A Bayesian CapturendashRecapture Population ModelWith Simultaneous Estimation of Heterogeneity

Ross CORKREY Steve BROOKS David LUSSEAU Kim PARSONS John W DURBANPhilip S HAMMOND and Paul M THOMPSON

We develop a Bayesian capturendashrecapture model that provides estimates of abundance as well as time-varying and heterogeneous survivaland capture probability distributions The model uses a state-space approach by incorporating an underlying population model and anobservation model and here is applied to photo-identification data to estimate trends in the abundance and survival of a population ofbottlenose dolphins (Tursiops truncatus) in northeast Scotland Novel features of the model include simultaneous estimation of time-varyingsurvival and capture probability distributions estimation of heterogeneity effects for survival and capture use of separate data to inflate thenumber of identified animals to the total abundance and integration of separate observations of the same animals from right and left sidephotographs A Bayesian approach using Markov chain Monte Carlo methods allows for uncertainty in measurement and parameters andsimulations confirm the modelrsquos validity

KEY WORDS Abundance Logits Photo-identification Survival Trends

1 INTRODUCTION

Estimates of changes in abundance are a fundamental re-quirement of most wildlife conservation and managementprograms (Williams Nichols and Conroy 2001) Capturendashrecapture methodology provides an important tool for esti-mating key demographic parameters such as survival but itcan also be used to obtain abundance In traditional capturendashrecapture (Seber 2002) animals are initially captured taggedand then released On subsequent capture occasions individualsare recognized and new animals are tagged allowing construc-tion of a capture history for each Such parameters as survivaland abundance are then derived from the capture histories Inthe case of species with individually distinct natural markingscapture can correspond to photographing distinctive features(Wuumlrsig and Jefferson 1990) These photo-identification tech-niques are especially useful in situations where data must becollected nonintrusively to monitor the status of populations

A difficulty with capturendashrecapture is that individuals mayhave differing catchabilities manifesting as heterogeneousand time-varying capture probabilities (Chao 1987 Hammond1986) The chance of observing an animal at a particular pointin time and space also depends on the effort expended But ef-fort may be unimportant as long as each animal has an equalchance of capture at some point in its range during a sam-pling period One effect of capture heterogeneity is to producenegatively biased estimates of abundance (Pollock NicholsBrownie and Hines 1990 Cormack 1972) because an ani-mal with a higher capture probability will be more likely tobe caught on the first occasion resulting in a decreased averagecapture probability on the second occasion and an underesti-mate of the total marked population If the individual captureprobabilities are uncorrelated between sampling periods then

Ross Corkrey is Senior Research Fellow Tasmanian Institute of Agri-cultural Research University of Tasmania Tasmania Australia (E-mailscorkreyutaseduau) Steve Brooks is Professor Statistical Laboratory Cen-tre for Mathematical Sciences Cambridge UK David Lusseau is PostdoctoralFellow Department of Biology Dalhousie University Halifax Nova ScotiaCanada Kim Parsons and John W Durban are Research Biologists NationalOceanic and Atmospheric Administration Seattle WA Philip S Hammond isProfessor Sea Mammal Research Unit Gatty Marine Research Institute Uni-versity of St Andrews St Andrews Fife UK Paul M Thompson is Professorand Chair in Zoology School of Biological Sciences University of AberdeenCromarty Ross-shire UK This modeling work was supported by a grant fromthe Leverhulme Trust Field data were collected with support from WDCSDETR ASAB BES Talisman Energy (UK) Ltd and ChevronTexaco

no bias is expected (Pollock et al 1990) but it seems unlikelythat the individual capture probabilities remain constant overtime Heterogeneity may also result in biased estimates of sur-vival probability (Pollock et al 1990)

