabundanceforagivendepletiontechnique.HavingestimatedtheexpectedHPDcredibleintervallengthformultipledepletiontechniquesandforvariousnumbersofdepletionpasses,theresearchercancomparerelativesamplingefcienciesofthetechniques.Further,whensite-specicabundanceisofinterest,examinationoftheHPDcredibleintervallengthsfordifferentnumbersofdepletionpasses(foragivendepletiontechnique)providesinformationabouthowmanydepletionpassestoconductinthefuture,givendesiredlevelsofprecisionintheestimationofabundance.Suchsamplingefciencyinformationisessentialwhenconsideringfuturesamplingefforts,particularlywhennaturalresourcedepartmentsndthemselvesbeingaskedtodomorewithless.Understandingwhyaparticulardepletiontechniqueisorisnotefcientisalsoimportant.Asevidencedbythecrabdepletionvesselcomparison,differenttechniquescanhaveverydifferentmeancatchabilities.Weseemajorreductionsinthestandarddeviationofabundanceestimatesoverarangeofabundancevaluesascatchabilityincreases(FigureD-1,AppendixD),whichwerecalculatedusingthehierarchicaldepletionmodel.Thesamplingefcienciesofthevesselsinthebluecrabdepletionsappeartied,atleastinlargepart,totheirmeancatchabilities.However,evenwithhighermeancatchabilities,techniquesthataremorevariableincatchabilityfromsitetositemaybelessefcientthantechniqueswithlessvariablecatchabilitiesacrosssites. 30

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Inadditiontomeancatchability,ourmethodaccountsforvariabilityinatechnique'scatchabilitybecauseevaluationof( 2 )involvesthedensity[tjYt,a]whichinuencesthedistributionofatechnique'scatchvectors.Ourapplicationofthismethodologytothebluecrabdepletiondatasetshowsstrikingcontrastamongthesamplingefcienciesofthevevesselsincludedinthestudy.Indeed,thesediscrepanciesgenerallycoincidewiththeestimatedcatchabilities.Vesselstwoandfourhaveconsiderablyhighercatchabilitiesthantheotherthree(.40and.37,respectively,versus.20,.19and.12).Accordingtoouranalyses,samplingwithvesselstwoandfourismoreinformativeaboutlocalabundance,asevidencedbytheirconsiderablyshorterexpectedHPDlengths.Additionally,notethatourmethodaccountsfortheuncertaintyincatchability(notjustmeancatchability)whenevaluatingatechnique'sefciency.Thisisapparentbytherelativelyhighersamplingefciencyestimatedforvesselfourthanvesseltwo,atleastformorethantwodepletionpasses.Althoughvesseltwohasahighermeancatchability,vesselfourhaslessuncertaintyinthatmean,andformorethantwodepletionpassesthevariabilityofvesselfour'sabundanceestimateswillbelower.Havingthistypeofinformationaboutdepletiontechniquesamplingefciencycanbeextremelyvaluablewhenmakingdecisionsaboutfuturesampling.However,ourmethoddoesnotprovidedirectinformationaboutexpectedvariabilityofatotal-abundanceestimatewhenthestudyareaextendsbeyondthesurveysites.Althoughsuchacomparisonispossiblewiththemethodsdescribedabove,theimplementationwouldbecomputationallyintensive.AnotherdrawbackisthattheapproximationusedincalculatingtheHPDcredibleintervallengthl(~y,t,)doesnotaccountfortheuncertaintyintheparameterst,t,1and2,fort=1,2,...,T.ThisimpliesthatthecredibleintervalsobtainedusingtheapproximationmaybeshorterthanthoseobtainedusingthefullyBayesianimplementation.However,withoutaclosedformposteriordistribution,eachsimulateddatavectorwouldyieldaposteriordistributiononly 31

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throughadvancedcomputation(likelyfullMCMC),whichcanbecomputerintensive,particularlyforthemanysimulationsrequiredtoapproximate( 2 ).Witheconomicpressuresdecreasingavailablefundingfornaturalresourceandsheriesmonitoring,considerationofefciencyatthesamplingstageisessential.Inthismanuscript,wehavepresentedamethodtodosothatcomplementsexistingBayesianhierarchicalmodelswhichhavebecomeprevalentinecologicalstudies,particularlythoseinvolvingcatchabilityordetectabilityconsiderations.Wehaveconnectedtothehierarchicalmodelingframeworktheconceptofsamplesizeanalysisinordertoinformdepletionstudies,butamultitudeofapplicationsconnectingthesetwomethodologiescouldyieldusefulinformationinmanycontexts.Withnaturalresourceandsheriesmonitoringprogramsnotbeingtheonlyrealmsfeelingeconomicpressures,samplesizeconsiderationsinhierarchicalmodelingcanbeapowerfultoolinsamplingdesignanddecision-making. Table2-1. Estimatedaverage95%HPDcredibleintervallengthsforsiteabundanceforthevevesselsusedtosurveybluecrabandfornumberofdepletionpassesp=2,3,...,10. Vessel#DepletionPasses12345 2844.7174.0854.2204.8772.33650.7124.0626.689.1679.04520.498.2498.546.5602.65435.480.4424.227.8541.06379.069.8365.919.1486.47325.662.7327.913.0434.98286.154.9293.310.0394.49260.949.0273.88.0360.610235.945.1255.66.2326.6 Table2-2. Posteriormeansforcatchabilityofthevevesselsandthestandarderrorsofthoseestimates. Vessel12345 MeanCatchabilityEstimate.20.40.19.37.12(StandardError)(.023)(.031)(.026)(.028)(.026) 32

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CHAPTER3OPTIMALALLOCATIONOFSAMPLINGEFFORTINDEPLETIONSURVEYS 3.1BackgroundTotalabundanceestimationofanimalpopulationsisanessentialpartofconservation,monitoringandmanagementofnaturalresources.Oftenestimationofanimalabundanceismadedifcultbyimperfectdetectability,i.e.animalsmaybepresentbutgoundetectedduringasurvey.Samplingmethodssuchasdepletion(alsoknownasremoval)surveysaimtoestimateabundancewhileaccountingforimperfectdetectability.Depletionsurveysinvolvesuccessiveanimalremovalsfromaspeciedareasuchthatdiminishingnumbersinlaterremovalsallowtheestimationofbothlocalabundanceanddetectionrate.Sitesatwhichmultipledepletionpassesareconducted(henceforthmultiple-passsites)primarilyprovideinformationaboutdetectabilityandoftenconstituteonlyaportionofalargerdesignedsurvey.Thesemultiple-passsitesprimarilyprovideinformationaboutdetectability.Knowledgeofdetectabilityallowstheresearchertoinatecountsatsiteswhereasingledepletionwasconducted(henceforthsingle-passsites).Otherwise,withimperfectdetectability,abundanceestimatesbasedonlyontheobservedcountswouldbebiased.Anexampleofsuchasurveyistheannualstockassessmentofbluecrab(Callinectessapidus)intheChesapeakeBay.Thereresearchersconductmultiple-passdepletionsatasmallsetofsitesandsingle-passdepletionsatamuchlargersetofsiteschosenbystratiedrandomsampling(Sharov,etal.2003).Informationaboutdetectabilityfromthemultiple-passsitesiscombinedwithcountsatthesingle-passsitestoyieldatotalabundanceestimateofthebluecrabpopulation.Insuchasamplingdesign,researchersmustchoosetheallocationofeffortappliedtothemultiple-passsites.Inthismanuscriptwedescribemethodologyforoptimizingtheallocationofthesamplingeffortatmultiple-passsites.Hereoptimizingeffortallocationinvolvesdetermininghowmanydepletionpassesshouldbeconductedpermultiple-pass 33

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site,anaspectofsamplingundercontroloftheresearcher.Atthemultiple-passsites,fewerdepletionpassespersiteclearlypermitsmoretotalsitestobesurveyedforaxedamountofsamplingeffort.However,theoptimalcombinationofnumberofpassespersiteandnumberofmultiple-passsitesisnotobvious.Undertheassumptionthatefcientestimationofdetectability(whichwecallcatchabilityanddenebelow)istheobjectiveofthemultiple-passdepletionsurveys,weconsidermaximizingtheFisherinformationofthecatchabilityparameterwithrespecttothesurveydesignwithinthestandardmodelingframeworkfordepletionsurveys.Numerousmodelsfordepletionsurveyshavebeenimplemented(LeslieandDavis,1939;Moran,1951;Zippin,1956).Theyhaveappearedinmanyforms(Seber,1982;Pollock1991)andrecentlyhavebeenimplementedasaclassofhierarchicalmodels(Wyatt,2002;Dorazioetal.2005).Commontomanyofthesewidelyvaryingmodels,however,isthatthenumberofanimalscaughtismodeledasabinomialprocesswithsizeequaltothenumberofanimalsatthesiteandprobabilityparameterequaltocatchability.Henceformallywedenecatchabilitytobetheprobabilityananimaliscaughtduringapassofthedepletionsurveyandweassumethroughoutthisisconstantforeveryanimal.Thesequentialbinomialprocessdescribingmultiple-passdepletionscanbeequivalentlymodeledasastructuredmultinomialdistributionwithcategoryprobabilitiesderivedfromthesinglecatchabilityparameter(RoyleandDorazio,2006).Usingthismultinomialcharacterizationweproceedtoconsiderhowthesamplingdecisionofthenumberofmultiple-passsitesversusthenumberofdepletionpassespersiteinuencesknowledgeofthecatchabilityparameter.WequantifyknowledgeaboutthecatchabilityparameterusingFisherinformation.MaximizingFisherinformationisequivalenttominimizingthevarianceofthebestunbiasedestimator(CasellaandBerger,2002).Implementingasamplingdesignthatincreases,inexpectation,theFisherinformationoftheparameterofinterestwilllikelyyieldestimatesoftheparameterofinterestwithsmallvarianceandhighprecision. 34

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Increasedprecisionaboutthecatchabilityparameterwillultimatelyaccompanyincreasedprecisionoftotalabundanceestimatesthatdependonknowledgeofcatchability.Indepletionsurveys,theprocessofestimatingtotalabundancerequiresknowledgeaboutthecatchabilityparameterwhichisobtainedbydepletionsatthemultiple-passsites. 3.2Methods 3.2.1TheModelWebeginbysupposingtheresearcherwillallocatetotaleffortformultiple-passsitesbyspecifyingsurveystooccuratImultiple-passsites,eachwithptotaldepletionpasses.Thisresultsintotalefforte=Ip.LetYi=fyi1,yi2,...,yipgdenotethedepletionobservationsatsiteiwherei=1,2,...,I.Weassumeeachyij,j=1,2,...,p,isaconditionalbinomialrandomvariable.ThatisyijBinomial(Nij,q) (3)whereNijisthenumberofanimalsavailabletobecaughtatsiteifordepletionpassjandqiscatchability.ThatisNi1=NiandNij=Ni)]TJ /F6 7.97 Tf 13.62 15.21 Td[(j)]TJ /F3 7.97 Tf 6.58 0 Td[(1Xk=1Nik (3)forj=2,3,...,p.Weassumecatchabilityqisconstantacrosssitesandforalldepletionpasseswithinasiteandthatanimalsarecaughtindependentofotheranimals.Underthisspecication,thevectorofpcatchesatsitei,alongwiththenumberofanimalsthatarenevercaught,yi(p+1),isamultinomialrandomvector(RoyleandDorazio,2006).Thatisfyi1,yi2,...,yip,yi(p+1)gMultinomial(Ni,f(q)) (3) 35

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wheref(q)=ff1(q),f2(q),...,fp(q),fp+1(q)gwithfk(q)=q(1)]TJ /F5 11.955 Tf 11.95 0 Td[(q)k)]TJ /F3 7.97 Tf 6.58 0 Td[(1 (3)fork=1,2,...,pandfp+1(q)=(1)]TJ /F5 11.955 Tf 13.04 0 Td[(q)p.Notethatyi(p+1)isunobservable.Thismultinomialcharacterizationwithstructuredprobabilityvectorf(q)willproveconvenientlaterbutinitscurrentformisnotadequatetomodelthedepletiondatabecauseoftheunobservedyi(p+1).TheabovemultinomialmodelinvolveslackofknowledgeaboutbothNiandq.Sanathanan(1972)consideredsuchascenarioandproposedamodiedmultinomialinvolvingsubsetsofthecategoriesofthefullmultinomialdistribution.Inourcase,thisinvolvesmodelingonlythepobservedcategories(thedepletioncatches).ThemodiedmultinomialmodelforthedepletionobservationsatsiteiisconditionalonTi=pXk=1yik (3)andisYijTiMultinomial(Ti,f(q)) (3)wheref(q)=ff1(q),f2(q),...,fp(q)g,fk(q)=fk(q) Ppm=1fm(q)=q(1)]TJ /F5 11.955 Tf 11.96 0 Td[(q)k)]TJ /F3 7.97 Tf 6.59 0 Td[(1 Ppm=1q(1)]TJ /F5 11.955 Tf 11.96 0 Td[(q)m)]TJ /F3 7.97 Tf 6.59 0 Td[(1=q(1)]TJ /F5 11.955 Tf 11.95 0 Td[(q)k)]TJ /F3 7.97 Tf 6.59 0 Td[(1 1)]TJ /F7 11.955 Tf 11.96 0 Td[((1)]TJ /F5 11.955 Tf 11.96 0 Td[(q)p (3)fork=1,2,...,p.NotethatTiisobservedwhichalleviatestheproblemofanunknownsizeparameteralthoughlaterweaccountforthefactthatitisarandomandnotunderthecontroloftheresearcher.Nowweintroducesomenotationforconvenience.Let[X]denotethemarginalprobabilitymassfunction(p.m.f.)orprobabilitydensityfunction(p.d.f.)oftherandomvariable(r.v.)Xand[XjY]denotetheconditionalp.m.f.orp.d.f.ofXgiventher.v.Y.Further,denotethep.m.f.orp.d.f.ofX,whichdependsonxed(butpossibly 36

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unknown)parameters,as[X;].Also,wewillusethenotation[X;jY]whenwewishtoemphasizethattheconditionaldistributionofar.v.XgivenYdependsonparameters.Next,considerthedistributionofTi.ConsideringtherelationshipbetweenthebinomialandmultinomialdistributionsweseeTi,beingasumofcategoriesfrom( 3 ),hasdistributionTiBinomial(Ni,1)]TJ /F7 11.955 Tf 11.96 0 Td[((1)]TJ /F5 11.955 Tf 11.96 0 Td[(q)p) (3)fori=1,2,...,I(CasellaandBerger2002,p181).Then,usingournotation,weconsidertheconditionalmultinomialformulationofthedataYi;qjTiandthedistributionofthetotalTi;Ni,q.Weusethisalternateformulationbyrstconsideringthedata'ssingle-sitelikelihood[Yi;qjTi].Then,inasecondstep,weconsidertheeffectofTibeingrandomandnotxedbytheresearcherasisthecasewithmuchbinomialormultinomialdata.Howeveraswewillseeinthenextsection,thistwostepviewofdepletionsurveysyieldsatractablesolutionandaccountsfortherandomnessinTiasweconsidertheFisherinformationofthecatchabilityparameterq. 3.2.2InformationDerivationWenowderivetheFisherinformationI(q)forthecatchabilityparameterq.Intheprevioussection,wepresentedthealternateformulationforadepletionmodelatasinglesiteYi;qjTi.Thisformulationprovidestwoadvantages.First,thecatchabilityparameterqistheloneunknownparameterinastructuredmultinomialdistributionofsizeTiforwhichweobservecountsfromeachcategory.RecallconsideringfYi;yi(p+1)gasamultinomialwithsizeNiinvolvesanunobservedcategory.Second,becauseFisherinformationissimplyanexpectation,wecanapplythelawofiteratedexpectationstoaccountfortherandomnessinTiwhilemaintainingtractability.Thatis,werstcalculate 37

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expectationwithrespecttotheconditionalmultinomialdistributionofsizeTi,Yi;qjTi,andthenwithrespecttothedistributionofTi,Ti;Ni,q.FisherinformationinourcaseisdenedasI(q)=E"@ @qlog(L(qjY))2# (3)whereL(qjY)isthelikelihoodfunctionofqgiventheallthemultiple-passcatchdataY=fY1,Y2,...,YIg.InthiscaseL(qjY)=IYi=1[Yi;qjTi]. (3)Next,sincethemultinomialprobabilitydistributionisamemberoftheexponentialfamily,wehaveI(q)=)]TJ /F1 11.955 Tf 9.3 0 Td[(E@2 @q2log(L(qjY)) (3)(CasellaandBerger2002,p.338).Lastly,applyingthelawofiteratedexpectationweobtainI(q)=)]TJ /F1 11.955 Tf 9.3 0 Td[(ETEY@2 @q2log(L(qjY))T (3)whereT=fT1,T2,...,TIgistheconditioningvector.Forclarity,thesubscriptnotationETandEYdenotesexpectationwithrespecttothedensitiesIYi=1[Ti;Ni,q] (3)andIYi=1[Yi;qjTi], (3)respectively.Workingwith( 3 )isstraightforwardandtractableaswenowsee. 38

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Bysubstitutingforthelikelihoodfunctionusing( 3 )wehaveI(q)=)]TJ /F1 11.955 Tf 9.3 0 Td[(E"E @2 @q2log IYi=1[Yi;qjTi]!T!#. (3)Next,wesubstitutetheaforementionedp.m.f.sinto( 3 )andproceedwiththederivation.ThatisI(q)=)]TJ /F1 11.955 Tf 9.3 0 Td[(E"E @2 @q2log IYi=1Ti! yi1!yi2!yip!(f1(q))yi1(f2(q))yi2)]TJ /F5 11.955 Tf 5.48 -9.68 Td[(fp(q)yip!!#=)]TJ /F1 11.955 Tf 9.3 0 Td[(E"E @2 @q2IXi=1log(Ti!))]TJ /F1 11.955 Tf 11.95 0 Td[(log(yi1!))]TJ /F1 11.955 Tf 11.95 0 Td[(log(yi2!))-222()]TJ /F1 11.955 Tf 40.52 0 Td[(log(yip!)+yi1log(f1(q))+yi2log(f2(q))++yiplog)]TJ /F5 11.955 Tf 5.48 -9.68 Td[(fp(q)!# (3)Realizingtermsfreeofqwillzerooutwhenweapplythepartialderivativeswithrespecttoq,wehaveI(q)=)]TJ /F1 11.955 Tf 9.3 0 Td[(E"E @2 @q2IXi=1yi1log(f1(q))+yi2log(f2(q))+yi1log)]TJ /F5 11.955 Tf 5.48 -9.68 Td[(fp(q)!#. (3)Substitutionusing( 3 )givesusI(q)=)]TJ /F1 11.955 Tf 9.29 0 Td[(E"E @2 @q2IXi=1(yi1logq 1)]TJ /F7 11.955 Tf 11.95 0 Td[((1)]TJ /F5 11.955 Tf 11.95 0 Td[(q)p+yi2logq(1)]TJ /F5 11.955 Tf 11.96 0 Td[(q) 1)]TJ /F7 11.955 Tf 11.96 0 Td[((1)]TJ /F5 11.955 Tf 11.96 0 Td[(q)p++yiplogq(1)]TJ /F5 11.955 Tf 11.96 0 Td[(q)p)]TJ /F3 7.97 Tf 6.59 0 Td[(1 1)]TJ /F7 11.955 Tf 11.95 0 Td[((1)]TJ /F5 11.955 Tf 11.95 0 Td[(q)p)!# (3)andapplyingstandardlogarithmicidentitieswehaveI(q)=)]TJ /F1 11.955 Tf 9.3 0 Td[(E"E @2 @q2IXi=1(yi1log(q))]TJ /F5 11.955 Tf 11.96 0 Td[(yi1log(1)]TJ /F7 11.955 Tf 11.96 0 Td[((1)]TJ /F5 11.955 Tf 11.96 0 Td[(q)p)+yi2log(q)+yi2log(1)]TJ /F5 11.955 Tf 11.95 0 Td[(q))]TJ /F5 11.955 Tf 11.96 0 Td[(yi2log(1)]TJ /F7 11.955 Tf 11.96 0 Td[((1)]TJ /F5 11.955 Tf 11.96 0 Td[(q)p)+yi3log(q)+2yi3log(1)]TJ /F5 11.955 Tf 11.96 0 Td[(q))]TJ /F5 11.955 Tf 11.95 0 Td[(yi3log(1)]TJ /F7 11.955 Tf 11.95 0 Td[((1)]TJ /F5 11.955 Tf 11.95 0 Td[(q)p) 39

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++yiplog(q)+(p)]TJ /F7 11.955 Tf 11.96 0 Td[(1)yiplog(1)]TJ /F5 11.955 Tf 11.95 0 Td[(q))]TJ /F5 11.955 Tf 11.95 0 Td[(yiplog(1)]TJ /F7 11.955 Tf 11.95 0 Td[((1)]TJ /F5 11.955 Tf 11.95 0 Td[(q)p))!#. (3)GroupingtermswehaveI(q)=)]TJ /F1 11.955 Tf 9.29 0 Td[(E"E @2 @q2IXi=1([yi1+yi2++yip]log(q)+[yi2+2yi3++(p)]TJ /F7 11.955 Tf 11.96 0 Td[(1)yip]log(1)]TJ /F5 11.955 Tf 11.95 0 Td[(q))]TJ /F7 11.955 Tf 11.95 0 Td[([yi1+yi2++yip]log(1)]TJ /F7 11.955 Tf 11.95 0 Td[((1)]TJ /F5 11.955 Tf 11.95 0 Td[(q)p))!# (3)whichyields,bydenitionofTi,I(q)=)]TJ /F1 11.955 Tf 9.3 0 Td[(E"E @2 @q2IXi=1(Tilog(q)+[yi2+2yi3++(p)]TJ /F7 11.955 Tf 11.96 0 Td[(1)yip]log(1)]TJ /F5 11.955 Tf 11.96 0 Td[(q))]TJ /F5 11.955 Tf 11.95 0 Td[(Tilog(1)]TJ /F7 11.955 Tf 11.96 0 Td[((1)]TJ /F5 11.955 Tf 11.96 0 Td[(q)p))!#. (3)Passingthepartialderivativeoperatorthroughthesummationanddistributing,andthenapplyingthesecondorderpartialderivativesofAppendixAandcollectingnegativesignswehaveI(q)=)]TJ /F1 11.955 Tf 9.3 0 Td[(E"E IXi=1(Ti)]TJ /F7 11.955 Tf 9.29 0 Td[(1 q2+[yi2+2yi3++(p)]TJ /F7 11.955 Tf 11.95 0 Td[(1)yip])]TJ /F7 11.955 Tf 9.3 0 Td[(1 (1)]TJ /F5 11.955 Tf 11.95 0 Td[(q)2)]TJ /F5 11.955 Tf 9.3 0 Td[(Ti)]TJ /F5 11.955 Tf 9.3 0 Td[(p(p)]TJ /F7 11.955 Tf 11.96 0 Td[(1)(1)]TJ /F5 11.955 Tf 11.95 0 Td[(q)p)]TJ /F3 7.97 Tf 6.59 0 Td[(2(1)]TJ /F7 11.955 Tf 11.95 0 Td[((1)]TJ /F5 11.955 Tf 11.95 0 Td[(q)p))]TJ /F5 11.955 Tf 11.95 0 Td[(p2(1)]TJ /F5 11.955 Tf 11.96 0 Td[(q)2p)]TJ /F3 7.97 Tf 6.59 0 Td[(2 (1)]TJ /F7 11.955 Tf 11.95 0 Td[((1)]TJ /F5 11.955 Tf 11.95 0 Td[(q)p)2)!#. (3)Wenowtaketheinnerexpectationwithrespecttotheproductofindependent,conditionaljointmultinomialdensityQIi=1[Yi;qjTi].ThisyieldsI(q)=)]TJ /F1 11.955 Tf 11.95 0 Td[(E"IXi=1()]TJ /F5 11.955 Tf 9.3 0 Td[(Ti q2)]TJ /F5 11.955 Tf 11.95 0 Td[(Ti q(1)]TJ /F5 11.955 Tf 11.96 0 Td[(q)+2q(1)]TJ /F5 11.955 Tf 11.95 0 Td[(q)2++(p)]TJ /F7 11.955 Tf 11.96 0 Td[(1)q(1)]TJ /F5 11.955 Tf 11.96 0 Td[(q)p)]TJ /F3 7.97 Tf 6.58 0 Td[(1 1)]TJ /F7 11.955 Tf 11.96 0 Td[((1)]TJ /F5 11.955 Tf 11.96 0 Td[(q)p!1 (1)]TJ /F5 11.955 Tf 11.96 0 Td[(q)2+Tip(p)]TJ /F7 11.955 Tf 11.95 0 Td[(1)(1)]TJ /F5 11.955 Tf 11.96 0 Td[(q)p)]TJ /F3 7.97 Tf 6.58 0 Td[(2(1)]TJ /F7 11.955 Tf 11.96 0 Td[((1)]TJ /F5 11.955 Tf 11.96 0 Td[(q)p)+p2(1)]TJ /F5 11.955 Tf 11.95 0 Td[(q)2p)]TJ /F3 7.97 Tf 6.58 0 Td[(2 (1)]TJ /F7 11.955 Tf 11.96 0 Td[((1)]TJ /F5 11.955 Tf 11.96 0 Td[(q)p)2)#. (3) 40

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Next,wenotethatwecanfactorouttheTiwhichyieldsI(q)=)]TJ /F1 11.955 Tf 11.96 0 Td[(E"IXi=1Ti()]TJ /F7 11.955 Tf 9.3 0 Td[(1 q2)]TJ /F4 11.955 Tf 11.95 20.44 Td[( q(1)]TJ /F5 11.955 Tf 11.95 0 Td[(q)+2q(1)]TJ /F5 11.955 Tf 11.96 0 Td[(q)2++(p)]TJ /F7 11.955 Tf 11.95 0 Td[(1)q(1)]TJ /F5 11.955 Tf 11.95 0 Td[(q)p)]TJ /F3 7.97 Tf 6.59 0 Td[(1 1)]TJ /F7 11.955 Tf 11.96 0 Td[((1)]TJ /F5 11.955 Tf 11.96 0 Td[(q)p!1 (1)]TJ /F5 11.955 Tf 11.96 0 Td[(q)2+p(p)]TJ /F7 11.955 Tf 11.96 0 Td[(1)(1)]TJ /F5 11.955 Tf 11.96 0 Td[(q)p)]TJ /F3 7.97 Tf 6.59 0 Td[(2(1)]TJ /F7 11.955 Tf 11.96 0 Td[((1)]TJ /F5 11.955 Tf 11.96 0 Td[(q)p)+p2(1)]TJ /F5 11.955 Tf 11.95 0 Td[(q)2p)]TJ /F3 7.97 Tf 6.59 0 Td[(2 (1)]TJ /F7 11.955 Tf 11.95 0 Td[((1)]TJ /F5 11.955 Tf 11.95 0 Td[(q)p)2)#=)]TJ /F1 11.955 Tf 9.29 0 Td[(E"()]TJ /F7 11.955 Tf 9.3 0 Td[(1 q2)]TJ /F4 11.955 Tf 11.95 20.44 Td[( q(1)]TJ /F5 11.955 Tf 11.95 0 Td[(q)+2q(1)]TJ /F5 11.955 Tf 11.96 0 Td[(q)2++(p)]TJ /F7 11.955 Tf 11.95 0 Td[(1)q(1)]TJ /F5 11.955 Tf 11.95 0 Td[(q)p)]TJ /F3 7.97 Tf 6.59 0 Td[(1 1)]TJ /F7 11.955 Tf 11.96 0 Td[((1)]TJ /F5 11.955 Tf 11.96 0 Td[(q)p!1 (1)]TJ /F5 11.955 Tf 11.96 0 Td[(q)2+p(p)]TJ /F7 11.955 Tf 11.96 0 Td[(1)(1)]TJ /F5 11.955 Tf 11.96 0 Td[(q)p)]TJ /F3 7.97 Tf 6.59 0 Td[(2(1)]TJ /F7 11.955 Tf 11.96 0 Td[((1)]TJ /F5 11.955 Tf 11.96 0 Td[(q)p)+p2(1)]TJ /F5 11.955 Tf 11.95 0 Td[(q)2p)]TJ /F3 7.97 Tf 6.59 0 Td[(2 (1)]TJ /F7 11.955 Tf 11.95 0 Td[((1)]TJ /F5 11.955 Tf 11.95 0 Td[(q)p)2)IXi=1Ti#. (3)Notingthattheonlyrandomcomponentsof( 3 )aretheTis,weobtainI(q)=)]TJ /F4 11.955 Tf 9.3 20.45 Td[(()]TJ /F7 11.955 Tf 9.3 0 Td[(1 q2)]TJ /F4 11.955 Tf 11.96 20.45 Td[( q(1)]TJ /F5 11.955 Tf 11.96 0 Td[(q)+2q(1)]TJ /F5 11.955 Tf 11.95 0 Td[(q)2++(p)]TJ /F7 11.955 Tf 11.96 0 Td[(1)q(1)]TJ /F5 11.955 Tf 11.95 0 Td[(q)p)]TJ /F3 7.97 Tf 6.59 0 Td[(1 (1)]TJ /F7 11.955 Tf 11.96 0 Td[((1)]TJ /F5 11.955 Tf 11.96 0 Td[(q)p)(1)]TJ /F5 11.955 Tf 11.96 0 Td[(q)2!+p(p)]TJ /F7 11.955 Tf 11.95 0 Td[(1)(1)]TJ /F5 11.955 Tf 11.96 0 Td[(q)p)]TJ /F3 7.97 Tf 6.58 0 Td[(2(1)]TJ /F7 11.955 Tf 11.96 0 Td[((1)]TJ /F5 11.955 Tf 11.96 0 Td[(q)p)+p2(1)]TJ /F5 11.955 Tf 11.96 0 Td[(q)2p)]TJ /F3 7.97 Tf 6.58 0 Td[(2 (1)]TJ /F7 11.955 Tf 11.96 0 Td[((1)]TJ /F5 11.955 Tf 11.96 0 Td[(q)p)2)IXi=1E(Ti). (3)RecallTiBinomial(Ni,1)]TJ /F7 11.955 Tf 11.96 0 Td[((1)]TJ /F5 11.955 Tf 11.96 0 Td[(q)p)),sothatE(Ti)=Ni(1)]TJ /F7 11.955 Tf 12.18 0 Td[((1)]TJ /F5 11.955 Tf 12.17 0 Td[(q)p)inexpression( 3 )andwehaveI(q)=)]TJ /F4 11.955 Tf 9.3 20.45 Td[(()]TJ /F7 11.955 Tf 9.29 0 Td[(1 q2)]TJ /F4 11.955 Tf 11.96 20.45 Td[( q(1)]TJ /F5 11.955 Tf 11.95 0 Td[(q)+2q(1)]TJ /F5 11.955 Tf 11.96 0 Td[(q)2++(p)]TJ /F7 11.955 Tf 11.95 0 Td[(1)q(1)]TJ /F5 11.955 Tf 11.95 0 Td[(q)p)]TJ /F3 7.97 Tf 6.59 0 Td[(1 (1)]TJ /F7 11.955 Tf 11.95 0 Td[((1)]TJ /F5 11.955 Tf 11.95 0 Td[(q)p)(1)]TJ /F5 11.955 Tf 11.96 0 Td[(q)2!+p(p)]TJ /F7 11.955 Tf 11.95 0 Td[(1)(1)]TJ /F5 11.955 Tf 11.95 0 Td[(q)p)]TJ /F3 7.97 Tf 6.59 0 Td[(2(1)]TJ /F7 11.955 Tf 11.96 0 Td[((1)]TJ /F5 11.955 Tf 11.96 0 Td[(q)p)+p2(1)]TJ /F5 11.955 Tf 11.96 0 Td[(q)2p)]TJ /F3 7.97 Tf 6.59 0 Td[(2 (1)]TJ /F7 11.955 Tf 11.96 0 Td[((1)]TJ /F5 11.955 Tf 11.96 0 Td[(q)p)2)IXi=1Ni(1)]TJ /F7 11.955 Tf 11.95 0 Td[((1)]TJ /F5 11.955 Tf 11.95 0 Td[(q)p)=)]TJ /F4 11.955 Tf 9.3 20.45 Td[(()]TJ /F7 11.955 Tf 9.3 0 Td[((1)]TJ /F7 11.955 Tf 11.95 0 Td[((1)]TJ /F5 11.955 Tf 11.95 0 Td[(q)p) q2)]TJ /F4 11.955 Tf 11.96 20.45 Td[( q(1)]TJ /F5 11.955 Tf 11.95 0 Td[(q)+2q(1)]TJ /F5 11.955 Tf 11.96 0 Td[(q)2++(p)]TJ /F7 11.955 Tf 11.95 0 Td[(1)q(1)]TJ /F5 11.955 Tf 11.95 0 Td[(q)p)]TJ /F3 7.97 Tf 6.59 0 Td[(1 (1)]TJ /F5 11.955 Tf 11.96 0 Td[(q)2! 41

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+p(p)]TJ /F7 11.955 Tf 11.95 0 Td[(1)(1)]TJ /F5 11.955 Tf 11.95 0 Td[(q)p)]TJ /F3 7.97 Tf 6.59 0 Td[(2(1)]TJ /F7 11.955 Tf 11.96 0 Td[((1)]TJ /F5 11.955 Tf 11.96 0 Td[(q)p)+p2(1)]TJ /F5 11.955 Tf 11.96 0 Td[(q)2p)]TJ /F3 7.97 Tf 6.59 0 Td[(2 (1)]TJ /F7 11.955 Tf 11.95 .01 Td[((1)]TJ /F5 11.955 Tf 11.95 .01 Td[(q)p))IXi=1Ni (3)LettingN=PIi=1Ni IwehaveI(q)=)]TJ /F4 11.955 Tf 9.3 20.44 Td[(()]TJ /F7 11.955 Tf 9.3 0 Td[((1)]TJ /F7 11.955 Tf 11.95 0 Td[((1)]TJ /F5 11.955 Tf 11.95 0 Td[(q)p) q2)]TJ /F4 11.955 Tf 11.96 20.44 Td[( q(1)]TJ /F5 11.955 Tf 11.95 0 Td[(q)+2q(1)]TJ /F5 11.955 Tf 11.96 0 Td[(q)2++(p)]TJ /F7 11.955 Tf 11.95 0 Td[(1)q(1)]TJ /F5 11.955 Tf 11.95 0 Td[(q)p)]TJ /F3 7.97 Tf 6.59 0 Td[(1 (1)]TJ /F5 11.955 Tf 11.96 0 Td[(q)2!+p(p)]TJ /F7 11.955 Tf 11.95 0 Td[(1)(1)]TJ /F5 11.955 Tf 11.96 0 Td[(q)p)]TJ /F3 7.97 Tf 6.58 0 Td[(2(1)]TJ /F7 11.955 Tf 11.96 0 Td[((1)]TJ /F5 11.955 Tf 11.96 0 Td[(q)p)+p2(1)]TJ /F5 11.955 Tf 11.96 0 Td[(q)2p)]TJ /F3 7.97 Tf 6.58 0 Td[(2 (1)]TJ /F7 11.955 Tf 11.95 0 Td[((1)]TJ /F5 11.955 Tf 11.95 0 Td[(q)p))(IN). (3)RecallingtotalefforteistheproductofthenumberofdepletionsitesIandthenumberoftowspersite,p,thatis,e=Ip,wesubstituteforIin( 3 )sothatI(q)=)]TJ /F4 11.955 Tf 9.3 20.44 Td[(()]TJ /F7 11.955 Tf 9.3 0 Td[((1)]TJ /F7 11.955 Tf 11.95 0 Td[((1)]TJ /F5 11.955 Tf 11.95 0 Td[(q)p) q2)]TJ /F4 11.955 Tf 11.96 20.44 Td[( q(1)]TJ /F5 11.955 Tf 11.95 0 Td[(q)+2q(1)]TJ /F5 11.955 Tf 11.96 0 Td[(q)2++(p)]TJ /F7 11.955 Tf 11.95 0 Td[(1)q(1)]TJ /F5 11.955 Tf 11.95 0 Td[(q)p)]TJ /F3 7.97 Tf 6.59 0 Td[(1 (1)]TJ /F5 11.955 Tf 11.96 0 Td[(q)2!+p(p)]TJ /F7 11.955 Tf 11.95 0 Td[(1)(1)]TJ /F5 11.955 Tf 11.96 0 Td[(q)p)]TJ /F3 7.97 Tf 6.58 0 Td[(2(1)]TJ /F7 11.955 Tf 11.96 0 Td[((1)]TJ /F5 11.955 Tf 11.96 0 Td[(q)p)+p2(1)]TJ /F5 11.955 Tf 11.96 0 Td[(q)2p)]TJ /F3 7.97 Tf 6.58 0 Td[(2 (1)]TJ /F7 11.955 Tf 11.95 0 Td[((1)]TJ /F5 11.955 Tf 11.95 0 Td[(q)p))e pN=((1)]TJ /F7 11.955 Tf 11.96 0 Td[((1)]TJ /F5 11.955 Tf 11.96 0 Td[(q)p) pq2+ q(1)]TJ /F5 11.955 Tf 11.95 0 Td[(q)+2q(1)]TJ /F5 11.955 Tf 11.96 0 Td[(q)2++(p)]TJ /F7 11.955 Tf 11.95 0 Td[(1)q(1)]TJ /F5 11.955 Tf 11.95 0 Td[(q)p)]TJ /F3 7.97 Tf 6.59 0 Td[(1 p(1)]TJ /F5 11.955 Tf 11.95 0 Td[(q)2!)]TJ /F4 11.955 Tf 11.95 16.86 Td[(p(p)]TJ /F7 11.955 Tf 11.95 0 Td[(1)(1)]TJ /F5 11.955 Tf 11.96 0 Td[(q)p)]TJ /F3 7.97 Tf 6.58 0 Td[(2(1)]TJ /F7 11.955 Tf 11.96 0 Td[((1)]TJ /F5 11.955 Tf 11.96 0 Td[(q)p)+p2(1)]TJ /F5 11.955 Tf 11.96 0 Td[(q)2p)]TJ /F3 7.97 Tf 6.58 0 Td[(2 p(1)]TJ /F7 11.955 Tf 11.95 0 Td[((1)]TJ /F5 11.955 Tf 11.95 0 Td[(q)p)))]TJ /F5 11.955 Tf 5.48 -9.68 Td[(eN. (3)Maximizinginformationabouttheparameterqequatestomaximizingthebracketedfunctionofqandpinexpression( 3 ),foragivenlevelofefforteandassumingthatNisnotunderthecontroloftheresearcher.Thebracketedfunctionofqandpinexpression( 3 )canbecalculatedoveranegridofqvaluesandforintegervaluesofpandmaximaforeachvalueofqobtained.Thatis,oneobtainsthenumberofdepletion 42

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passespmaximizinginformationgainaboutthecatchabilityparameterq,forgivenvaluesofqandexed. 3.3ResultsOptimizationofeffortallocationwithreferencetothenumberofdepletionpassespersite,p,foragivencatchabilityvalueqisaccomplishedbymaximizingthebracketedpartofexpression( 3 ).Wecalculatedthisvalueforeachpairofcatchabilityparametersq=.05,.10,...,.95andnumbersofdepletionpassespersitep=2,3,...,81.TheresultingvalueforeachpaircanbefoundinTablesB-1andB-2(AppendixB).Increasingeffortmerelyallowsmoretotalsitestobesurveyed,increasingoverallinformationaboutq.TheoptimalnumberofdepletionpassespersiteforvariousvaluesofqissummarizedinTables3-1and3-2.. 3.4DiscussionInthismanuscriptwehaveconsideredtheproblemofoptimizingtheallocationofeffort(e)inmultiple-passdepletionsurveys.OptimizationofeffortinvolvesspecifyingthenumberofdepletionpassespersitethatmaximizestheFisherinformationaboutthecatchabilityparameterq.Withaxedamountofavailableeffortforthemultiple-passsites,specifyingthenumberofpassespersite(p)alsospeciesI,thenumberofmultiple-passsites,duetotherelationshipe=Ipdescribedabove.Allocationofeffortthatincreasesinformationaboutqultimatelyresultsinimprovementsinprecisionoftotalabundanceestimatesthatincludeaccountingforuncertaintyinq.Althoughourinterestisoptimaleffortallocation,expression( 3 )anditsderivationprovidegeneralinsightsonotherissuesindepletionsurveys.First,expression( 3 )highlightsthe(notsurprising)relationshipbetweeneffortallocationandinformationaboutq.However,weseetheformofthisrelationshipexplicitlyinexpression( 3 ):atalllevelsofeffort,theexpectedinformationgainaboutqincreaseslinearlywitheffort.Forexample,regardlessofqorpadoublingofeffortresultsinanexpecteddoublingoftheinformationweobtainaboutq.Insomecasesthismayjustifyincreasesineffort 43