Heterogeneous capture probability is usually considered anuisance variable that must be estimated to obtain unbiased es-timates of other parameters However the underlying causesof innate heterogeneous capture probabilities are not well un-derstood and may be of interest because they may provide in-sight into individual differences in range use feeding strategiesor other behaviors They may arise from variable responses tocapture (Hammond 1990) social structuring within the popu-lation (Wilson 1995) nonoverlapping individual ranges (Wil-son 1995) or temporary migration from the study area (White-head 2001a) Sampling methods themselves may introduceheterogeneity and as such the estimation of unequal cap-ture probabilities may facilitate the development of improvedsampling techniques Estimates of survival probability maybe biased by such unequal capture probabilities (Buckland1982 1990) although the effect may be small (Carothers 1979)Methods for dealing with heterogeneous capture include fittingspecialized closed population models (Otis Burnham Whiteand Anderson 1978) jackknife (Burnham and Overton 1979)and coverage estimators (Chao Lee and Jeng 1992) modifiedtrapping design (Pollock et al 1990) stratification of estimatesby covariates such as age or sex (Seber 2002) the use of co-variates (Huggins 1989) selective analysis of capture histories(Hammond 1990) and the use of beta-binomial models (Do-razio and Royle 2003) logit models (Coull and Agresti 1999Pledger 2000) and Rasch models (Fienberg Johnson andJunker 1999) Heterogeneity in survival also may occur The bi-ological factors underlying heterogeneity in survival probabilityare of interest because they represent the response of differentindividuals to their environment

Where natural marks are used to identify individuals thoseanimals lacking marks are by definition not identifiable (Pol-lock et al 1990) for example distinctive notches on dorsal finsof cetaceans may be present only on older animals in the popu-lation This means that only part of the population is catchableBut this problem can be dealt with by inflation using other data

copy 2008 American Statistical AssociationJournal of the American Statistical Association

September 2008 Vol 103 No 483 Applications and Case StudiesDOI 101198016214507000001256

948

Corkrey et al Bayesian Recapture Population Model 949

to estimate the proportion of the sampled population having dis-tinctive marks (Seber 2002) or by double sampling (Pollock etal 1990)

To address the aforementioned issues we construct a modelincorporating heterogeneity in capture and survival that makesuse of the available data consisting of left-side and right-sidephotographic series Individuals are regarded as captured af-ter they have been identified by a specific type of dorsal finmarking called ldquonicksrdquo that are visible in these photographsBecause these nicks appear on animals only as they maturewe include a nicking-recruitment process in the model How-ever we lack knowledge of the probability of having nicks andthe rate of acquiring nicks Rather than adopt vague priors forthese probabilities we include separate data sources that pro-vide these quantities We also make use of the survival andcapture information from the recapture model to obtain an esti-mate of the population size Because some priors remain vaguewe assess the effect of prior specification using simulated dataWe apply the method to photo-identification data from a nat-urally marked bottlenose dolphin (Tursiops truncatus) popula-tion from the Moray Firth Scotland This is the only residentcoastal population of bottlenose dolphins in the North Sea anda Special Area of Conservation has been established to pro-tect these animals in response to the 1992 EC Habitats Direc-tive (Council Directive 9243EEC Scottish Natural Heritage1995) The UK government needs information on trends inthe abundance of this population to report on the success of thisconservation program to the European Union

In this article we develop a general framework for estimat-ing the total dolphin population size by embedding a standardcapturendashrecapture model within a state-space model that de-scribes the evolution of population size over time We begin inSection 2 by describing the data available In Section 3 we dis-cuss implementation of the model In Section 4 we discuss theBayesian approach including the use of Markov chain MonteCarlo (MCMC) techniques the specification of priors and asimulation study In Section 5 we provide the results of ouranalysis and discuss the implications for the Moray Firth bot-tlenose dolphin (Tursiops truncatus) population We concludewith discussion of the wider implications of our research andthe utility of our approach for other wild animal populations

2 DATA COLLECTION AND FORMAT

Between 1990 and 2002 boat-based surveys were con-ducted both in the inner Moray Firth and along adjacent coastsFull details of survey protocols have been provided by (Wil-son Thompson and Hammond 1997 Wilson Hammond andThompson 1999) The number of trips made varied amongyears and areas Most trips were made between May and Sep-tember with each trip encountering one or more groups of dol-phins Because the data collection effort in winter was low andavailable only for the early years of the study these data areexcluded from the analysis Between 1990 and 1999 trips fol-lowed a fixed route (Wilson et al 1997) but from 2000 onwardopportunistic surveys were conducted in an attempt to increasethe number of dolphin groups sighted