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allocatedtomultiple-passsurveys,sayiftheresearcherhadpreviouslybelievedthatdiminishingreturnsresultfromaddingmoremultiple-passdepletionsitesbeyondacertainthreshold.Asecondandpotentiallymoreimportantobservationaboutexpression( 3 )isthatinformationalsoscaleslinearlywiththeN,themeannumberofanimalsatthemultiple-passsites.Thismayhaveenormousimplicationsonsamplingdesigntoestimatetotalabundance,particularlyiftheresearcherhasknowledgeaboutwheresubpopulationstendtohavelargenumbers.Recallingtheprimarypurposeofthemultiple-passsitesistolearnaboutthecatchabilityparameterq,itmaybemuchmoreinformativeforestimatingtotalabundancetoselectsiteswithknownhighnumbersofanimalstoconductthemultiple-passdepletionsurveys.Theresearcherwouldlikely,inthatcase,disregardtheabundanceestimatesatthosebiasedly-chosensitesbuttheimprovementintotalabundanceestimationprecisioncouldbedramatic,withoutincreasesineffort.Indeed,knowledgeofthepopulationthatallowsak-foldincreaseinNisequal,inexpectation,toak-foldincreaseinsamplingeffortatthemultiple-passsites.Theapparentrelationshipbetweenthenumberofanimalsavailabletobecaught(whichisinitiallyNi)andinformationaboutqsuggeststhatanadaptivesamplingapproachmayfurtheroptimizeinformationgain.Afewinitialdepletionsatasiteimmediatelyyieldinformationaboutabundanceatthatsiteandtheexibilityofchoosingtoremainandcontinuesamplingatthatsiteversussamplingelsewheremayprovetobeanefcientsamplingstrategy.Smallcatchesintherstfewdepletionpasseslikelysuggestasmallabundanceatthatsite,whichmaypromptanadaptivesamplertogoelsewhere.Alternately,largeinitialcatchesmaypromptthesamplertoremain.Suchanapproachwouldlikelyincreasethetotalnumberofanimalscaughtandpotentiallytheinformationgainabouttheparameterq. 44

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AlthoughtherelationshipbetweeninformationgainandNisstriking,wemusttemperourreactiontotheoreticalresultswithpragmatism.Forexample,atacitassumptionofourmodelisthatcatchabilityisconstantacrossthegradientofanimalabundance.Althoughthismaybereasonable,withoutmultiple-passdepletionsatarangeofabundanceswewillbeunabletotestthatassumption.Therefore,althoughweseetheapparentimprovementinprecisionresultingfromchoosingmultiple-passsitesthathavehighanimalabundances,theresearchershouldremainawareoftheassumptionsmadeinthemodeling.Clearlyitispreferabletoinvokeasamplingdesignthatwilllikelyyieldinformationusefulinassessingthevalidityofmodelingassumption.Analobservationabouttheutilityofexpression( 3 )resultsagainfromtherelationshipbetweenefforteandinformationaboutq.Throughoutthismanuscriptwehaveassumedthattheeffortallocatedtothemultiple-passsurveysissimplytheproductofthenumberofmultiple-passsitesandthenumberofdepletionpassesateachofthosesites.Thatis,e=Ip.However,formanypopulationsitmaybethecasethatagreatdealofeffortisspenttravelingfromonesitetoanotherorperhapssettingupnettingtoensureaclosedpopulationforthesampling.Forexample,theChesapeakeBaybluecrabareclusteredinspaceanddepletionsurveysareconductedviadredgingusingalargeseavessel.Therefore,itrequiressomeefforttotravelfromonereasonablemultiple-passdepletionsitetoanother.Withknowledgeofthelikelycostoftravelbetweenmultiple-passsites,wemaybeinclinedtore-optimizeoursamplingdesigntoincludethisinformation.Forexample,ifthetotaleffortrequiredtoconductamultiple-passdepletionsurveyatonesiteisp+c,wherecisthecostfortravelingbetweensitesthenweobtaintheneweffortrelationshipe=I(p+c).Substitutinge p+cforIintoexpression( 3 )allowsouroptimizationtoaccountforeffortinvolvingthetravelbetweenmultiple-passdepletionsites.Clearlydoingsowillresultinatendency 45

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toconductmoredepletionpassesatfewersites.Analpointisthatthevaluecmustrepresentadditionaleffortinunitsofdepletionpasses,theunitswithwhichwehavemeasuredeffortthroughoutthismanuscript.Thatis,onemustestimatehowmanydepletionpassescanbeconductedwithanequivalentamountofcostcomparedtothecostrequiredto,forexample,travelbetweendepletionsites.Inadditiontoinsightprovidedfromthederivationandformof( 3 ),wenowconsidertheresultsoftheoptimization.Informationaboutqappearshighesttowardtheboundariesofthe(0,1)intervalonwhichqlies.However,wehaveseenlargeanderraticallybehavedvariancesinabundanceestimatesobtainedfromdepletionsurveyswithsmallq(Chapter2).Whenqissmall,itsvariancemaybelikewisesmall(henceitsinformationvaluelarge)mainlybecauseofthelowerboundaryeffect.However,giventhatabundanceestimationistheultimategoal,therelativelylargerexpectedinformationvalueswhenqissmallarenotindicativethatdepletionsurveysbasedonsmallerqaremoreefcient.IndeedthegeneraltrendnotedinChapter3isthatincreasesinqimproveestimatesofabundanceindepletionsurveys.Hencetheresultsofthisoptimizationmerelyprovideanoptimalnumberofdepletionpassespersiteforaparticularqvalueandshouldnotbeinterpretedaloneasajusticationforselectingadepletionmethodwithalargerorsmallerq.Althoughwehaveseentheutilityofconsideringinformationaboutqindeterminingeffortallocationindepletionsurveys,wehavenotdiscussedhowmuchefforttoapplytothemultiple-passsitesversusthesingle-passsites.Ananalogouscomputationmaximizinginformationaboutpopulation-levelparametersinahierarchicalmodelmayyieldinsightintothefullsurveydesign.Inparticular,thehierarchicaldepletionmodelsofWyatt(2002)andDorazioetal.(2005)containpopulation-levelparametersusefulindescribinganimaldensityandhenceabundancethroughoutthehabitat.Maximizationoftheinformationaboutwithrespecttotheallocationofefforttomultiple-passversus 46

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single-passsitesmayyieldimportantinformationforsurveydesign,furtherhelpingtheresearchergainprecisioninestimatinganimalabundancewithoutincreasesineffort.Also,wehaveseentheoptimalallocationofeffortatthemultiple-passsitesindeeddependsonthecatchabilityparameterqitself.Therefore,forthisoptimizationtobeofusetheresearchermusthavesomeknowledgeofthecatchabilityofthedepletionmethodemployedduringplanning.However,withevenvagueknowledgeaboutcatchabilityasmallrangeofdepletionpassescanbeidentied.Further,withinformationfrompreviousdepletionsurveystoinformfuturesurveydesigns,previousestimatesofqalongwithourresultswillbeusefulindeterminingallocationofeffortatthemultiple-passsites.Certainlyinformingsurveydesignusingevenaroughestimateofqalongwithourresultsispreferabletoanarbitrarydecisionaboutallocationofeffort.Inthismanuscriptwehavepresentedananalysisusefulforthedesignofdepletionsurveys.Ourresultsprovidetheoptimalallocationofeffortforthemultiple-passsitesinabroaderdepletionsurveywheretheobjectiveofthemultiple-passsurveysislearnaboutthecatchabilityordetectabilityparameterq.Inthiscaseallocationofeffortreferstothedecisionofhowmanydepletionpassestoconductpersiteversushowmanymultiple-passsitestosurvey.Wehavediscussedseveralusefulinsightsapparentfromexpression( 3 )inadditiontomaximizingthefunctiontooptimizeeffortallocation.Wehavefurtherdiscussedhowtoaccountforeffortthatmustbeexpendedforeachmultiple-passsiteinadditiontotheactualdepletionpasses(e.g.theeffortoftravelingbetweensites).Wefeeltheutilityofsuchananalysis,whichaimstoultimatelymaximizeprecisionofabundanceestimatesforagivenlevelofeffort,isanimportantstepinthemanagementorconservationofanimalpopulations.Asweseenaturalresourcemonitoringbudgetsforsuchmonitoringdecreaseandyetasdemandformanyoftheseresourcesincreases,optimizationofsurveydesignbecomesincreasinglyimportant. 47

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Table3-1. Optimalnumberofdepletionpassespersiteforcatchabilityvalueslessthan.50. Catchabilityq.05.10.15.20.25.30.35.40.45.50 Optimalp8038251814119876 Table3-2. Optimalnumberofdepletionpassespersiteforcatchabilityvaluesgreaterthan.50. Catchabilityq.55.60.65.70.75.80.85.90.95 Optimalp554433322 48

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CHAPTER4ROBUSTTOTALABUNDANCEESTIMATIONINDEPLETIONSURVEYS 4.1BackgroundModelingandestimationofanimalabundancearecentraltomanagementandconservationofimportantpopulations.However,twoprocessesoftencomplicateabundanceestimation.First,animaldetectionisoftenlessthanperfectmeaningthatallanimalsatasamplinglocationmaynotbecounted.Thesecondinvolvesthewayinwhichanimalsdistributeinspace.Althoughsometimesanimalsdistributeinspaceinawaywell-describedbyknownmodels,othertimesthatisnotthecase.Animalbehavior,heterogeneousanimalhabitatandotherfactorsinuencethedistributionoftheseanimalsinsuchawaythatknownabundancemodelsmayinsufcientlydescribethatdistribution.Thesecondcomplicationinvolvingpatternsofanimaldistributioninspaceisbyitselfhandledeffectivelybydesign-basedestimationmethods.Whendetectionisnotanissue,i.e.theprobabilityofdetectingorcapturinganimalsatsamplinglocationsisone,design-basedmethodsprovideareasonableframeworkforabundanceestimation.Design-basedestimationreliesuponasamplingdesign(e.g.simplerandomsamplingorstratiedrandomsampling)andtheprobabilisticstructureinducedbythatsamplingdesigntoestimatetotalabundance.Thisapproachdoesnotrelyonspecicationofamodeldescribingtheabundanceprocessorthecountsofanimalsatthesurveysites;countsofanimalsareassumedtobexed(perhapsunknown)values,notrandomvariables.Further,theunbiasednessofdesign-basedestimatorsisnotinuencedbyspatialpatternsofanimals(Thompson1992).Inotherwords,theunbiasednessofdesign-basedestimatorsdoesnotrelyoninformationaboutthepopulationitself(Thompson1992).However,manydesignsandtheircorrespondingestimatorscanintegrateinformationaboutthepopulationtoincreaseprecisionofestimates.Probability-proportional-to-sizesamplingisanexampleofadesign-basedmethod 49

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implementinginformationaboutthepopulationtodecreasetheuncertaintyinestimates(Thompson1992).However,formanyanimalpopulationsdetectionisnotperfect.Whendetectionisnotperfectcomplicationsariseindesign-basedestimationintheformofbiasedabundanceestimates(ThompsonandSeber1994).Inthissituation,design-basedmethodsmustbemodiedtoaccountforthisbias.Inthismanuscriptwepresentmethodologycombiningtheadvantagesofdesign-basedestimationwithanotherestimationframework,model-basedestimation,whichwedescribebelow.Weconsidertheproblemofabundanceestimationwhendepletionsurveysarethesamplingmethodofchoice.Depletionsurveysinvolvesequentiallyremovinganimalsfromaspecicsamplingsite(withoutreplacement)whichyieldsinformationusefulinestimatingbothlocalabundance(atthatsite)anddetectionrate.Acomplicationarises,however,whenthegoalofthesurveyisabundanceestimationoverastudyregionlargerthanthesamplingsites.Inthiscasemethodologymustbeemployedtocoherentlytranslateinformationaboutabundanceanddetectionrateobtainedduringthedepletionsinordertoestimatetotalabundance.Analternativeapproachtoestimatingtotalabundanceoverthestudyregionandalsoapplicabletodepletionsurveysismodel-basedestimation.Indepletionsurveys,model-basedestimationreliesonmodelingobservedcountsastheresultofdetectionandabundanceprocesses(Wyatt2002;Dorazioetal.2005).Thatis,theunderlying,unobservedcountofanimalsatasamplingsiteisconsideredarandomvariablegovernedbyanunobservableprocess.Byassumingamodelstructure,observedcountsallowtheresearchertolearnabouttheunderlyingabundanceprocessandhencedescribeaspectsofthepopulationofanimalsresultingfromthatabundanceprocess.Althoughthemodel-basedapproachprovidesaframeworktoaccountforimperfectdetection,estimatesandinferencesderivedfrommodel-basedapproachesmaybesensitivetospecicationoftheappropriatemodelfortheabundanceprocess.Hence 50

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neitherdesign-basednormodel-basedmethodsaloneareappropriateforabundanceestimationwhenimperfectdetectionofananimalpopulationiscoupledwithapoorlyunderstoodabundanceprocess.Analternativetostrictlymodel-basedordesign-basedestimationisahybridapproachwhichincorporatesadvantagesofbothestimationregimes(Little2004).Inthismanuscript,wepresentsuchahybridestimatoroftotalabundancethattakesadvantageofamodelfordetectability(orcatchability)whilesimultaneouslyincorporatingdesign-basedmethodsforabundanceestimation.WediscusstheimplementationofthisestimatoraspartofaBayesianhierarchicalmodelandconductasimulationstudydemonstratingtherobustnessoftheestimatorascomparedtoatypical,fullymodel-basedapproach.ThroughoutwerefertotheutilityofthisestimatorwithrespecttotheannualstockassessmentoftheChesapeakeBaybluecrabCallinectessapidusanditssamplingdesign.Finallywediscussthegeneralityofourestimatorinsurveysaimedattotalabundanceestimationinwhichtheabundanceprocessispoorlyunderstoodandimperfectcatchabilitycomplicatesthesamplingprocess. 4.2MethodsWeassumeasamplingdesignconsistingoftwotypesofdepletionsites.First,asmallsetofsitesischosenatwhichtoconductmultiple-passdepletions.CallthissetofsitesM;wedonotassumethesesitesarerandomlyselectedfromthepopulationofpotentialdepletionsites,P.IndeedwetreatthesitesinMdifferentlyfromtheothersitesasweshallsee.WeassumethattheobjectiveofdepletionsatMistolearnaboutcatchability,notabundance.AteachsiteinMweassumeddepletionpassesareconducted,althoughthisassumptioniseasilyrelaxed.Alargersetofsingle-passdepletionsites,S,ismeanttoyieldinformationabouttheabundanceofthepopulation.Thereforeweassumethosesitesarechosenbysomeprobabilisticsamplingdesigninwhichsiteinclusionprobabilitiesi,i2Sareknownaswellasjointinclusionprobabilitiesij,i,j2S,i6=j.Examplesincludesimplerandom 51

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samplingandstratiedrandomsampling,thelatterofwhichisusedintheannualstockassessmentoftheChesapeakeBaybluecrab(Sharovetal.2003).Forconveniencewedenethenotation[X]tobetheprobabilitymassfunction(p.m.f.)orprobabilitydensityfunction(p.d.f.)ofXand[XjY]tobetheconditionalmassordensityfunctionofXgivenY.Regardlessoftypeofsite,wealsosupposethereareNianimalsattheithsite,i2P,andthatthecatchabilityatthatsiteisqi.Atthemultiple-passsitesweoutlineafullhierarchicalmodelforabundanceandcatchability,whereasatthesingle-passsiteswerequireonlythecountobservationmodel.Webeginbydescribingthesub-modelthatappliestothemultiple-passsites. 4.2.1Multiple-PassSiteModelWeappealtoexistinghierarchicaldepletionmodelsforthesetofmultiple-passsitesM.Wyatt(2002)andDorazioetal.(2005)discusstheuseofhierarchicaldepletionmodelsandweadapttheirmethodstothesitesinM.Thehierarchicalmodelisbasedondescribingthreeprocessesinunison.FirstthehierarchicalmodeladdressestheabundanceprocessbyconsideringlocalabundanceNiasconnectedtoanunderlying,unobservableanimaldensityprocessdescribinghowdensityvariesthroughspace.Nextweaccountforvariablecatchabilityfromsitetositebyconsideringthecatchabilityatsitei,qi,tobearandomdrawfromadistributionofcatchabilities.FinallythemodelconsiderstheobservationordataprocessatsiteiconditionalonNiandqi.Modelingoftheabundanceprocessinvolvesspecifyingadistributionfortheabundanceatsitei,Ni,fori2M.Thisdistributionisconditionalonahigher-levelparameterthatdescribesexpectedanimalabundanceatthatsite,i,andhasp.m.f.[Niji].Inturntheishavep.d.f.[ij]wherearethehyperparameters.Notethatwecoulddeneanequivalentmodelinwhichiistheexpectedanimaldensityatsitei,whichwouldbeusefulifdepletionsitesvariedinsize.However,withoutlossofgenerality,weassumeallsitesarethesamesizesothatiissimplyexpectedabundanceatsitei.TheresultinghierarchicalabundancemodelforNiandiisthe 52

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jointdensity[Niji][ij].Animportantnotehereisthatspecifyingahierarchicalabundancemodelisnotnecessary.InsteadonemayassumetheNisarexed,unknownparametersinafrequentistsettingorinaBayesiansettingonemayplaceonthemacommon,non-informativepriordistributionsuchasadiscreteuniformdistribution.Weincludetheoptionofplacingahierarchicalabundancemodelforexibility.Thenextprocessofinterestisthecatchabilityprocess.Wyatt(2002)conductedaseriesofsimulationssuggestingreductionsinthebiasofabundanceestimateswhencorrectlyaccountingforsite-to-sitevariationincatchability.Therefore,wesupposetheqi,i2Mareindependentandidenticallydistributedwithp.d.f.[qij]wherearetheparametersoftheprocessgoverningcatchabilityatallsites.Forexample,knowledgeofyieldsknowledgeaboutthemeancatchabilityoverallsitesaswellashowmuchcatchabilitytendstovaryfromonesitetoanother.ConditionalonNiandqiwespecifyamodelfortheobservedcounts.LetYi1,Yi2,...,Yidbethedobserveddepletioncountsatsitei.ThenwesupposeYi1jNi,qiBinomial(Ni,qi) (4)andYijjYi1,Yi2,...,Yi,j)]TJ /F3 7.97 Tf 6.58 0 Td[(1,Ni,qiBinomial(Nij,qi) (4)whereNij=Ni)]TJ /F6 7.97 Tf 13.63 15.21 Td[(j)]TJ /F3 7.97 Tf 6.59 0 Td[(1Xk=1Yikforj=2,3,...,d.AlthoughwehavenotspeciedparticulardistributionsforNiji,ijandqij,wedoassumeabinomialsamplingmodelforYijjNij,qi.Wedothistobeconsistentwiththemodelatthesingle-passsites,wherewealsoassumeabinomial 53

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samplingprocess.Awidevarietyofdepletionmodelsusethebinomialmodel(Moran1951;Wyatt2002;Dorazioetal.2005). 4.2.2Single-PassSitesTodenearobusttotalabundanceestimatorweavoidspecifyinganabundancemodelforthesingle-passsitesandinsteadassumetheNi,i2PnM,arexedbutunobservablequantities.Wethereforerelyondesign-basedapproacheswhichallowestimationandinferencetobeconductedwithoutspecifyingparametricabundancemodels.Generallyindesign-basedestimationstandarderrorsoftheestimatorsandcondenceintervalsderivefromthedesignusedtoselectthesamplingsites.Uncertaintyintotalabundanceresultsnotfromuncertaintyinmodelparametersasinmodel-basedestimationbutratherfromthesamplingvariabilityinherentinrandomlyselectingastrictsubsetofsitesatwhichtosample(Thompson1992).IfthexedquantitiesNi,i2Swereobserved,onecouldestimatetotalanimalabundanceT=Xi2PNi (4)withtheHorvitz-Thompsonestimator(HorvitzandThompson1952;Thompson1992):bT=Xi2SNi i (4)whereiistheprobabilitysiteiinincludedinthesample,Pr(i2S).Inthecaseofperfectcatchability,theNi,i2S,areobservedandbTisanunbiasedestimatorofTwithrespecttothesamplingdesign(HorvitzandThompson1952).However,withimperfectcatchabilitytheNisareunobservedandexpression( 4 )cannotbeuseddirectly.WithimperfectcatchabilityweonlyobserveYi1Ni,i2S.However,wecircumventthiscomplicationbyassumingtheobservedYi1resultsfromabinomialprocessconditionaloncatchabilityandabundanceatsitei.Wehenceforthdropthe 54

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secondsubscriptofYi1atthesingle-passsitesforconvenienceandsupposeYiBinomial(Ni,qi) (4)whereNiandqiarestilltheabundanceandcatchabilityatsitei,foralli2S. 4.2.2.1PartialpopulationestimatorUnderthisassumptionwenowconsiderthepropertiesofaHorvitz-ThompsontypeestimatorbasedontheobservationsYiasopposedtotheunobservedabundancesNi.Considersuchapartialpopulationestimatorofthesameformas( 4 ):bTp=Xi2SYi i (4)wherebTpisanabundanceestimatorbasedontheobservablefYi:i2Sg.BecauseofimperfectcatchabilitybTpisabiasedestimatoroftotalabundanceT,however,calculatingitsexpectationbTpwillallowustoadjustforthisbiaslater.FollowingThompson(1992)weobtaintheexpectationof( 4 ).Forclarity,letEM,EDjMandEDdenoteexpectationswithrespecttothemodel,thedesignconditionalonthemodelandthedesignmarginally.ExpectationwithrespecttothemodelincludesthebinomialsamplingprocessfortherandomvariablesYi,i2S,whichisconditionaloncatchability,andalsothedistributionfortheqi.WedistinguishamongtheseusingEYjqandEq,respectively.Next,weintroducethesetofbernoullirandomvariablesZ=fZi:i2Pgwhichrelatetothedesign.ThatisZiBernoulli(i) (4)whereZi=1denotesi2SandZi=0denotesi=2S.Also,letY=fYi:i2Pg.Then,theconditionalexpectationofthepartialpopulationestimatorisEDjM[bTpjY]=EDjM"Xi2PYiZi iY# 55

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=Xi2PYi iEDjM[ZijYi]=Xi2PYi ii=Xi2PYi=Tp (4)sinceEDjM[ZijYi]=EDjM[Zi]=i.Now,theYiareinfactrealizationsofabinomialprocesswithmeansEM[YijNi,qi]=Niqi.Further,Niandqiareindependentforalli2P.Dening=EM[qi]andlettingEM[Ni]=iwehave,unconditionally,EM[Yi]=EN,q)]TJ /F1 11.955 Tf 5.48 -9.68 Td[(EYjN,q[YijNi,qi]=iandthusEM[bTp]=Xi2PiwherePi2Pisthemeantotalabundanceofthesuperprocessthatgeneratedtheparticularrealizationsampled.Sinceourinterestisinthesampledrealizationandnotthesuperpopulation,weshallconsidertheNiasxed,i.e.weconditionontheNiandconsiderestimatorsof( 4 ).ConsideringtheexpectationofTpwithrespecttothemodel(andnowconsideringNiasxedandnotrandom)wehaveEM[Tp]=EM"Xi2PYi#=Xi2PEM(Yi)=Xi2PEq[EYjq(Yi)jqi]=Xi2PEq(Niqi)=Xi2PNi=T. (4) 56

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Hencewehaverelatedtheestimatorin( 4 )tototalabundanceT.Ifweknewthemeancatchability,wecouldestimateTusingtheestimatorbT=bTp whichfollowstheworkofThompsonandSeber(1994).However,wefocusonthecasewhereisestimatedbybyieldingtheestimatorbT=bTp b (4)whichmotivatesourworkinsubsequentsections. 4.2.2.2VarianceofbTpAsinitsexpectation,thevarianceofbTpisinuencedbytwocomponents,thesamplingdesignandthethevariabilityinducedbecauseateachsiteYiisnotthenumberofanimalsbutrathersome(unknown)proportionofit.Then,consideringthelawofconditionalvariance,wehavevar(bTp)=EMhvarDjM(bTpjY)i+varMhEDjM(bTpjY)i (4)wheresubscriptsonthevariancedenotewithrespecttowhatdistributionthevarianceiscalculatedandaredenedanalogouslytotheexpectationsubscripts.However,appealingtoHorvitzandThompson(1952)weknowvarDjM(bTpjY)=Xi2P1)]TJ /F10 11.955 Tf 11.96 0 Td[(i iY2i+Xi2PXj6=iij)]TJ /F10 11.955 Tf 11.96 0 Td[(ij ijYiYj (4)whereij=Pr(i2S,j2S),i6=jarethejointsiteinclusionprobabilities.Alsoby( 4 )weknowEDjM[bTpjY]=Xi2PYi. (4) 57

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ContinuingwehavevarM Xi2PYi!=Xi2PvarM(Yi)=Xi2PEqvarYjq(Yijqi)+EYjq(Yijqi)=Xi2PNiE[q(1)]TJ /F5 11.955 Tf 11.95 0 Td[(q)]+N2ivar(qi)=Xi2PNi+N2i2q=Xi2PNi+2qXi2PN2i (4)wheretherstequalityresultsfromtheindependenceoftheYisandwherewehavedened=E[qi(1)]TJ /F5 11.955 Tf 12.54 0 Td[(qi)]and2q=var(qi),recallingtheqiareidenticallydistributed.Finally,EM[varDjM(bTpjY)]=EM"Xi2P1)]TJ /F10 11.955 Tf 11.95 0 Td[(i iY2i+Xi2PXj6=iij)]TJ /F10 11.955 Tf 11.95 0 Td[(ij ijYiYj#=Xi2P1)]TJ /F10 11.955 Tf 11.96 0 Td[(i iEM(Y2i)+Xi2PXj6=iij)]TJ /F10 11.955 Tf 11.96 0 Td[(ij ijEM(YiYj)=Xi2P1)]TJ /F10 11.955 Tf 11.96 0 Td[(i ivarM(Yi)+(Ni)2+Xi2PXj6=iij)]TJ /F10 11.955 Tf 11.95 0 Td[(ij ijNiNj2=Xi2P1)]TJ /F10 11.955 Tf 11.96 0 Td[(i iNi+N2i2q+N2i2+Xi2PXj6=iij)]TJ /F10 11.955 Tf 11.95 0 Td[(ij ijNiNj2 (4)Applying( 4 )and( 4 )wehavevar(bTp)=Xi2PNi+2qXi2PN2i+Xi2P1)]TJ /F10 11.955 Tf 11.96 0 Td[(i iNi+N2i2q+N2i2+Xi2PXj6=iij)]TJ /F10 11.955 Tf 11.95 0 Td[(ij ijNiNj2="Xi2Pi+1)]TJ /F10 11.955 Tf 11.96 0 Td[(i iNi#+2q"Xi2Pi+1)]TJ /F10 11.955 Tf 11.96 0 Td[(i iN2i#+2Xi2PN2i i 58

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)]TJ /F10 11.955 Tf 11.96 0 Td[(2Xi2PN2i+2"Xi2PXj6=iij)]TJ /F10 11.955 Tf 11.96 0 Td[(ij ijNiNj#=Xi2PNi i+(2q+2)Xi2PN2i i)]TJ /F10 11.955 Tf 11.95 0 Td[(2Xi2PN2i+2Xi2PXj6=iij)]TJ /F10 11.955 Tf 11.95 0 Td[(ij ijNiNj, (4)thevarianceofthepartialpopulationestimator. 4.2.2.3EstimatingthevarianceofbTpIntheprevioussectionwederivedthevarianceoftheestimatorbTp.Inthissectionwepresentanestimatorforvar(bTp)whichinvolvesobservedcountsandcanbeestimatedwithasamplefromaknowndesign.Theconsistencyofourvarianceestimator,aswewillsee,reliesontheconsistencyofestimatorsforthefollowingfunctionsofq:2,2qand.Therefore,webeginbysupposingwehaveconsistentestimatorsb,b2qandbfor,2qand,respectively.Additionally,fornotationalconvenienceweassumetheestimatorbconsistentlyestimates=E(q2).Manyestimationtechniquesyieldconsistentestimates,includingMaximumLikelihoodestimation(CasellaandBerger2002,p.470)andBayesianestimation(Gelmanetal.2004,p.107).Further,werelyonthefollowingpropositionthroughout,assertedbyCasellaandBerger(2002,p.233):Proposition1Ifbisaconsistentestimatorofthenh(b)isaconsistentestimatorofh()foranycontinuousfunctionh.Nowthevarianceestimatorwewishtoconsideriscvar(bTp)=b bXi2SYi 2i+"Xi2SY2i 2i)]TJ /F4 11.955 Tf 11.96 20.45 Td[( 1)]TJ /F4 11.955 Tf 14.44 11.25 Td[(b b!Xi2SYi 2i#)]TJ /F4 11.955 Tf 12.49 0 Td[(b2"1 bXi2SY2i i)]TJ /F4 11.955 Tf 11.96 16.85 Td[(1 b)]TJ /F7 11.955 Tf 13.53 8.08 Td[(1 bXi2SYi i#+Xi2SXj6=iij)]TJ /F10 11.955 Tf 11.95 0 Td[(ij ijijYiYj. (4) 59

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Weshowtheconsistencyof( 4 )byconsideringitsasymptoticexpectationandforconvenienceweconsidereachtermindividually.Also,letthesymbols_=and_denoteasymptoticequalityandasymptoticdistribution,respectively.Expectationof( 4 )istakenwithrespecttothedesignandthemodel,asbefore,butnowalsowithrespecttothedistributionoftheestimatorsb,b2q,bandb.WedenoteexpectationwithrespecttothedistributionoftheseestimatorsbyE.Notethattheseestimatorsarefunctionsofdatafromthemultiple-passsitesonly,meaningtheyareindependentofrandomvariablesinvolvedwiththesingle-passsites.Thenforthersttermof( 4 )wehaveE b bXi2SYi 2i!=E b b!E Xi2SYi 2i!=E b b!E Xi2PYiZi 2i!=E b b!Xi2PE(Yi)E(Zi) 2i=E b b!Xi2PNii 2i=E b b!Xi2PNi i (4)andbyapplyingProposition1alongwithSlutsky'sTheorem(CasellaandBerger2002,p.239)wehaveE b bXi2SYi 2i!_= Xi2PNi i=Xi2PNi i (4)whichisthersttermof( 4 ).NextweconsiderE"Xi2SY2i 2i)]TJ /F4 11.955 Tf 11.95 20.45 Td[( 1)]TJ /F4 11.955 Tf 14.44 11.25 Td[(b b!Xi2SYi 2i# 60

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=E"Xi2PY2iZi 2i)]TJ /F4 11.955 Tf 11.96 20.44 Td[( 1)]TJ /F4 11.955 Tf 14.44 11.24 Td[(b b!Xi2PYiZi 2i#=Xi2PE(Y2i)E(Zi) 2i)]TJ /F4 11.955 Tf 11.96 20.44 Td[("1)]TJ /F1 11.955 Tf 11.95 0 Td[(E b b!#Xi2PE(Yi)E(Zi) 2i=Xi2PNi+N2i2q+N2i2i 2i)]TJ /F4 11.955 Tf 11.95 20.45 Td[("1)]TJ /F1 11.955 Tf 11.96 0 Td[(E b b!#Xi2PNii 2i=Xi2PNi+N2i2q+N2i2 i)]TJ /F4 11.955 Tf 11.96 20.45 Td[("1)]TJ /F1 11.955 Tf 11.96 0 Td[(E b b!#Xi2PNi iwhichimplies,againappealingtoProposition1andSlutsky'sTheorem,E"Xi2SY2i 2i)]TJ /F4 11.955 Tf 11.96 20.44 Td[( 1)]TJ /F4 11.955 Tf 14.44 11.24 Td[(b b!Xi2SYi 2i#_=Xi2PNi+N2i2q+N2i2 i)]TJ /F4 11.955 Tf 11.95 16.86 Td[(1)]TJ /F10 11.955 Tf 13.15 8.09 Td[( Xi2PNi i=Xi2PNi+N2i2q+N2i2 i)]TJ /F7 11.955 Tf 11.95 -.17 Td[(()]TJ /F10 11.955 Tf 11.96 0 Td[()Xi2PNi i=(2+2)Xi2PN2i i (4)since)]TJ /F10 11.955 Tf 11.95 0 Td[(=.Asymptoticexpectationofthethirdterm( 4 )isobtainedsimilarlyto( 4 ).ThereforeitiseasilyseenthatE"1 bXi2SY2i i)]TJ /F4 11.955 Tf 11.95 16.85 Td[(1 b)]TJ /F7 11.955 Tf 13.53 8.08 Td[(1 bXi2SYi i#_="1 Xi2P[Ni+N2i2q+N2i2]#)]TJ /F4 11.955 Tf 11.96 16.85 Td[(1 )]TJ /F7 11.955 Tf 13.54 8.09 Td[(1 Xi2PNi=)]TJ /F10 11.955 Tf 11.95 0 Td[(+ Xi2PNi+2q+2 Xi2PNi=Xi2PN2i 61

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sothatE b2"1 bXi2SY2i i)]TJ /F4 11.955 Tf 11.96 16.86 Td[(1 b)]TJ /F7 11.955 Tf 13.53 8.09 Td[(1 bXi2SYi i#!_=2Xi2PN2i. (4)Finally,considertheexpectationofthelasttermof( 4 ).Todoso,letZijBernoulli(ij)bethejointinclusionrandomvariableforsitesi,j2Psuchthati6=j.ThenwehaveE"Xi2SXj6=iij)]TJ /F10 11.955 Tf 11.96 0 Td[(ij ijijYiYj#=Xi2PXj6=iij)]TJ /F10 11.955 Tf 11.96 0 Td[(ij ijijE(Yi)E(Yj)E(Zij)=Xi2PXj6=iij)]TJ /F10 11.955 Tf 11.96 0 Td[(ij ijij2NiNjij=2Xi2PXj6=iij)]TJ /F10 11.955 Tf 11.95 0 Td[(ij ijNiNj (4)Compilingthefourconsistentlyestimatedtermsweseethatcvar(bTp)consistentlyestimatesvar(bTp)whichwetakeadvantageofinthenextsection. 4.2.3TheModelWehavethusfardescribedpiecesofamodelusefulforestimatingtotalabundanceusingdepletionsurveydata.Wehavesuggestedageneralhierarchicalstructureusefulformodelingthemultiple-passsitesandwehavediscussedadesign-basedapproachtothesingle-passsites.Inthissectionwediscusstheprocessofcoherentlycombiningtheseframeworksintoasinglemodel.Little(2004)assumedeachcountdividedbytheinclusionprobabilityatthatsite,Yi i, (4)isanormalrandomvariable.Instead,followingRosen(1972a,b)andSen(1988)weappealtolargesampletheoryandhave,forsufcientlymanysinglepasssites,bTp=Xi2SYi i_Normal(E(bTp),var(bTp)). (4) 62

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Now,recallE(bTp)=EM[EDjM(bTpjY)]=EM(Tp)=T.Substitutionyieldsausefulform:bTp_Normal(T,var(bTp)). (4)Next,deningp=Twehaveourestimandofinterest:p =T. (4)Finally,notethatimplementationofbTpintoamodelinvolvesconsideringbTpasasingledatumusefulforestimatingp.However,theindividualsingle-siteobservations,alongwiththesiteinclusionprobabilitiesandthevariousestimatedfunctionsofq(thequantitiesof( 4 ))allowustoestimatevar(bTp)using( 4 ).Henceimplementationofourmethod,whichincludesspecicationofadistributionforbTp,involvesreplacingitstruevariancevar(bTp)withitsestimatorcvar(bTp),expression( 4 ).Thecompletemodelincludesimplementationofthesingle-passsitedataasjustdescribedandalsothemultiple-passsitesthroughthehierarchicaldepletionmodeldescribedinsection2.1.Thesetwosourcesofinformationcombinetoallowestimationofthequantityofinterest,T,aswellascondenceintervalsorcredibleintervalsforT.InthenextsectionwedescribetheprocessofestimationoftheparametersofthemodelusingBayesianmethods. 4.2.4BayesianEstimationofModelParametersOneoptionfortheestimationoftheparametersofahierarchicaldepletionmodelistheBayesianframework(Wyatt2002).Wenowdescribethatprocesswithrespecttothemodeloutlinedabove.Forconvenienceconsiderthefollowingvectorsofparametersinvolvingthemultiple-passsites:q=fqi:i2MgN=fNi:i2Mg=fi:i2Mg. 63

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Similarly,letYm=fYij:i2Mandj=1,2,...,dgbethemultiple-passsitedepletioncounts.Then,appealingtosection2.1,wehavethefollowingmultivariatedensityfunctions:[YmjN,q]=Yi2MpYj=1[YijjNij,qi][Nj]=Yi2M[Niji][j]=Yi2M[ij][qj]=Yi2M[qij]. (4)Next,thedistributionofbTpdependsontheparameterpbutalsoontheparameters,2q,andwhichdescribevariousaspectsofthedistributionoftheqis.However,theparameters,2q,andaresimplyfunctionsofthecatchabilityparameters.Additionally,implementationofourmethodinvolvesapplyingthevarianceestimatorcvar(bTp),whichdependsonthesingle-sitecountsYs=fYi:i2Sg.Therefore,althoughtheHorvitz-ThompsonsumbTphasdensityfunction[bTpjp,],expression( 4 ),weemphasizethevarianceapproximationbyultimatelyusingthedensityfunction[bTpjp,,Ys].Notethatwehaveonlyspeciedparticulardistributions(e.g.anormaldistributionforbTp)forYmjN,qandbTpjp,,Ys.Theotherscanbespeciedbytheresearcherbasedonaparticularapplicationofourmethod.Byplacingpriordistributionsonthehighestlevelparameters,,andp,namelythedensities[],[]and[p]weuseBayes'Theoremtoobtainthejointposteriordistribution:[,,p,q,N,jYm,Ys]/[YmjN,q][Nj][j][qj][bTpjp,,Ys][][][p]. (4)EstimationoftotalabundanceT=p (4) 64

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isaccomplishedbyobtainingtheposteriordistributionoftheparametersofinterest,whichinvolvesintegratingovertheuncertaintyoftheotherparamtersinthemodel.Inthiscase,recalling1 =f()forsomefunctionf,theposteriordistributionofinterestis[p,jYm,Ys]/ZZZZ[YmjN,q][Nj][j][qj][bTpjp,,Ys][][][p]dNdddq. (4)HenceameanestimateofTjYm,Ysisobtainedasfollows:E[T=pf()jYm,Ys]=ZZpf()[p,jYm,Ys]dpd. (4)andsimilarlyonemaycomputethevarianceorothermomentsofthisdistribution.Typicallytheintegrationsabovecannotbecomputedinclosedform.However,useofnumericalmethodssuchasMarkovChainMonteCarlo(MCMC)allowsimulateddrawsfromeachdistributiondescribedabove.UseofMCMCmethodsalsoallowstheresearchertoexplorevariousaspectsofinterestofthedistributionsabove,includingobtainingmeans,variancesandcredibleintervals. 4.2.5AsymptoticDistributionoftheHybridAbundanceEstimatorWehaveoutlinedaconsistenttotalabundanceestimatoranddescribedhowtoobtainameanestimateE[T=pf()jYm,Ys],conditionalonthedata.ThedescriptionreliedonalargesampleapproximationforthedistributionoftheHorvitz-Thompsonsumbasedonthesingle-passsites,ofwhichtherearetypicallymany.Itdidnot,however,relyonalarge-sampleapproximationfortheestimandf()=1=.Thisisimportantbecauseinmanycases,includingthebluecrabsamplingdesign,therearemanyfewermultiple-passsiteswhichprovidethedatausefulinestimatingfunctionsofcatchabilityq. 65

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Despitebeingabletoconductinferenceaboutabundancewithoutassumingasymptoticdistributionsforcatchability-relatedestimandsresultingfromthemultiple-passsites,thereismerittoconsideringthefullasymptoticdistributionofourestimator.Firstconsiderourestimatorasaproduct,namelybf()bp (4)wherebf()isaBayesianestimatorof1=andbpisaBayesianestimatorofp.Now,recallfromabovethatbpderivesfromthesingle-passsitedatawhilebf()derivesfromthemultiple-passdata,makingthemindependent.Further,beingBayesianpointestimators,theyhaveasymptoticallynormaldistributions(Gelmanetel.2004).Letbf()_Normal(f,2f)andbp_Normal(p,2p). (4)Now,thedistributionoftheproductoftwoasymptoticallynormalandindependentrandom-variablesisitselfasymptoticallynormal(Barndorff-NielsenandCox1989,p.42).Thenwehavebf()bp_Normal(pf,2p2f+2f2p). (4)Expression( 4 )providesuswithanasymptoticpartitioningofthevarianceofourtotalabundanceestimatorbf()bpwhichisusefulindeterminingwhataspectsofourdatacontributemosttothevarianceinourtotalabundanceestimate.Forexample,byestimatingthecomponentsofthevariancein( 4 )onecanidentifydisproportionatelylargecontributionstothetotalvariance(e.g.uncertaintyincatchability)anddesignfuturesurveystodecreasethatvariance. 66