When bottlenose dolphins were encountered photographswere taken of their dorsal fins using an single-lens reflux cam-era with a telephoto lens (Wilson et al 1999) Both 35-mm

transparency and digital cameras were used during the studywith all photos subjected to the same strict grading of photo-graphic quality (described in Wilson et al 1999) Only high-quality pictures were used to minimize errors in identification(Stevick Palsboslashll Smith Bravington and Hammond 2001Forcada and Aguilar 2000)

Individual bottlenose dolphins are identifiable by markingsalong the whole of the dorsal fin (see Fig 1) Such fin markshave been shown to be of varying persistence and reliability(Wilson et al 1999) therefore we restricted our data set onthose dolphins with nicks to the posterior edge of the dorsalfin which have been shown to be relatively permanent andto accumulate over time The accrual of individual nicks overtime changes the dorsal edge profile but reliable identificationcan still be obtained if the rest of the dorsal edge remains un-affected Estimated nicked proportions are determined from aseparate data set comprising the number of animals encounteredin each trip and the number of those that were nicked

On each occasion animals could be photographed from theleft side the right side or both sides Individuals photographedfrom one side can be identified at later times from photographsof either side because the dorsal trailing edge is visible fromeither side This means that differences in left and right captureprobability can be related to animal behavior and boat differ-ences and left and right capture probabilities are of potentialuse in sampling programs A total of 13 individuals are onlyknown from the left side and another 13 are only known fromthe right side These individuals are included in the calculationsince there is no evidence that they are not members of the pop-ulation

Each animal is assigned a unique identification code and arecord made of the date and location it was observed The ob-servations of a single animal are represented as a capture historyconsisting of a series of 0rsquos and 1rsquos where 1 indicates that theanimal was observed in the corresponding time period and 0 in-dicates the animal was not observed The entire data set consistsof a matrix in which each row corresponds to a capture historyfor a single animal and each column represents one calendaryear Separate history matrices are constructed as described ear-lier for photoidentification series of the left and right sides Inaddition a bilateral history is constructed from both the left andright side series in which 0 1 2 and 3 indicate that the animalwas unobserved or observed from the left side right side orboth sides on each occasion We denote the capture status foranimal d at time j as hdj hdj isin [0123]

3 MODEL CONSTRUCTION

The overall model comprises a series of submodels that re-late different elements of the observed (and indeed unobserved)data to common population parameters We begin with a de-scription of the component of the model that divides the pop-ulation into two distinct classes those animals that are nickedand thus identifiable and those that are not nicked Using a stan-dard capturendashrecapture model the size of the nicked populationcan be determined and from this the total population size canbe determined by estimating the proportion of nicked animals

950 Journal of the American Statistical Association September 2008

Figure 1 Example of dorsal fin showing nicks to the trailing edge Insets show the changes in the dorsal fin for this animal over time withthe earliest photograph on the left and the latest on the right The background image corresponds to the third inset from the right Apparentchanges in shape of the inset images are the result of differences in camera perspective The effect of these nicks is to produce a serrated edgethe whole of which is used to confirm the identity of the animal rather than just a selected few nicks Other markings such as skin lesions arenot considered reliable enough to be used in this study Minor changes in some nicks may occur over time but as long as other nicks on thedorsal edge remain intact identification can proceed

31 Modeling the Nicked Population

We begin with the assumption that young animals are ini-tially unnicked but that as they mature they will accumu-late nicks as they increasingly engage in social interactionsThis is a reasonable assumption in light of the social develop-ment of juvenile bottlenose dolphins (Mann and Smuts 1999Haase 2000) The animals can be considered as being recruitedfrom the unnicked population to the nicked population Once inthe nicked population individuals become observable and can-not return to the unmarked population

We denote the number of dolphins recruited from the un-nicked population to the nicked population at time 1 le j le J

as Bj and the number of nicked animals that survived betweentimes j minus 1 and j as Sj Thus the nicked population size attime j is given by Sj + Bj that is the current population com-prises new recruits and surviving former members The nickedpopulation is denoted by Nj = Sj + Bj

We assume that previously nicked animals survive from timej minus1 to time j with probability θj = exp(α+βj )(1+exp(α+βj )) where α denotes a global survival tendency and βj de-notes a time-varying survival component Thus the number ofsurviving animals Sj is given a binomial distribution so thatSj sim Bin(Sjminus1 + Bjminus1 θjminus1)