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4.3AFullyModel-BasedEstimatorInasimulationstudydescribedlaterweconsideredthebehaviorofourhybridestimatorversusafullymodel-based,Bayesiantotalabundanceestimator.Wyatt(2002)describedsuchaBayesianestimatoroftotalabundanceindepletionsurveys.Inthissectionwebrieydescribehisapproach.Recallthatourmultiple-passsitemodelconsistedofafullhierarchicalmodelforthedepletionsurveys.Forthefullymodel-basedestimator,suchamodelisappliedtoallsites,multiple-passandsingle-pass.LettingN=fNi:i2M[Sg,q=fqi:i2M[Sg,and=fi:i2M[Sgconsidertheprocessmodels:[YjN,q],[qj],[Nj]and[j]whereY=Ym[Ys.Thesedensitiesrepresentthesequentialbinomialdatamodel,thecatchabilityprocessmodel,themodelforthesitecountsandtheunderlyinganimalabundancemodel,respectively,andaredenedasfollows:[YjN,q]=Yi2MpYj=1[YijjNij,qi]Yi2S[YijNi,qi][Nj]=Yi2M[S[Niji][j]=Yi2M[S[ij][qj]=Yi2M[S[qij]. (4)Clearlyanimportantdifferencebetweenthismodelandourmodelisthatfullyparametricassumptionsaremadeatallsampledsites,includingS.Nextweassignpriordistributionstothehighestlevelparameters,namelyand.Callthese[]and[],respectively.Then,usingBayes'Theoremweobtainthejointposteriordistributionofalltheparametersofinterest:[N,,,q,jY]/[YjN,q][Nj][j][qj][][] (4) 67

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Estimationoftotalabundanceusing( 4 )involvesobtainingtheposteriordistributionoftheparametersgoverningtheabundanceprocess,namelyjY.Totalabundanceoverthestudyareaisthensomefunctiong()oftheseparameters.Forexample,if=f1,2gand1 2istheaverageanimaldensity(numberperunitarea)overthestudyregion,theng()=A1 2=T,whereAistotalarea,istheestimandofinterest.ThenT=g()isestimatedbycalculatingthemarginalposteriordistributionofjYas[jY]/ZZZZ[YjN,q][Nj][j][qj][][]dNdqdd. (4)andintegratingtheestimandoverthisdensityyieldstheexpectedvalueofT,conditionalonthedata:E[T=g()jY]=Zg()[jY]d. (4)Onecansimilarlyobtainthevarianceoftheestimate,howeverinapplicationtypicallytheaboveintegralswillnotbeavailableinclosedformbutMCMCmethodswillyieldanapproximationof( 4 )alongwithsimulatedcredibleintervalsofT. 4.4SimulationStudyHavingdescribedourhybridabundanceestimatoranditsimplementationinaBayesianframeworkalongwithafullymodel-basedBayesianestimator,wenowdiscussasimulationstudyconductedtocomparethetwoestimators.Thescenariounderwhichwewishtocomparethetwoestimatorsisinthecaseofmis-specicationoftheabundanceprocessmodel.Recallfromtheprevioussectionthatestimationoftotalabundanceinafullymodel-basedframeworkinvolvesestimatingaparticular 68

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functionofthemodelparametersasin( 4 ).Thoseparametersofinterest,namely,areconnectedtothedata(observations)throughastrictmodelingframeworkdenedbythefullhierarchicalmodel.Whenthecomponentsofthatmodelarecorrectlyspecied,estimatesandcredibleintervalswillbeaccurate.However,ifthemodelingframeworkcontainsmis-specications,estimationandinferencemaysuffer,whichmotivatedourtwosetsofsimulations:oneforeachestimatorunderabundancemodelmis-specication.Wewishedtoascertaintherobustnessofourhybridestimatorversusthefullymodel-basedBayesianestimatorundermodelmis-specication.Inparticular,weconsideredthecaseofmis-specicationoftheabundanceprocesssub-modelwhilecorrectlyspecifyingtheothersub-models(e.g.thecatchabilityprocessmodel).Toconsiderthisbehavior,thestudyinvolvediterativelysimulatingtheoreticalpopulationsusingoneabundanceprocessmodelandthenestimatingtheabundanceofeachtheoreticalpopulationusingadifferentabundanceprocessmodel,alongwithsimulateddepletiondata.Recallspecicationofthefullmodelinvolvingourhybridestimatorincludedtheoptionofspecifyingahierarchicalabundanceprocessmodelforthemultiple-passsites(expression( 4 )).Inthecaseofthefullymodel-basedBayesianestimator,theabundanceprocessmodelappliestoallsites(expression( 4 )).Toevaluatetheperformanceofthetwoestimators,weconsideredcredibleintervalcoverage.Thatis,foreachsimulatedpopulation,95%credibleintervalswerecomputedusingeachestimator.Thenwecatalogedwhetherthesimulatedtotalabundancefellwithineachestimator's95%credibleinterval.Foraneffectiveestimator,approximately95%ofthesimulatedtotalabundancesshouldfallwithintheirrespectivecredibleintervals.Inourcase,deviationfromthiscoveragelevelindicatesanestimatorsuffersfromthemis-specicationoftheabundanceprocessmodelbecauseallothermodelingassumptionsheld. 69

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Conductingsuchasimulationstudyinvolvesspecifyingprocessmodels,includingtheirrespectiveparameters,inordertogeneratetheoreticalpopulations.Further,onemustspecifyasamplingdesignwhichyieldsthetheoreticaldatausedtoestimatetotalabundanceinconcertwiththemodels.Ingeneral,wechoseprocessmodelsandparameterswhichmimickedtheworkofWyatt(2002)andDorazioetal.(2005),inthecaseofmodelchoice,andparameterizedthembasedonourexperiencefromanalyzingdepletionsurveydataoftheChesapeakeBaybluecrabpopulation.Forsimplicity,wechoseallsites,boththemultiple-passsitesandthesingle-passsites,usingsimplerandomsampling.Wenowfullyoutlinethedetailsofthesimulationstudy. 4.4.1SimulatingPopulationsandDepletions-DataGeneratingModelForeachiterationforeachestimatoratheoreticalpopulationwassimulatedfromaknownabundanceprocess.ThesimulatingabundanceprocesswasaPoisson/Uniformmixturedistribution,thatisNijiPoisson(500i)iUniform(0,1)fori=1,2,...,10000sites.Inthiscase,iisthedensity(numberperunitarea)oftheanimalsatsitei,whereeachsiteis500unitsofarea,sotheabundanceNiisPoissonwithmean500i.Fromthepopulationof10000sites,werandomlysampled,withoutreplacement,10multiple-passsites(M)and500single-passsites(S).Individualsitecatchabilitiesqiforsitesi2M[Sweredrawnrandomlyfromahierarchicalbetadistribution,namelyqij,Beta(,) 70

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wherethehigher-levelcatchabilityparametersandweredrawnfromrespectivenormaldistributions,namelyNormal(6,2)Normal(18,2).Withanabundance,Ni,andacatchability,qi,foreachsitei2M,wesimulatedadepletionsurveyusing12passesateachsite.Theresultingdatawererandomlydrawnasfollow:YijBinomial(Nij,qi)forj=1,2,...,12,whereasbeforeNi1=NiandNij=Ni)]TJ /F6 7.97 Tf 13.63 15.21 Td[(j)]TJ /F3 7.97 Tf 6.59 0 Td[(1Xk=1Yik.Thechoiceof12depletionpassesatthesitesinMresultedfromtheresultsofChapter3.Catchesatthesingle-passsites,i2S,wereconductedsimilarlybutwithonlyasinglebinomialtrialYiBinomial(Ni,qi)fori2S.Henceforeachiteration,weobtainafullsimulatedpopulationalongwithasimulateddepletionsurveyusefulforestimatingthetotalabundanceofthatpopulation. 4.4.2EstimationofPopulationAbundance-DataAnalysisModelWenowdiscusstheestimationoftotalabundanceforeachsimulatedpopulation.Toaddressthegoaloftestingrobustnessofourhybridestimatorversusthefullymodel-basedestimatortomis-specicationoftheabundanceprocessmodel,themodelusedtoanalyzethedepletiondataforeachsimulatedpopulationdidnotmatchthemodelusedtogeneratetheunderlyingcounts.Inparticular,whereasunderlying 71

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populationdensitiesweregeneratedfromaUniform(0,1)distribution,thecorrespondingabundanceprocessmodelforthedataanalysiswasincorrectlyassumedtobeagammadistribution.Thatis,thehierarchicalabundancemodelforeachestimatorisNijiPoisson(500i)ij1,2Gamma(1,2).Catchabilitiesweremodeledaccuratelyasfollows:qij,Beta(,).Similarlythebinomialdataprocessmodelwasspeciedidenticallytothegeneratingmodel:YijjNij,qiBinomial(Nij,qi)fori2M,j=1,2,...,12forbothestimators.Becausethefullymodel-basedestimatorrequiresmodelingofallsites,weadditionallyspeciedthesingle-trialbinomialmodelYijNi,qiBinomial(Ni,qi)fori2Sforthemodel-basedestimator.Forthehybridestimator,thesitesSwereincorporatedthroughtheirHorvitz-ThompsonsumasdescribedintheMethodssection.Thatis,Xi2SYi iNormal+ T,v(,,i,ij,Ys) (4)wherehasbeenreplacedby+ asaresultofthechoiceofaBeta(,)distributionforthecatchabilitiesandv(,,i,ij,Ys)isthevarianceestimator( 4 )calculatedwiththesimulatedsingle-passdataYs. 72

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Becauseofourchoiceofsamplingdesign,thesiteinclusionprobabilitieswerei=500 10000=0.05ij=500 10000499 9999=0.002495250fori,j2P,i6=j.BecauseeachestimatorisimplementedwithinaBayesianhierarchicalmodel,priordistributionsareneededonthehighestlevelparametersineachcase.Forthehybridestimator,priordistributionswererequiredfortheparameters,,1,2andp.Therstfouroftheseweregivendiffusegammapriors,namely1,2,,Gamma(.001,.001)whilepwasgivendiffusenormalpriorwithmeanandvariancebelow:pNormal(555,11010). (4)Forthefullymodel-basedestimator,priordistributionswererequiredfortheparameters,,1and2.Ideallythesamepriordistributionswouldhavebeenusedforthefullymodel-basedestimator.However,tofacilitateMarkovchainconvergencerates,weplacedthefollowingpriorsontheparameters:Gamma(.2,.1)Gamma(1.8,.3)1Gamma(.1,.1)1Gamma(.1,.1). (4)Theabovepriordistributionsweremoreconcentratedaroundthecorrectparameterspacethanthepriorsplacedonthehybridestimator.Additionallyinthecaseofthefullymodel-basedestimatorweimplementedtheequivalent,marginalizeddepletionmodel 73

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describedbyRoyleandDorazio(2008)tofurtherimprovechainconvergencerates.Finally,withbothmodelingframeworksfullyspecied,wediscussimplementation.WeimplementedtheabovesimulationstudyusingthestatisticalprogramminglanguageRv.2.10.1(RDevelopmentCoreTeam2009)andthestatisticalprogramWinBUGSv.1.4.3(Lunnetal.2000)byusingtheRpackageR2WinBUGS(Sturtz,Ligges,andGelman2005).Weconducted500iterationsimplementingthefullymodel-basedestimatorwithaburn-inof180000drawsforeachiterationandanadditional60000drawsthinnedtoevery10thdrawtoreduceautocorrelationinthechains.Thisgaveus6000simulateddrawsfromtheposteriordistributionofinterest.Forourproposedestimatorweranatotalof2000iterations.Foreachiterationweallowedaburn-inof15000drawsandthendrewanadditional30000draws,alsothinningtoevery10thdraw,yielding3000simulateddrawsfromtheposteriordistributionofinterest.Foreachestimatorandforeachiterationofthesimulation,weranthreeindependentchainsfromdifferentstartingvaluestoaidincharacterizingbehaviorofthechainslater.MoredetailedinformationincludingstartingvaluesisprovidedwiththecomputercodeavailableinsectionC.3ofAppendixC.Foreachiterationofeachestimatorweobtainedthe95%credibleintervalcomprisedofthevaluesbetweenthe2.5thand97.5thpercentilesofthesimulateddrawsfromtheposteriordistribution.Asametrictocomparethetwoestimators,wecalculatedcredibleintervalcoverage,i.e.whatproportionoftheiterationsinwhichthe95%credibleintervalcontainedthesimulatedtotalabundance10000Xi=1Ni.AdditionallyweattemptedtoascertaintheconvergenceoftheMarkovchainsbyvisualinspection;anyiterationsinwhichtheMarkovchainsdemonstratedpoorbehaviorwereexcludedfromthestudy.Fortheremainingchains,theBrooks-Gelman-Rubin(BGR)convergencediagnostic(BrooksandGelman1998)wascalculatedandchainswith 74

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BGRdiagnosticvaluesinexcessof1.1werealsoexcludedfromthesimulationstudy.Inthecaseofthefullymodel-basedsimulations,Markovchaininspectionwasconductedonthechainsmeanttoconvergetothemarginalposteriordistributionofaverageanimaldensity.Forourmethod,theinspectionoccurredontheMarkovchainsfortheestimand+ .Finally,becausethissimulationstudywasconductedtoconsiderbehaviorunderamis-speciedabundanceprocessmodel,weonlyconductedsimulationsunderthisscenario.Inthesituationwherethedataanalysismodelfullymatchesthedatageneratingmodel,thefullymodel-basedBayesianestimatorwillyieldaccurateestimatesandcredibleintervals(Gelmanetal.2004).Ourhybridestimatorwillalsoyieldaccurateinferencesforthesamereasonsaslongasenoughsingle-passsitesaresampledtojustifythelarge-samplenormalapproximation.Undermodelmis-specicationwehavenosuchguarantees. 4.5SimulationResultsVisualinspectionoftheMarkovchainsrelatedtoourhybridestimatorproducednoproblematiciterations.CalculationoftheBGRdiagnosticvalueyieldedthreeoutof2000iterationswithBGRvalueslargerthan1.1.Theseiterationswerethusexcludedfromthesimulationstudy.VisualinspectionoftheMarkovchainsforthefullymodel-basedestimatoridentied140of500iterationsthatdemonstratedquestionableconvergenceandthoseiterationswereexcludedfromthestudy.Severalexamplesofpoorly-behavedMarkovchainsappearinFigureE-1inAppendixE.AnadditionalnineiterationshadcalculatedBGRdiagnosticvaluesinexcessof1.1andwerealsoexcluded.Ofthe1997iterationsofourhybridestimatorretainedforthestudy,themeantotalabundanceforthesimulatedpopulationswas2500273andthemeanofourabundanceestimateswas2566126.Henceforthesimulationstudyourestimatorexhibitedapositivebiasof+2.57%.Ofthe351iterationsofthefullymodel-basedestimatorretainedforourstudy, 75

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themeansimulatedpopulationabundancewas2498177andthemeanabundanceestimatewas2458372,yieldinganegativebiasof-1.62%.Regardingcoverageproportions,ofthe1997iterationsretainedinthesimulationstudy,in95.04%ofthemthe95%credibleintervalcontainedthesimulatedtotalabundanceT.Ofthe351iterationsretainedforthefullymodel-basedestimator,78.35%ofthe95%credibleintervalscontainedthesimulatedtotalabundanceT.FullresultsaresummarizedinTables4-1and4-2. 4.6DiscussionInthismanuscriptwehavepresentedandimplementedahybridtotalabundanceestimatorfordepletionsurveysthatincorporatesadvantagesofbothmodel-basedanddesign-basedestimation.Ourframeworkalsoaccountsforsite-to-sitevariabilityincatchabilitywhichhaspreviouslybeenshowntoreducebiasoftotalabundanceestimates(Wyatt2002).Inadditiontothemodel-basedframework,wehavetakenadvantageofaspectsofdesign-basedestimationwhichallowedustoavoiddescribingwithamodelapoorlyunderstoodabundanceprocessl.Insteadwehaveappealedtodesign-basedresultsthatusetheHorvitz-Thompsonestimator.ByapplyingtheHorvitz-Thompsonestimatorourmethodologyisimmediatelyapplicabletodepletionsurveysinwhichthelocationswereselectedfromawidevarietyofdesigns.However,ourapproachisnotrestrictedtotheHorvitz-Thompsonestimator.Otherdesign-basedestimatorsforwhichtheasymptoticdistributionisknown(e.g.theHansen-Hurwitzestimator,Thompson1992)maybeimplementedwithinthegeneralframeworkweoutlinedabove.Inourmethodologythesingle-passsiteobservationsenterthemodelthroughtheasymptoticdistributionoftheirweightedsum,alleviatingtheneedtospecifyanabundancemodelatthosesites.Finally,althoughwehaveonlyconsiderednitepopulationsampling,theinclusionofinnitepopulationanaloguestotheHorvitz-Thompsonestimator(Cordy1993)isalsopossible.Estimationinaninnite 76

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populationsettingwouldinvolvereplacingsiteanimalabundanceswithanimaldensitiesatpointsinspace.Withinourframeworkwehaveincludedasafundamentalpartthebinomialmodelfordepletionobservations(thedatamodel).Thebinomialmodelappliesnotonlytothemultiple-passsitesbutalsotothesingle-passsitesandthereforeispartofthederivationofthevarianceofbTpinsection2.2.1.Thebinomialcountmodelindepletionsurveyshasbeencentraltoearlydepletionmodels(Moran,1951;Zippin,1956)aswellasacomponentofmorerecenthierarchicaldepletionmodels(Wyatt,2002;Dorazioetal.2005).Althoughanymodel'sassumptionsshouldbeconsidered,ourspecicationofthebinomialmodelforthedepletionprocessisacommonone.Otherwiseourmethodologyisexibleenoughtoallowresearcherstochoosefromawidevarietyofmodelsfortheotherprocessesofinterest(e.g.thecatchabilityprocess).Insection2.1weallowforaninformativeabundancemodeltobespeciedatthemultiple-passsites,butwedonotrequireit.Itmaybepossiblethatmultiple-passsitesarechosenfromanareaatwhichtheabundanceprocessisunderstood,orchosenatsitesofknownhighabundance(aswediscussinChapter3).Inthiscasespecifyingamodelonlyforthosesitesmaybereasonable.Howeverwithenoughdepletionpassesatthemultiple-passsites,thedatagenerallyoverwhelmapoorlyspeciedabundancemodel.Inoursimulationstudyweassignedthesameincorrectabundancemodelatthemultiple-passsitesinourestimatorasinthefullymodel-baseddesign.Inthecaseofthesimulationstudy,mis-specicationoftheabundancemodelatthemultiple-passsitesinthehybridestimatorhadnoill-effectsoncredibleintervalcoveragewhileclearlythefullymodel-basedestimatorsuffered.Oursimulationstudy,designedtocomparetheperformanceofastandardmodel-basedestimatorandourmodel-assistedestimatorunderabundancemodelmis-specication,indicatedseveraladvantagestousingourestimator.Mostimportantistheimprovedperformanceincredibleintervalcoverageforourestimatorversusthe 77

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fullymodel-basedestimator.Ourmethodappearstoaccuratelyassessuncertaintyofestimatedtotalabundanceasevidencedbythe95.04%coverageofits95%credibleintervals.Thatisamajorimprovementoverthefullymodel-basedestimatorwhichhadacoverageoflessthan80%.Ingeneral,theill-performingMarkovchainsassociatedwiththefullymodel-basedestimatorconcernedusandwouldlikelyleadtheresearchertoattemptadifferentapproach.However,thatthepooroverallmodelperformanceisattributabletoamis-speciedabundancemodelisnotobvious.ThepoorperformanceoftheMarkovchainsledustoincorporatelongburn-inssothatwecouldonlycomplete500simulationiterations.OntheotherhandMarkovchainsassociatedwithourestimatoringeneralappearedtoconvergetotheirstationarydistributionsveryquickly.Thisisattributablenotonlytotherobustnessofourestimatorbutalsotothemanyfewerparametersinthemodelassociatedwithourestimator.Sincethesingle-passsitesenterthemodelthroughtheirHorvitz-Thompsononlyasingleparameter(p)isestimatedbasedonthoseobservations.Alternatelythemodel-basedapproachinvolvesestimatinganabundanceandcatchabilityvalueateachsingle-passsite.Thereforethroughoursimulationstudywenotedmultipleadvantagesofourestimatorversusastandardmodel-basedestimator,atleastinthecaseofabundancemodelmis-specication.Thebiasbothestimatorsexhibitedinthesimulationstudylikelyresultsfromthemis-specicationoftheabundanceprocessmodel.Ofcourseourestimatorreliedonlyuponthismodelspecicationforthemultiple-passsites.Consistencyoftheourtotalabundanceestimatorhingesonconsistencyofthecatchabilityparameters.Consistencymaybecompromisedbymodelmis-specication,whichresultsinthesmallbiasweseeinthesimulations.However,withourestimatorweobservedexcellentcredibleintervalcoveragedespitetheabundancemodelmis-specication.Additionally,inpracticeourestimatorcanbeimplementedbyspecifyinganon-informativediscreteuniformdistributionfortheabundancesatthemultiple-passsites.Suchaspecicationmayresultinmorerobustpointestimation.However,tobeconsistentacrossboth 78

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typesofestimatorsinthesimulationstudyweappliedthesameincorrectabundanceprocessmodelforbothcases.Finally,althoughwedesiredtomaintainconsistencyacrossestimatorsinthesimulationstudy(e.g.byplacingthesamepriordistributionsonparameterscommontobothmodels),thepoorperformanceofMarkovchainsininitialsimulationsofthefullymodel-basedestimatorpromptedustotightenpriordistributionsinthecorrectparametersspaceforthemodel-basedestimator.Theseinconsistencieswerenecessaryforcomputationalreasonshoweveranysuchinconsistentspecicationwasmadesoastoprovideanadvantagetothefullymodel-basedestimator.Itisimportanttonotethatthemethodologypresentedinthismanuscriptallowstheresearchertoobtainnotonlyameanandvarianceforatotalabundanceestimatorbutalsoprovidesacoherentframeworkwithwhichtheresearchercanapproximatethefulldistributionoftheabundanceestimatorusingstandardnumericalmethods(e.g.MCMC).Althoughwepresentthefullasymptoticdistributionoftheestimatorandsuggestitsutilityinsection2.4,obtainingthe(MCMCsimulated)posteriordistributionoftotalabundancedoesnotrequirealargesampleassumptionforthemultiple-passsites.Asinthecaseofthebluecrabdataset,researchersmayconductmultiple-passsurveysatconsiderablyfewersitesthanthesingle-passsurveys,andourmethodologyaccountsforthis.Regardingtheasymptoticdistributionofourhybridestimator,beingabletoconsiderthecomponentsofvariabilityofthetotalabundanceestimateisextremelyvaluable.Thecomponentsofthevarianceinexpression( 4 )areestimatedduringtheimplementationofourmethod,andtheirrelativemagnitudesmayinformfuturesamplingstrategies.Forexample,analysisofthecomponentsofvariancemaysuggesttheinclusionofmoremultiple-passsites,andhenceanexpecteddecreasein2fwhichcharacterizesuncertaintyaboutthecatchabilityprocess,mayimprovetheprecisionoftotalabundanceestimatesinsubsequentyears. 79

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Althoughwehaveprovidedevidencefortheutilityofourtotalabundanceestimatorinthecaseofuncertaintyintheabundanceprocessmodel,therearelimitationstoitsuse.Primarily,ourestimatorisaimedsolelyatestimatingtotalabundanceandnotestimatingtheabundancesatthesingle-passsitesS.Alternativemethodsmaybeapplicableincaseswherelocalabundanceestimationisemphasized,includingBayesiannon-parametricapproaches(Dorazioetal.2008).Additionallyforsamplingdesignsinwhichtheproportionofmultiple-passsitesislarge,usingtheabundanceestimatesatthemultiple-passsitesmaybeimportant;suchasituationishandledwithBayesiannon-parametricmethods.Usingsuchmethodsprovidesadditionalinsightintolocalabundanceprocessesand,similartoourmethod,doesnotrequirespecicationofanabundancemodel.HoweverimplementationofsuchmethodsmaybemoreinvolvedforresearchersnotfamiliarwithBayesiannon-parametricmethods.FuturecomparisonsoftheperformanceofourhybridestimatorversusBayesiannon-parametricabundanceestimatorsmayprovidevaluableinsights.Whenestimatingtotalabundance,animportantconsiderationisdeningthestudyareaofinterest.Withaprioriknowledgeofthearealextentofthepopulation,thestudyareaiseasilydeterminedandfurtheristheareafromwhichsiteshavepositiveprobabilityofbeingincludedinthesampleS.Withoutknowledgeofthearealextentofthepopulationitislessclearhowtoproceed.Byrestrictingsamplingtoastrictsubsetoftheareaofthepopulation,moreintensesamplingoccurswithinthesamplingregion(i.e.moreareasampledrelativetothetotalstudyarea),howevermembersofthepopulationresidingoutsideofthesamplingareawillnotbeconsideredwhenestimatingtotalabundance.Alternately,bydeningastudyareamuchlargerthantheactualarealextentofthepopulation,thesamplingperunitareawillbelessintensealthoughtheareaoverwhichoneconductsinferenceislarger.Therefore,applyingaprioriknowledgewhendeningthetotalstudyareaorevenconductingindependentsamplingtoinformthisareamayimproveabundanceestimatesintheend. 80

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Oftenmultipledepletionmethodsareusedwithinastudy.Forexample,someareaofalakemaybesampledforabundanceofsomepopulationofinterestbyonegroupofresearchersandtherestofthelakebyanothergroup.However,estimatesofabundanceacrosstheentirelakemaybeofinterest.Insuchacaseandunderthesameassumptionsmadeabove,ourestimatormaybeappliedtoeachindividualmethodandarea.Thatis,eachdepletionmethodhasitsownpopulationofsitesfromwhichthesingle-passsitesarechosen,aswellasmultiple-passsitestoinformcatchabilityprocess.AswesawinChapter2,itmaybeimportanttoconsiderdifferencesincatchabilityfromonedepletionmethodtoanother.However,byassumingsite-to-siteindependenceofthedepletions,andifthestudyareasofthevariousdepletionmethodspartitionthetotalareaofinterest,thenusingourestimatorindividuallyforeachdepletionmethodanditssurveydatayieldsanabundanceestimateofthetotalarea.Thatis,becauseoftheindependenceassumptionofthedepletionsurveysfromsitetosite,anestimateofthetotalabundanceacrossalldepletionmethodsissimplythesumoftheindividualestimates(obtainedbyourestimator)andthevarianceofthissumislikewisethesumoftheindividualvariances.Thisiseasilyaccommodatedwithinahierarchicalmodelaccountingforthedifferentdepletionmethods.Finallyourcompletemodelwhichincludesahierarchicalmodelforthemultiple-passsitesforthecatchabilityprocessandadesign-basedresulttoincorporatealargesetofsingle-passsitesmayproveusefulinotherstudiesofabundanceinwhichdetectabilityorcatchabilityisimperfect.Wehavepresentedacoherentstructurethatdoesnotrequirespecicationofanabundancemodel,accountsforimperfectdetectionanduncertaintyinestimatingthatdetectionprobabilityandprovidesabundanceestimates.Futureworkmaysuggestothersamplingparadigms,whicharemeanttoinformdetectabilityandabundance(e.g.mark-recapture),couldbeimplementedinamodelsuchasours. 81

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HavingdescribedmethodsforincreasingsamplingefciencyindepletionsurveysinChapters2and3,inthischapterwehaveconfrontedtheproblemofestimatingtotalabundanceusingdepletiondata.Wehaveconsideredthedifcultyofdescribingabundancesofanimalsusingexplicitmodelsandpresentedanestimatorfortotalabundancethatdoesnotrequiretheresearchertodoso.Wehaveconsideredimplementationofourmethodologywithinahierarchicalmodelwhichyieldscoherentestimateswithrespecttothesimultaneousestimationofcatchabilityandabundance.Wehavecomparedourestimatortoananalogous,fullymodel-basedestimatorinasimulationstudyandprovidedacasewherethemodel-basedestimatorperformspoorlybutourestimatorperformswell.Wehavealsodiscussedotherutilitiesoftheestimatorandconsideredextensionsofourapproach.Inpractice,robustestimationtechniquessuchasoursmayprovevitalinresearchinvolvingmanypopulationswhoseabundanceprocessesarepoorlyunderstood. Table4-1. Partialresultsfromasimulationstudycomparingourhybridabundanceestimatortoafullymodel-basedestimator. Estimator#SimulationsMeanTMeanofEstimatesofT Hybrid199725002732566126Model-based36024981772458372 Table4-2. Additionalresultsfromasimulationstudycomparingourhybridabundanceestimatortoafullymodel-basedestimator. Proportionof95%CIsMean%BiasProportionofRejectedEstimatorContainingTofEstimatesIterations Hybrid.9504+2.57%.0015Model-based.7835-1.62%.28 82

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CHAPTER5CONCLUSIONInthisdissertationwehaveconsideredtheproblemoftotalabundanceestimationwhendepletionorremovalsamplingisemployed.Chapters2and3presentedandappliedmethodsusefulinthedesignofadepletionsurveyofanimalabundance.Inparticular,Chapter2consideredmethodsusefulforcomparingthesamplingefciencyofvariousdepletiontechniqueswhichmayinformthedecisionofwhatvessel,gear-typeorpersonnelshouldbeusedinafuturesurvey.Chapter3focusedontheallocationofsamplingeffortindepletionsurveys.Fordesignswhereasmallsetofintensivelysampledsites(thesitesatwhichmultipledepletionsareconducted)informthedetectabilityorcatchabilityprocessandaugmentaseparatesetofsitesatwhichonlyasingledepletionisconducted,weconsideredtheproblemofhowbesttoconducttheintensivesampling.ByconsideringtheexpectedFisherinformationinasample,welookedattherelationshipbetweeninformationaboutthecatchabilityparameterqandthenumberofdepletionpassespermultiple-passsite.Asidefromobtainingoptimalnumbersofdepletionpassespermultiple-passsiteversustotalnumberofsitesforvariouslevelsofcatchability,thederivationoftheFisherinformationitselfyieldedusefulinformationabouthowotheraspectsofsamplingatthemultiple-passsitesimpacttheinformationaboutcatchability.Finally,inChapter4weconsiderednotthesamplingframeworkbutratherestimationoftotalabundance.Wepresentedanoveltotalabundanceestimatorthatborrowsfromtwocommonstatisticalestimationframeworks,design-basedandmodel-basedestimation.Althoughthisdissertationconfrontsproblemsinvolvingdepletionsurveys,whichareacommonformofsamplinginsheriesscienceandotherecologicalorconservationelds,Chapters2,3and4alsocontainanimportantstatisticaltheme.Jointlytheyconsidertheentirestatisticalprocessofdesigningasurveyandusingthatinformationtoobtainrobustestimatesofaquantityofinterest.Chapters2and3addressincreasing 83

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orevenmaximizingtheinformationobtainedduringthesamplingprocessbyimprovedsurveydesignwhereasChapter4presentsanestimatorthatrequireslittleinformationaboutthepopulationofinterest(appealingtodesign-basedestimationprinciples)whilestillincorporatingcertainadvantagesofmodel-basedestimation.Perhapsthemostimportantofthesemodel-basedadvantagesisthattheestimatorcanaccountforsite-to-sitevariationincatchabilityatthemultiple-passsiteswheretheobjectiveistoobtaininformationaboutthecatchabilityprocess.Notcorrectlyaccountingforsite-to-sitevariationincatchabilityhaspreviouslybeenshowntobiasestimatesindepletionsurveys(Wyatt2002).Finally,theestimatorcoherentlydealswiththecommonsituationinsurveysinvolvingimperfectcatchabilityordetectabilityinwhichthecatchabilityordetectabilityrateisunknownandmustbeestimatedaspartoftheanalysis.Hencetheworkpresentedinthisdissertationisusefulforabundanceestimationbeginningwiththeeffectivedesignofadepletionsurveyandendingwithcoherentestimationofabundanceusingthesurveyresults.Theresearchconductedforthisdissertationalongwiththeresultswehavepresented,particularlyinChapters3and4,havesuggestedseveralnewavenuesofresearch.First,Chapter3considersonlytheallocationofaportionofthetotalsamplingeffortinadepletionsurveyoftotalabundance.Althoughourresultsareusefulforallocatingthemultiple-passeffortinadepletionsurveyundercertainassumptions,wedidnotaddresshowtotalsamplingeffort(i.e.theeffortforbothmultiple-passsitesandsingle-passsites)shouldbeallocated.However,theframeworkofFisherinformationappliedinChapter3maybeusefulinmaximizingtotalexpectedinformationaboutparametersrelateddirectlytoabundance(suchasmeananimaldensity)withrespecttotheallocationofthetotalsamplingeffort.Suchananalysiscouldsuggesthowmanysingle-passsitesshouldbeincludedinadepletionsurvey,howmanysingle-passsites,andnallyhowmanydepletionpassespermultiple-passsite.Also,recallingthatthederivationofFisherinformationisbasedonthelikelihoodfunctionoftheparameter(s) 84

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ofinterest,alikelihoodfunctionbasedonamorerealisticdepletionmodel(e.g.onethatincludessite-to-sitevariationincatchability)mayfurtherimprovedesignsofdepletionsurveys.Thehybriddesign-based/model-basedestimatorofChapter4wasderivedtoestimatetotalabundancebasedondepletionsurveydata.However,itislikelythatthehybridframeworkpresentedinChapter4couldbeextendedtoothersurveydesignsforpopulationsofinterestthataresubjecttoimperfectdetection.Ourtotalabundanceestimatorwasmotivatedbytheneedtocorrectlyaccountforuncertaintyinthecatchabilityordetectabilityprocess,allowingitsuncertaintytopropagateultimatelytotheuncertaintyassociatedwithtotalabundanceestimates.Additionallyweaccomplishthisundertheassumptionthatcatchabilityordetectabilityratesareunknownandmustbeestimated,asisthetypicalcaseinpractice.Whileourframeworkconnectsaexibleandrealistichierarchicalmodelforthemultiple-passdepletionsitestoabroadersurveyoftotalanimalabundance,thehierarchicalmultiple-passdepletionmodelcouldbereplacedwithothersuchmodelsdescribinganimaldetectionandthereforeaccountforothertypesofabundancesurveys.SimultaneouslytheothersurveysitesarecurrentlyincludedcollectivelyasaHorvitz-Thompsonsumbutotherdesign-basedestimatorscouldbeimplementedaswell.WediscusssuchexibilitiesinthenalsectionofChapter4.Inadditiontoparticularresearchavenuessuggestedbyourwork,itisimportanttoaddressquestionsrelatedtomodelingassumptions.Althoughmodelingassumptionsshouldbecarefullyconsidered(andinsomecasestested)onacase-by-casebasis,wealsocontendthatitisimportanttoconsidertheimpactoftheseassumptionsmorebroadly.Thismaybeaccomplishedbydetermininghowrobustestimatesofquantitiesofinterestaretoviolationsofvariousmodelassumptions.Forexample,commontotheresearchinthisdissertationistheassumptionthatcatchabilityofanimalsatasiteremainsconstantthroughoutallpassesofadepletionsurveyatthatsite.Thereforeit 85

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isimportanttoconsiderwhetherviolationofthisassumptionintroducesproblematicbiasesintototalabundanceestimates.OnesuchscenarioisconsideredbyMantyniemietal.(2005)whosuggestthatbiasedabundanceestimatesdoresultfromimproperlyneglectingtoaccountforvariablecatchabilitybyanimal.However,moresuchstudiesareneededtoclassifyresultingbiasesunderanumberofincorrectdepletionmodelspecications.Recallingthatwecanviewadepletionsurveyatasinglesiteasapartiallyobservedmultinomialrandomvector(Chapter3),wecanuseapplyanumberofmultinomial-relatedhypothesisteststoascertainhowwellamodeltsagivensetofdepletions(ReadandCressie1988).Resultingdepletionmodelscouldinvolvetherelaxationofmanyassumptionsrelatedtocatchability,forexamplethatcatchabilityisconstantforalldepletionpasseswithinasite.However,theobviousquestionthatarisesrelatestowhetherornotamorecomplexmodel(i.e.amodelwithmoreparametersresultingfromrelaxingcatchabilityassumptions)canactuallybeteffectivelywithagivendataset.Althoughinrealitycatchabilitymaychangeduringthesuccessivedepletionpasseswithinasite,totakeadvantageofthisrealizationwemusthavethetypeofdatatotsuchamodel.Hence,anotheravenueofresearchmayinvolvedetermininghowtocollectdatathatwouldbeusefulinttingmodelswithvariousassumptionsaboutcatchability.Inthecasewhereassumptionsarenotmetandnotaccountingforthesedeviationsfromtheassumptionsisimportant,alternatedepletionmodelsmaybesubstitutedeasilyinourdataanalysismethodsofChapters2and4.Thatis,whereaswepresentedourgeneralmethodologywhichincludedstandardhierarchicaldepletionmodels,i.e.thoseofWyatt(2002)andDorazioetal.(2005),thesamegeneralprinciplescaneasilybeappliedwhileincludingadepletionmodelwhichaccountsforvariousdeviationsfromstandardassumptions.Forexample,adepletionmodelallowingforvariablecatchabilityfromonepasstoanotherwithinasitecouldbesubstitutedforourmodelwhichonly 86

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accommodatedsite-to-sitevariationincatchability.Indeed,intheexamplecasewherecatchabilityishigherintheinitialdepletionpassatasitethansubsequentpasses,thiswillhaveprofoundeffectswhenestimatingtotalabundanceusingdatafromadesignsimilartotheoneusedinChapter4.However,assumingonehasthedatatotamodelallowingforvariablewithin-sitecatchability,applyingthatmodelwithinourestimationmethodwillbestraightforwardandwillyieldresultsbasedonrealisticmodelingofthecatchabilityprocess.Inthisdissertationwehaveattemptedtocomplementtheexistingbodyofresearchindepletionsurveysbyconsideringalternateestimationmethods(asinChapter4)ormethodstooptimizesamplinginsomeway(Chapters2and3).Wehavealsopresentednovelstatisticalperspectivesthatmayprovideinsightsorevensolutionstootherabundanceestimationproblemsnotinvolvingdepletionsampling.Wehaveconsideredwaystomaketheestimationmorepreciseaswellasmorerobust,beginningwiththedesignofdepletionsurveysoftotalabundanceandendingwitharobustabundanceestimator.Improvingtheprecisionandrobustnessoftotalabundanceestimatesmayimprovestudiesinmanyeldsofscience,notjustenvironmentalandecologicalstudies. 87

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APPENDIXAPARTIALDERIVATIVESUSEDINCHAPTER3Thefollowingpartialderivativesarenecessaryinthederivationoftheinformationaboutthecatchabilityparameterq,I(q),inChapter3.@2 @q2(log(q))=@ @q1 q=)]TJ /F7 11.955 Tf 9.3 0 Td[(1 q2@2 @q2(log(1)]TJ /F5 11.955 Tf 11.96 0 Td[(q))=@ @q)]TJ /F7 11.955 Tf 9.3 0 Td[(1 1)]TJ /F5 11.955 Tf 11.96 0 Td[(q=)]TJ /F7 11.955 Tf 9.29 0 Td[(1 (1)]TJ /F5 11.955 Tf 11.95 0 Td[(q)2()]TJ /F7 11.955 Tf 9.3 0 Td[(1)()]TJ /F7 11.955 Tf 9.3 0 Td[(1)=)]TJ /F7 11.955 Tf 9.29 0 Td[(1 (1)]TJ /F5 11.955 Tf 11.95 0 Td[(q)2@2 @q2(log(1)]TJ /F7 11.955 Tf 11.96 0 Td[((1)]TJ /F5 11.955 Tf 11.96 0 Td[(q)p))=@ @q 1 (1)]TJ /F7 11.955 Tf 11.96 0 Td[((1)]TJ /F5 11.955 Tf 11.96 0 Td[(q)p)()]TJ /F5 11.955 Tf 9.3 0 Td[(p)(1)]TJ /F5 11.955 Tf 11.96 0 Td[(q)p)]TJ /F3 7.97 Tf 6.58 0 Td[(1()]TJ /F7 11.955 Tf 9.3 0 Td[(1)!="(1)]TJ /F7 11.955 Tf 11.96 0 Td[((1)]TJ /F5 11.955 Tf 11.96 0 Td[(q)p))]TJ /F5 11.955 Tf 5.48 -9.68 Td[(p(p)]TJ /F7 11.955 Tf 11.95 0 Td[(1)(1)]TJ /F5 11.955 Tf 11.95 0 Td[(q)p)]TJ /F3 7.97 Tf 6.59 0 Td[(2()]TJ /F7 11.955 Tf 9.3 0 Td[(1))]TJ /F5 11.955 Tf 11.96 0 Td[(p(1)]TJ /F5 11.955 Tf 11.95 0 Td[(q)p)]TJ /F3 7.97 Tf 6.59 0 Td[(1()]TJ /F5 11.955 Tf 9.3 0 Td[(p(1)]TJ /F5 11.955 Tf 11.96 0 Td[(q)p)]TJ /F3 7.97 Tf 6.59 0 Td[(1()]TJ /F7 11.955 Tf 9.3 0 Td[(1)#(1)]TJ /F7 11.955 Tf 11.95 0 Td[((1)]TJ /F5 11.955 Tf 11.95 0 Td[(q)p)2=)]TJ /F5 11.955 Tf 9.3 0 Td[(p(p)]TJ /F7 11.955 Tf 11.95 0 Td[(1)(1)]TJ /F5 11.955 Tf 11.95 0 Td[(q)p)]TJ /F3 7.97 Tf 6.59 0 Td[(2(1)]TJ /F7 11.955 Tf 11.95 0 Td[((1)]TJ /F5 11.955 Tf 11.95 0 Td[(q)p))]TJ /F5 11.955 Tf 11.95 0 Td[(p2(1)]TJ /F5 11.955 Tf 11.96 0 Td[(q)2p)]TJ /F3 7.97 Tf 6.59 0 Td[(2 (1)]TJ /F7 11.955 Tf 11.96 0 Td[((1)]TJ /F5 11.955 Tf 11.96 0 Td[(q)p)2 88

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APPENDIXBTABLEOFINFORMATIONVALUESBYCATCHABILITYANDNUMBEROFDEPLETIONPASSES TableB-1. Fisherinformationvaluesforcombinationsofcatchabilityandnumberofdepletionpassespersiteforcatchabilityvalueslessthanorequalto.50.CatchabilityValue #passes.050.100.150.200.250.300.350.400.450.500 20.0130.0290.0480.0690.0950.1260.1630.2080.2640.33330.0350.0740.1170.1650.2190.2800.3480.4260.5160.61940.0640.1310.2020.2760.3540.4360.5220.6120.7070.80850.1000.1990.2980.3940.4880.5800.6670.7500.8290.90560.1420.2760.3990.5120.6130.7020.7770.8390.8900.93270.1900.3580.5040.6260.7240.7990.8520.8860.9050.91380.2430.4450.6080.7310.8170.8710.8970.9010.8900.87190.3000.5360.7090.8260.8940.9200.9160.8910.8550.817100.3620.6280.8060.9100.9520.9490.9160.8660.8120.760110.4270.7210.8980.9820.9950.9620.9020.8320.7640.705120.4960.8130.9831.0421.0240.9610.8780.7930.7170.655130.5680.9051.0611.0901.0400.9500.8480.7530.6720.609140.6420.9941.1321.1281.0450.9310.8150.7130.6300.568150.7191.0811.1941.1551.0420.9070.7800.6750.5920.531160.7971.1651.2491.1741.0320.8800.7460.6380.5570.499170.8771.2451.2971.1851.0170.8510.7120.6050.5260.470180.9571.3221.3371.1890.9970.8200.6790.5740.4980.444191.0391.3941.3701.1870.9750.7900.6480.5450.4720.421201.1221.4631.3971.1800.9500.7600.6190.5190.4490.400211.2051.5271.4181.1690.9240.7310.5920.4950.4270.381221.2881.5861.4331.1540.8980.7040.5670.4730.4080.364231.3711.6411.4431.1370.8720.6770.5430.4520.3900.348241.4541.6921.4491.1180.8450.6520.5210.4340.3740.333251.5371.7381.4501.0970.8190.6280.5010.4160.3590.320261.6191.7801.4481.0750.7940.6050.4820.4010.3450.308271.7001.8181.4421.0520.7690.5840.4650.3860.3330.296281.7801.8521.4341.0290.7460.5640.4480.3720.3210.286291.8601.8811.4231.0060.7230.5450.4330.3590.3100.276301.9381.9071.4100.9820.7010.5280.4180.3470.2990.267312.0151.9291.3960.9590.6810.5110.4050.3360.2900.258322.0901.9481.3790.9360.6610.4950.3920.3260.2810.250332.1651.9641.3620.9140.6420.4800.3810.3160.2720.242342.2371.9761.3440.8920.6240.4660.3690.3060.2640.235352.3081.9861.3240.8700.6070.4530.3590.2980.2570.229 89

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TableB-1.Continued. CatchabilityValue#passes.050.100.150.200.250.300.350.400.450.500 362.3781.9931.3040.8500.5910.4410.3490.2890.2490.222372.4461.9971.2840.8290.5750.4290.3390.2820.2430.216382.5111.9991.2640.8100.5600.4180.3300.2740.2360.211392.5761.9981.2430.7910.5460.4070.3220.2670.2300.205402.6381.9961.2220.7730.5330.3970.3140.2600.2240.200412.6981.9921.2010.7550.5200.3870.3060.2540.2190.195422.7571.9861.1800.7380.5080.3780.2990.2480.2140.190432.8131.9781.1600.7220.4960.3690.2920.2420.2090.186442.8681.9691.1400.7060.4850.3610.2850.2370.2040.182452.9211.9581.1200.6910.4740.3530.2790.2310.2000.178462.9711.9471.1000.6770.4640.3450.2730.2260.1950.174473.0201.9341.0810.6630.4540.3380.2670.2220.1910.170483.0671.9211.0620.6490.4440.3310.2620.2170.1870.167493.1121.9061.0430.6360.4350.3240.2560.2130.1830.163503.1551.8911.0250.6240.4270.3170.2510.2080.1800.160513.1961.8751.0070.6120.4180.3110.2460.2040.1760.157523.2351.8590.9900.6000.4100.3050.2420.2000.1730.154533.2721.8420.9730.5890.4020.2990.2370.1970.1690.151543.3081.8240.9570.5780.3950.2940.2330.1930.1660.148553.3421.8070.9410.5680.3880.2890.2280.1890.1630.145563.3731.7890.9250.5580.3810.2830.2240.1860.1600.143573.4041.7710.9100.5480.3740.2780.2200.1830.1580.140583.4321.7520.8950.5390.3680.2740.2170.1800.1550.138593.4591.7340.8810.5290.3620.2690.2130.1770.1520.136603.4841.7150.8670.5210.3560.2650.2090.1740.1500.133613.5071.6970.8530.5120.3500.2600.2060.1710.1470.131623.5291.6780.8400.5040.3440.2560.2030.1680.1450.129633.5491.6590.8270.4960.3390.2520.1990.1650.1430.127643.5671.6410.8140.4880.3330.2480.1960.1630.1400.125653.5841.6220.8020.4810.3280.2440.1930.1600.1380.123663.6001.6040.7900.4730.3230.2410.1900.1580.1360.121673.6141.5860.7790.4660.3180.2370.1870.1550.1340.119683.6271.5680.7670.4600.3140.2330.1850.1530.1320.118693.6391.5500.7560.4530.3090.2300.1820.1510.1300.116703.6491.5320.7460.4460.3050.2270.1790.1490.1280.114713.6581.5150.7350.4400.3000.2240.1770.1470.1260.113723.6661.4970.7250.4340.2960.2200.1740.1450.1250.111733.6721.4800.7160.4280.2920.2170.1720.1430.1230.110743.6781.4630.7060.4220.2880.2150.1700.1410.1210.108 90

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TableB-1.Continued. CatchabilityValue#passes.050.100.150.200.250.300.350.400.450.500 753.6821.4470.6970.4170.2840.2120.1670.1390.1200.107763.6851.4300.6880.4110.2810.2090.1650.1370.1180.105773.6871.4140.6790.4060.2770.2060.1630.1350.1170.104783.6881.3980.6700.4010.2740.2040.1610.1340.1150.103793.6881.3820.6620.3960.2700.2010.1590.1320.1140.101803.6881.3670.6530.3910.2670.1980.1570.1300.1120.100813.6861.3520.6450.3860.2630.1960.1550.1290.1110.099 TableB-2. Fisherinformationvaluesforcombinationsofcatchabilityandnumberofdepletionpassespersiteforcatchabilityvaluesgreaterthan.50. CatchabilityValue#passes.550.600.650.700.750.800.850.900.950 20.4210.5360.6880.8971.2001.6672.4644.0919.04830.7400.8851.0601.2811.5711.9782.6143.8117.23640.9171.0351.1681.3241.5201.7902.2163.0465.53050.9781.0511.1301.2221.3431.5221.8282.4644.43260.9660.9981.0351.0841.1611.2921.5352.0573.69370.9160.9190.9290.9551.0091.1141.3181.7643.16680.8500.8350.8300.8440.8870.9761.1531.5432.77090.7820.7570.7460.7540.7900.8681.0251.3722.462100.7180.6880.6740.6800.7110.7810.9231.2352.216110.6590.6280.6140.6180.6460.7100.8391.1222.015120.6080.5770.5630.5670.5930.6510.7691.0291.847130.5630.5340.5200.5230.5470.6010.7100.9501.705140.5240.4960.4830.4860.5080.5580.6590.8821.583150.4890.4630.4510.4540.4740.5210.6150.8231.477160.4590.4340.4230.4250.4440.4880.5770.7721.385170.4320.4080.3980.4000.4180.4600.5430.7261.304180.4080.3860.3760.3780.3950.4340.5130.6861.231190.3870.3650.3560.3580.3740.4110.4860.6501.166200.3670.3470.3380.3400.3560.3910.4610.6171.108210.3500.3310.3220.3240.3390.3720.4390.5881.055220.3340.3160.3070.3090.3230.3550.4190.5611.007230.3190.3020.2940.2960.3090.3400.4010.5370.964240.3060.2890.2820.2830.2960.3260.3840.5140.923250.2940.2780.2700.2720.2840.3120.3690.4940.886260.2830.2670.2600.2620.2740.3000.3550.4750.852 91

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TableB-2.Continued. CatchabilityValue#passes.550.600.650.700.750.800.850.900.950 270.2720.2570.2500.2520.2630.2890.3420.4570.821280.2620.2480.2420.2430.2540.2790.3300.4410.791290.2530.2390.2330.2350.2450.2690.3180.4260.764300.2450.2310.2250.2270.2370.2600.3080.4120.739310.2370.2240.2180.2190.2290.2520.2980.3980.715320.2300.2170.2110.2130.2220.2440.2880.3860.693330.2230.2100.2050.2060.2150.2370.2800.3740.672340.2160.2040.1990.2000.2090.2300.2710.3630.652350.2100.1980.1930.1940.2030.2230.2640.3530.633360.2040.1930.1880.1890.1980.2170.2560.3430.616370.1990.1880.1830.1840.1920.2110.2490.3340.599380.1930.1830.1780.1790.1870.2060.2430.3250.583390.1880.1780.1730.1740.1820.2000.2370.3170.568400.1840.1740.1690.1700.1780.1950.2310.3090.554410.1790.1690.1650.1660.1730.1910.2250.3010.541420.1750.1650.1610.1620.1690.1860.2200.2940.528430.1710.1610.1570.1580.1650.1820.2150.2870.515440.1670.1580.1540.1550.1620.1780.2100.2810.504450.1630.1540.1500.1510.1580.1740.2050.2740.492460.1600.1510.1470.1480.1550.1700.2010.2680.482470.1560.1480.1440.1450.1510.1660.1960.2630.472480.1530.1450.1410.1420.1480.1630.1920.2570.462490.1500.1420.1380.1390.1450.1590.1880.2520.452500.1470.1390.1350.1360.1420.1560.1850.2470.443510.1440.1360.1330.1330.1390.1530.1810.2420.435520.1410.1340.1300.1310.1370.1500.1770.2370.426530.1390.1310.1280.1280.1340.1470.1740.2330.418540.1360.1290.1250.1260.1320.1450.1710.2290.410550.1340.1260.1230.1240.1290.1420.1680.2240.403560.1310.1240.1210.1210.1270.1400.1650.2200.396570.1290.1220.1190.1190.1250.1370.1620.2170.389580.1270.1200.1170.1170.1230.1350.1590.2130.382590.1250.1180.1150.1150.1210.1320.1560.2090.376600.1220.1160.1130.1130.1190.1300.1540.2060.369610.1200.1140.1110.1120.1170.1280.1510.2020.363620.1180.1120.1090.1100.1150.1260.1490.1990.357630.1170.1100.1070.1080.1130.1240.1460.1960.352640.1150.1090.1060.1060.1110.1220.1440.1930.346 92

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TableB-2.Continued. CatchabilityValue#passes.550.600.650.700.750.800.850.900.950 650.1130.1070.1040.1050.1090.1200.1420.1900.341660.1110.1050.1020.1030.1080.1180.1400.1870.336670.1100.1040.1010.1020.1060.1170.1380.1840.331680.1080.1020.0990.1000.1050.1150.1360.1820.326690.1060.1010.0980.0990.1030.1130.1340.1790.321700.1050.0990.0970.0970.1020.1120.1320.1760.317710.1030.0980.0950.0960.1000.1100.1300.1740.312720.1020.0960.0940.0940.0990.1090.1280.1710.308730.1010.0950.0930.0930.0970.1070.1260.1690.304740.0990.0940.0910.0920.0960.1060.1250.1670.299750.0980.0930.0900.0910.0950.1040.1230.1650.295760.0970.0910.0890.0900.0940.1030.1210.1620.292770.0950.0900.0880.0880.0920.1010.1200.1600.288780.0940.0890.0870.0870.0910.1000.1180.1580.284790.0930.0880.0860.0860.0900.0990.1170.1560.281800.0920.0870.0850.0850.0890.0980.1150.1540.277810.0910.0860.0830.0840.0880.0960.1140.1520.274 93

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APPENDIXCRANDWINBUGSCODEUSEDTOIMPLEMENTTHEMETHODSOFCHAPTERS2,3AND4 C.1Chapter2Code ##ThisisanRprogramtodeterminewhatsamplesizes(#oftows)yieldwhatlengthcredibleintervals,bydepletionmethod.Inthiscase,therewerefivevesselsusedintheChesapeakeBaybluecrabdepletionsurveys.Wewishtocomparetheefficacyofthesefivevesselsbyconsideringthelengthsofthecredibleintervalsforlocalabundanceconditionalonpotentialoutcomesfromtheirsurveys.##Importposteriorparameterestimatesfortheparametersofinterest.InthiscasethisisoutputfromWinBUGSforthemodelwithvessel-specificcatchabilities.Thefollowingaretheposteriormeansandvariancesforthebetaparameters"a"and"b".ab.mean<-matrix(c(1.316,3.813,1.004,2.046,3.27,5.266,5.744,4.344,3.534,25.17),ncol=1)ab.sd<-matrix(c(.324,1.157,.235,.446,3.046,1.396,1.759,1.163,.788,21.87),ncol=1)ab.var<-round(ab.sd^2,3)##Nowweneedtotaketheposteriormeansandvariancesand 94

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usethemtoparameterizegammadistributionsfortheparametersforeachvessel'scatchabilitybetadistribution.Thisisahierarchicalmodelforeachvessel'scatchability,sowecancapturenotonlytheestimatedmeancatchabilityforeachvessel,butalsotheuncertaintyinthatmean.gamma.shape<-ab.mean^2/ab.vargamma.rate<-ab.mean/ab.vargamma.params<-cbind(gamma.shape,gamma.rate)##Rows1-5ofgammaparamsrepresentthealphaparameterofthebetadistributionforeachofthe5vesselsandthenrows6-10representthebetaparameterofthebetadistributionforeachofthe5vessels.##Draw1,000alphasandbetasforeachvesseltobeusedforthebetadistributionfortheircatchability.alpha.beta<-matrix(c(0),nrow=1000,ncol=10)for(iin1:10){alpha.beta[,i]<-rgamma(nrow(alpha.beta),shape=gamma.shape[i],rate=gamma.rate[i]) 95

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}##Thefirst5columnsofalpha.betarepresentthe1000drawsofthealphaparameterofthebetadistributionforvessels1through5,whilethelast5columnsrepresentthedrawsofthebetaparameterfromthebetadistributionforvessels1through5.Forexample,alpha.beta[3,100]representsthe100thrandomdrawofthe3rdvessel'salphaparameterwherethevessels'catchabilitiesareconsidered~beta(alpha,beta).##Usingthosegeneratedbetaparameters,nowdrawcatchabilitiesq<-matrix(c(0),ncol=5,nrow=nrow(alpha.beta))for(iin1:5){q[,i]<-rbeta(nrow(alpha.beta),shape1=alpha.beta[,(2*i-1)],shape2=alpha.beta[,(2*i)])}##Sonowwehaveabunchofq'stouse,butweneedN'saswell.##Generate1000N'sforeachvesselusingthehierarchicalprior:N~Poisson(theta*area),theta~uniform(0,.5).Forthecrabs,areaisalways550meterssquared.theta<-matrix(c(0),nrow=nrow(alpha.beta),ncol=5) 96

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N<-matrix(c(0),nrow=nrow(alpha.beta),ncol=5)for(iin1:5){theta[,i]<-runif(nrow(alpha.beta),0,.5)N[,i]<-rpois(1000,550*theta[,i])}##NowIhaveN'sandq'ssothatIcangeneratestringsofdataforeachvessel.data<-array(data=NA,dim=c(5,10,nrow(alpha.beta)),dimnames=c("Vessel","#ofTow","Simulation#"))for(iin1:5){s0<-matrix(c(N[,i]),nrow=nrow(alpha.beta),ncol=1)data[i,1,]<-rbinom(n=nrow(alpha.beta),size=s0,prob=q[,i])s1<-s0-data[i,1,]data[i,2,]<-rbinom(n=nrow(alpha.beta),size=s1,prob=q[,i])s2<-s1-data[i,2,]data[i,3,]<-rbinom(n=nrow(alpha.beta),size=s2,prob=q[,i])s3<-s2-data[i,3,] 97

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data[i,4,]<-rbinom(n=nrow(alpha.beta),size=s3,prob=q[,i])s4<-s3-data[i,4,]data[i,5,]<-rbinom(n=nrow(alpha.beta),size=s4,prob=q[,i])s5<-s4-data[i,5,]data[i,6,]<-rbinom(n=nrow(alpha.beta),size=s5,prob=q[,i])s6<-s5-data[i,6,]data[i,7,]<-rbinom(n=nrow(alpha.beta),size=s6,prob=q[,i])s7<-s6-data[i,7,]data[i,8,]<-rbinom(n=nrow(alpha.beta),size=s7,prob=q[,i])s8<-s7-data[i,8,]data[i,9,]<-rbinom(n=nrow(alpha.beta),size=s8,prob=q[,i])s9<-s8-data[i,9,]data[i,10,]<-rbinom(n=nrow(alpha.beta),size=s9,prob=q[,i])}##Codingofposterior:ThisistheposteriordistributionofabundanceN.Weallowvaryingareaa,althoughforthisdatasetonlyanaof550meterssquaredisused.Thisisthesingle-siteapplicationofequation(6.8).Noteweuse 98

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thelogarithmofquantitieswherenecessaryoutofcomputationalnecessity,andappropriatelyexponentiate.##Setpriorsandarea:a<-550lambda<-1kappa<-.5prob.N.temp<-matrix(c(0),nrow=nrow(alpha.beta),ncol=1)HPD.length<-array(data=NA,dim=c(5,10,nrow(alpha.beta)),dimnames=c("Vessel","#ofTow","Simulation#"))for(iin1:5){#i_thvesselfor(jin2:10){#jnumberoftowsfor(kin1:nrow(alpha.beta)){#ksimulationsT<-sum(data[i,1:j,k])support<-as.matrix(c(T:(T+1999)))for(linT:(T+1999)){#addupthefirst2000termsinthepmf,startingwiththeobservedtotalprob.N.temp[(l-T+1)]<-( 99

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lgamma(l+lambda)-lgamma(l-T+1)+l*log(a)-l*log(a+kappa)+lgamma(T+alpha.beta[k,i])+lgamma(j*l-j*data[i,1,k]-(j-1)*data[i,2,k]-max(j-2,0)*data[i,3,k]-max(j-3,0)*data[i,4,k]-max(j-4,0)*data[i,5,k]-max(j-5,0)*data[i,6,k]-max(j-6,0)*data[i,7,k]-max(j-7,0)*data[i,8,k]-max(j-8,0)*data[i,9,k]-max(j-9,0)*data[i,10,k]+alpha.beta[k,(5+i)])-lgamma(T+alpha.beta[k,i]+j*l-j*data[i,1,k]-(j-1)*data[i,2,k]-max(j-2,0)*data[i,3,k]-max(j-3,0)*data[i,4,k]-max(j-4,0)*data[i,5,k]-max(j-5,0)*data[i,6,k]-max(j-6,0)*data[i,7,k]-max(j-7,0)*data[i,8,k]-max(j-8,0)*data[i,9,k]-max(j-9,0)*data[i,10,k]+alpha.beta[k,(5+i)]))}prob.N.temp<-prob.N.temp-prob.N.temp[1]prob.N.temp<-exp(prob.N.temp)prob.N<-prob.N.temp/(sum(prob.N.temp))EN<-sum(prob.N*support)prob.N.sort<-sort(prob.N,decreasing=TRUE)cum.prob.N.sort<-cumsum(prob.N.sort)HPD.length[i,j,k]<-min(which(cum.prob.N.sort>=.95))} 100

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}}##HPD.length.meansHPD.length.real<-HPD.length[,2:10,]HPD.length.real2<-HPD.length.realfor(iin1:5){for(jin1:9){for(kin1:nrow(alpha.beta)){if(HPD.length.real2[i,j,k]>5000){HPD.length.real[i,,k]<-c(NA)}}}}HPD.length.means<-matrix(c(NA),ncol=5,nrow=9)for(iin1:5){for(jin1:9){ 101

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HPD.length.means[j,i]<-mean(HPD.length.real[i,j,],na.rm=TRUE)}}##HPD.length.meansrepresentsthemeanlengthsofthe95%highestposteriordensityintervals.Eachcolumnrepresentsavesselandeachrowrepresentsthenumberoftows,from2to10totaltows.Thisisconditionalontheinformationintheposteriordistributionaboutindividualvesselcatchabilitiesbutnotontheposteriordistributionofabundances.However,abundancedataaregeneratedfromamodelinwhichcrabdensitiesarelimitedto"reasonable"numbersbasedonpastinformation.Theanalysismodel,however,isnotquiteasrestrictive-seethepriors.Thedatagenerationmodelwasmorerestrictivesimplyforcomputationalconvenience.ThiscanhopefullybefixedlatersothatdataoflessrestrictedsizescanbehandledinR.##Theposteriorsofthisstudyassumethatthepriorsforeachvessel'scatchabilityaretheposteriorsfromthefullpreviousdataset.Anotherinterestingthingtodowouldbetogeneratethedatabasedonthevesselbutthenanalyzethedatawithoutusingtheinformativepriorsforthecatchabilityofthevesselofinterest. C.2Chapter3Code #ThefollowingcodeevaluatestheobjectivefunctionofChapter3 102

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#inordertofindtheoptimalnumberofdepletionpasses#persiteversusthetotalnumberofdepletionsitesq<-seq(from=.05,to=.95,by=.05)#gridofcatchabilitiest<-seq(from=2,to=81,by=1)#potentialnumbersofdepletionpasses/siteobj.fun<-matrix(c(0),nrow=length(q),ncol=length(t))obj.fun2<-matrix(c(0),nrow=length(q),ncol=length(t))test.fun1<-matrix(c(0),nrow=length(q),ncol=length(t))test.fun2<-matrix(c(0),nrow=length(q),ncol=length(t))#Loopstartshere-loopthroughcombinationsofqandtfor(iin1:length(q)){for(jin1:length(t)){c<-q[i]p<-t[j]temp.vec<-c()temp.sum<-0 103

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for(kin1:(t[j]-1)){temp.vec[k]<-k*q[i]*(1-q[i])^k}temp.sum<-sum(temp.vec)obj.fun[i,j]<-#ActualfunctionasperChapter3((1-(1-q[i])^t[j])/(t[j]*q[i]^2)+(temp.sum/(t[j]*(1-q[i])^2))-(((1-(1-q[i])^t[j])*t[j]*(t[j]-1)*(1-q[i])^(t[j]-2)+(t[j]^2*(1-q[i])^(2*t[j]-2)))/((1-(1-q[i])^t[j])*t[j])))}} C.3Chapter4CodeBelowrepresentstheRscriptforimplementingtheChapter4methods. #R2WinBUGSRscriptwhichgeneratesthesimulations#usedtoimplementthemethodsofChapter4.#install.packages("R2WinBUGS")#library(R2WinBUGS) 104

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#setwd("C:\\DocumentsandSettings\\Bohrmann\\Desktop\\Tommy'sStuff\\#CrabDepletion\\PhDWork\\March2011Work")nsims<-2000numbers<-matrix(c(0:500),ncol=1)#potentialabundancesatasiteJ<-as.numeric(c(0),nrow=510,ncol=1)J[1:10]<-12J[11:510]<-1I<-length(J)counts.single<-matrix(c(0),nrow=500,ncol=1)Catches<-matrix(c(NA),nrow=length(J),ncol=12)summary.sims<-data.frame(sim.number=c(1:nsims),total.true=0,t.hat.htbayes=0,sd.htbayes=0,sd.normal.approx=0,lower.htbayes95=0,upper.htbayes95=0,in.htbayes=1,R.hat.T=0)coda.list.T<-list()##Universalinitialvaluesfor3chains-a,bandcinit.gamma1.a<-.9init.gamma1.b<-4init.gamma1.c<-2init.gamma2.a<-1.1init.gamma2.b<-2 105

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init.gamma2.c<-3init.alpha.a<-1.2init.alpha.b<-.5init.alpha.c<-3init.beta.a<-3init.beta.b<-1.9init.beta.c<-3.5init.T.a<-500000init.T.b<-1000000init.T.c<-1500000for(iin1:nsims){##Datacreationdensities.true<-runif(n=10000,min=0,max=1)counts.true<-rpois(n=10000,lambda=500*densities.true)#Animalcountsin10000cellssurvey.sites<-sample(seq(1:10000),size=length(J),replace=FALSE)#Whichsitesgetsurveyedalpha.true<-rnorm(n=1,mean=6,sd=2^.5) 106

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#Randomlygeneratetheparameters#forthebetacatchabilitydist.beta.true<-rnorm(n=1,mean=18,sd=2^.5)q.true<-rbeta(n=510,shape1=alpha.true,shape2=beta.true)#Drawi.i.d.catchabilities#forthesitesforthis#simulation.counts.single<-rbinom(n=500,size=counts.true[survey.sites[1:500]],prob=q.true[1:500])#Single-passcatchesfor500sitescounts.multi<-matrix(c(0),nrow=10,ncol=12)#Matrixtoholdmultiple-passcatchesfor(jin1:12){#Multiple-passsitecatchescounts.multi[,j]<-rbinom(n=10,size=counts.true[survey.sites[501:510]]-rowSums(counts.multi),prob=q.true[501:510])}Catches[1:10,]<-counts.multiCatches[11:510,1]<-counts.singletotal.catch<-rowSums(Catches,na.rm=TRUE)sum.ys<-sum(counts.single) 107

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#HTestimatorfunctionssum.counts<-sum(counts.single)sum.counts.2<-sum(counts.single*counts.single)sum.temp<-matrix(c(0),nrow=500,ncol=1)for(kin1:499){sum.temp[k]<-sum(counts.single[k]*counts.single[(k+1):500])}sum.pairs<-sum(sum.temp)data.sim<-list("counts.multi","sum.counts","sum.counts.2","sum.pairs")#ListofdatatoreadintoWinBUGSinits.sim<-list(#Thesearethestartingvaluesreferredtointhetextlist(gamma1=init.gamma1.a,gamma2=init.gamma2.a,alpha=init.alpha.a,beta=init.beta.a,theta=runif(n=10,min=0,max=2),q=runif(n=10,min=.15,max=.4), 108

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N.total=round(runif(n=10,min=total.catch[1:10]+1,max=total.catch[1:10]+100),digits=0),T.unadj=init.T.a),list(gamma1=init.gamma1.b,gamma2=init.gamma2.b,alpha=init.alpha.b,beta=init.beta.b,theta=runif(n=10,min=0,max=2),q=runif(n=10,min=.15,max=.4),N.total=round(runif(n=10,min=total.catch[1:10]+1,max=total.catch[1:10]+100),digits=0),T.unadj=init.T.b),list(gamma1=init.gamma1.c,gamma2=init.gamma2.c,alpha=init.alpha.c,beta=init.beta.c,theta=runif(n=10,min=0,max=2),q=runif(n=10,min=.15,max=.4),N.total=round(runif(n=10,min=total.catch[1:10]+1,max=total.catch[1:10]+100),digits=0),T.unadj=init.T.c) 109

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)temp<-bugs(data=data.sim,inits=inits.sim,parameters.to.save=c("alpha","beta","T.adj","sigma2.T.hat","h","h.2"),model.file="VERSION2ofBayesianHTestimationflatterpriors030211.bug",n.chains=3,n.iter=15000,n.burnin=5000,n.thin=10,n.sims=3000,save.history=TRUE,program="WinBUGS",debug=FALSE)coda.list.T[[i]]<-temp$sims.list$T.hat.adjsummary.sims$total.true[i]<-sum(counts.true)#Keepingtrackofsummary.sims$t.hat.htbayes[i]<-temp$summary[3,1]#simulationresultssummary.sims$sd.htbayes[i]<-temp$summary[3,2]summary.sims$lower.htbayes95[i]<-temp$summary[3,3]summary.sims$upper.htbayes95[i]<-temp$summary[3,7]summary.sims$R.hat.T[i]<-temp$summary[3,8]summary.sims$sd.normal.approx[i]<-(temp$mean$sigma2.T.hat*temp$mean$h.2+(temp$sd$h)^2*(20*sum(counts.single))^2)^.5if(summary.sims$lower.htbayes95[i]>summary.sims$total.true[i]){summary.sims$in.htbayes[i]<-0}if(summary.sims$upper.htbayes95[i]
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}BelowrepresentstheWinBUGScodethattheaboveRcodecallsinordertoimplementthemethodsofChapter4. #WinBUGScodeofthecompletemodelasdescribedin#Chapter4.Thesamplingdesignforthesingle-#passsitesherewasSRSwith10000totalsites\#ofwhich500weresampledmodel;{for(iin1:10){#HierarchicalDepletionmodeltheta[i]~dgamma(gamma1,gamma2)lambda[i]<-500*theta[i]q[i]~dbeta(alpha,beta)N.total[i]~dpois(lambda[i])N[i,1]<-N.total[i]for(jin1:12){counts.multi[i,j]~dbin(q[i],N[i,j])N[i,j+1]<-N[i,j]-counts.multi[i,j]} 111

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}Eq<-alpha/(alpha+beta)#Estimandsofinterestvarq<-alpha*beta/(pow(alpha+beta,2)*(alpha+beta+1))Eq2<-pow(Eq,2)+varqEq.1q<-Eq-Eq2mu.T.hat<-sum.counts*20#Horvitz-Thompsonsumpi.ij<-500*499/(10000*9999)#Jointinclusionprobabilitysigma2.T.hat<-(Eq.1q/Eq*sum.counts*pow(20,2)+varq*(1/(Eq2*pow(.05,2))*sum.counts.2-(1/pow(Eq,2)-1/Eq)*pow(20,2)*sum.counts)+pow(Eq,2)*(.95/(Eq2*pow(.05,2))*sum.counts.2-(1/pow(Eq,2)-1/Eq)*pow(20,2)*.95*sum.counts)+2*(pi.ij-pow(.05,2))/(pow(.05,2)*pi.ij)*sum.pairs)tau.T.hat<-1/sigma2.T.hatmu.T.hat~dnorm(T.unadj,tau.T.hat)#Dist.ofH-Tsumh<-1/Eqh.2<-pow(h,2) 112

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T.adj<-T.unadj*halpha~dgamma(.001,.001)#Priorsbeta~dgamma(.001,.001)gamma1~dgamma(.001,.001)gamma2~dgamma(.001,.001)T.unadj~dnorm(500000,.0000000001)} 113

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APPENDIXDRELATIONSHIPBETWEENABUNDANCEANDCATCHABILITYANDPOSTERIORUNCERTAINTYOFABUNDANCEESTIMATESFigureD-1considerstherelationshipbetweenposteriorstandarddeviationofasiteabundanceestimateandtheactualsiteabundanceandcatchability. FigureD-1. Expectedposteriorstandarddeviationofsiteabundanceestimatesoveragridofabundancesandcatchabilities.Resultingstandarddeviationsarebasedonsix-passdepletionsurveys. 114

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APPENDIXESEVERALEXAMPLESOFPOORLYBEHAVEDMARKOVCHAINS FigureE-1. Severalexamplesofpoorly-behavedMarkovChainsfromtheMCMCsimulationsresultingfromapplyingthefullymodel-basedestimator.MarkovChainsrepresentdrawsfromthesimulatedposteriordistributionofthemeandensityparameter. 115

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REFERENCES Barndorff-Nielsen,O.E.andCox,D.R.(1989),AsymptoticTechniquesforUseinStatistics,NewYork:ChapmanandHall. Brooks,S.P.andGelman,A.(1998),Generalmethodsformonitoringconvergenceofiterativesimulations,JournalofComputationalandGraphicalStatistics,7,434. Casella,G.andBerger,R.L.(2001),StatisticalInference,California,USA:Duxbury. Cordy,C.B.(1993),AnextensionoftheHorvitz-Thompsontheoremtopointsamplingfromacontinuousuniverse,StatisticsandProbabilityLetters,18,353. Cressie,N.,Calder,C.A.,Clark,J.S.,Hoef,J.M.V.,andWikle,C.K.(2009),Accountingforuncertaintyinecologicalanalysis:Thestrengthsandlimitationsofhierarchicalstatisticalmodeling,EcologicalApplications,19,553. Dorazio,R.M.,Jelks,H.L.,andJordan,F.(2005),Improvingremoval-basedestimatesofabundancebysamplingapopulationofspatiallydistinctsubpopulations,Biomet-rics,61,1093. Dorazio,R.M.,Mukherjee,B.,Zhang,L.,Ghosh,M.,Jelks,H.L.,andJordan,F.(2008),ModelingunobservedsourcesofheterogeneityinanimalabundanceusingaDirichletprocessprior,Biometrics,64,635. Gelman,A.,Carlin,J.B.,Stern,H.S.,andRubin,D.B.(2004),BayesianDataAnalysis,BocaRaton,FL:ChapmanandHall. Horvitz,D.G.andThompson,D.J.(1952),Ageneralizationofsamplingwithoutreplacementfromaniteuniverse,JournaloftheAmericanStatisticalAssociation,47,663. Joseph,L.,Wolfson,D.B.,andDuBerger,R.(1995),Samplesizecalculationsforbinomialproportionsviahighestposteriordensityintervals,TheStatistician,44,143. Leslie,P.H.andDavis,D.H.S.(1939),Anattempttodeterminetheabsolutenumberofratsonagivenarea,JournalofAnimalEcology,8,94. Little,R.J.(2004),Tomodelornottomodel?Competingmodesofinferencefornitepopulationsampling,JournaloftheAmericanStatisticalAssociation,99,546. Lunn,D.,Thomas,A.,Best,N.,andSpiegelhalter,D.(2000),WinBUGSaBayesianmodellingframework:concepts,structure,andextensibility,StatisticsandComputing,10,325. Mantyniemi,S.,Arjas,E.,andRomakkaniemi,A.(2005),Bayesianremovalestimationofapopulationsizeunderunequalcatchability,CanadianJournalofFisheriesandAquaticSciences,62,291. 116

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Moran,P.A.P.(1951),Amathematicaltheoryofanimaltrapping,Biometrika,38,307. Pham-Gia,T.andTurkkan,N.(1992),SamplesizedeterminationinBayesiananalysis,TheStatistician,41,389. Pollock,K.H.(1991),Modellingcapture,recapture,andremovalstatisticsforestimationofdemographicparametersforshandwildlifepopulations:past,present,andfuture,JournaloftheAmericanStatisticalAssociation,86,225. RDevelopmentCoreTeam(2009),R:ALanguageandEnvironmentforStatisticalCom-puting,RFoundationforStatisticalComputing,Vienna,Austria,ISBN3-900051-07-0. Rago,P.J.,Weinberg,J.R.,andWeidman,C.(2006),Aspatialmodeltoestimategearefciencyandanimaldensityfromdepletionexperiments,CanadianJournalofFisheriesandAquaticSciences,63,2377. Read,T.R.C.andCressie,N.A.C.(1988),Goodness-of-tstatisticsfordiscretemultivariatedata,NewYork:Springer-Verlag. Rivot,E.,Prevost,E.,Cuzol,A.,Bagliniere,J.L.,andParent,E.(2008),HierarchicalBayesianmodellingwithhabitatandtimecovariatesforestimatingriverineshpopulationsizebysuccessiveremovalmethod,CanadianJournalofFisheriesandAquaticSciences,65,117. Rosen,B.(1972a),Asymptotictheoryforsuccessivesamplingwithvaryingprobabilitieswithoutreplacement,I,AnnalsofMathematicalStatistics,43,373. (1972b),Asymptotictheoryforsuccessivesamplingwithvaryingprobabilitieswithoutreplacement,II,AnnalsofMathematicalStatistics,43,748. Royle,J.A.andDorazio,R.M.(2006),Hierarchicalmodelsofanimalabundanceandoccurrence,JournalofAgriculturalBiologicalandEnvironmentalStatistics,11,249. (2008),Hierarchicalmodelingandinferenceinecology:Theanalysisofdatafrompopulations,metapopulationsandcommunities,USA:Elsevier. Sanathanan,L.(1972),Estimatingthesizeofamultinomialpopulation,TheAnnalsofMathematicalStatistics,43,142. Schaffner,L.C.andDiaz,R.J.(1988),Distributionandabundanceofoverwinteringbluecrabs,Callinectessapidus,inthelowerChesapeakeBay,Estuaries,11,68. Seber,G.A.F.(1982),Theestimationofanimalabundance,London:CharlesGrifnandCompanyLimited. Sen,P.K.(1988),HandbookofStatistics,Vol.6(Sampling),Amsterdam:ElsevierSciencePublishers,chap.Asymptoticsinnitepopulationsampling. 117

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Sharov,A.F.,Volstad,J.H.,Davis,G.R.,Davis,B.K.,Lipcius,R.N.,andMontane,M.M.(2003),Abundanceandexploitationrateofthebluecrab(Callinectessapidus)inChesapeakeBay,BulletinofMarineScience,72,543. Sturtz,S.,Ligges,U.,andGelman,A.(2005),R2WinBUGS:ApackageforrunningWinBUGSfromR,JournalofStatisticalSoftware,12,1. Thompson,S.K.(1992),Sampling,USA:JohnWileyandSons. Thompson,S.K.andSeber,G.A.F.(1994),Detectabilityinconventionalandadaptivesampling,Biometrics,50,712. Volstad,J.H.,Sharov,A.F.,Davis,G.,andDavis,B.(2000),Amethodforestimatingdredgecatchingefciencyforbluecrabs,Callinectessapidus,inChesapeakeBay,FisheriesBulletin,98,410. Wyatt,R.J.(2002),Estimatingriverineshpopulationsizefromsingle-andmultiple-passremovalsamplingusingahierarchicalmodel,CanadianJournalofFisheriesandAquaticSciences,59,695. Zippin,C.(1956),Anevaluationoftheremovalmethodofestimatinganimalpopulations,Biometrics,12,163. 118

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BIOGRAPHICALSKETCH ThomasFrancisBohrmannwasborninNashville,Tennessee,toWilliamandRosanneBohrmann.Thethirdoffourchildren,helivedinNashvilleuntilhisgraduationfromHume-FoggHighSchoolin2000.Subsequently,heobtainedaBachelorofScienceinmathematicsfromLaGrangeCollegeinLaGrange,Georgia,in2004.UpongraduationfromLaGrangeCollege,heimmediatelypursuedaMasterofArtsinteachingwithamajorinmathematicsfromtheMathematicsDepartmentattheUniversityofFloridainGainesville,Florida.Whilenishingthisdegreeinthefallof2006,hebeganayearofteachingMathematicsinGainesville'spublicschoolsystematthemiddleschoollevel.Afterayearofmiddleschoolteaching,intheFallof2007heenrolledinaninterdisciplinaryPh.D.programbetweentheSchoolofNaturalResourcesandEnvironmentandtheDepartmentofStatisticsattheUniversityofFlorida.In2009,heobtainedaMasterofStatisticsdegree.Hisgraduatetraininginvolvedstudiesintheeldsofmathematics,ecologyandstatistics,aswellastheforgingofThomasasalifelongfanoftheUniversityofFloridaGators.ThomasgraduatedwithhisPh.D.in2011andmovedtoAthens,Georgia,tobeginapostdoctoralpositionasaresearchstatisticianwiththeUnitedStatesEnvironmentalProtectionAgency. 119