We also assume that unnicked animals acquire nicks withprobability ωj so that the number of newly nicked animals has

a binomial distribution Bj sim Bin(Wjminus1 minus Njminus1 ωj ) where attime j Wj denotes the total population and Nj denotes thenicked population Given this definition ωj is the probabilityof acquiring nicks and also surviving We believe that the num-ber of newly nicked animals is relatively small however

We refer to this component of the model as the popula-tion model It provides the probability distributions P(Sj |Sjminus1

Bjminus1 θjminus1) and P(Bj |Wjminus1Njminus1ωj )

32 Modeling the Total Population

In this case study the total population size and its variationover time is the primary statistic of interest but the total pop-ulation comprises both nicked and unnicked animals We canhowever estimate the size of the total population from the sizeof the nicked population using data on the proportion of ani-mals in each group that have nicked dorsal fins This is possiblebecause although unnicked individuals cannot be reliably iden-tified over periods of months or years they are recognizableover a period of days (see Wilson et al 1999) This allows usto count the number of unnicked individuals captured on high-quality photographs in any one encounter and to estimate theproportion of nicked and unnicked animals in the group to pro-vide an inflation factor

Then the unnicked part of the total population is given byWj minus Nj If the probability of an individual having nicks is

Corkrey et al Bayesian Recapture Population Model 951

given by υj then the size of the unnicked population at timej can be modeled as a negative binomial distribution condi-tional on the size of the nicked population so that Wj minus Nj simNegBin(Nj υj ) This approach allows the use of an indepen-dent data set to inform the υj rather than relying on the re-capture data to estimate this parameter The negative binomialdistribution is preferred over a binomial distribution becausethe Nj rsquos already have been assigned a distribution in the popu-lation model

In each time period j suppose that there are Tj trips eachobserving twji animals for i = 1 Tj of which tnji are nickedNumerous trips are undertaken each year and the numbernicked is separately estimated on each encounter A binomialdistribution then can be used to describe the number of nickedanimals seen per trip so that tnji sim Bin(twji υj ) Thus the ob-servations tnji and twji allow us to estimate υj and thus the totalpopulation size given an estimate of the nicked population Werefer to this as the inflation model It provides the probabilitydistributions P(Wj |Sj Bj υj ) and P(tnji |twji υj )

33 The Observation Model

In any time period only a portion of the nicked populationis seen Examination of the data suggests that the left and rightcapture probabilities differ at the population level The prob-ability that an individual animal would be seen from the leftor right side s isin [LR] is given by ρjs = exp(δs + εjs)(1 +exp(δs + εjs)) where δs denotes a global capture tendency andεjs denotes a time-varying capture component

Then the number of nicked animals seen from side s njs fol-lows a binomial distribution njs sim Bin(Nj ρjs) where njL =sum

d hdj isin [13] and njR = sumd hdj isin [23] We refer to this

model as the observation model It provides the probability dis-tributions P(njL|Nj ρjL) and P(njR|Nj ρjR)

We consider the njL and njR as independent counts A dol-phin seen in one year from one side contributes to the prob-ability of observing that side and one seen from both sidescontributes to both probabilities The introduction of captureprobabilities for each side allows the combination of the twophotographic series that otherwise would have been analyzedseparately because they would be correlated It is also of inter-est to estimate these probabilities to assess the capture methodsin use We include the term εjs to allow for observed differ-ences between years in observing all dolphins from one sideversus the other These differences may have arisen from dif-ferences in boat work

34 The Recapture Model

The population and observation models require estimates ofthe parameters α βj δL εjL δR and εjR corresponding tothe survival and capture probabilities These probabilities canbe obtained from the capture history data To do this we cal-culate the likelihood of the entire capture history matrix whichis the product of the probabilities associated with each animalrsquosindividual history L(hdj )

To combine the photo-identification series from the left andright sides into a single bilateral analysis the model allows dif-ferent capture probabilities when animals are seen from eachside The number of animals seen also can differ between the

left and right sides Other parameters do not depend on the sidefrom which an animal is observed for example the populationsize and survival probability of an animal should not depend onwhich side is observed However the left-side capture proba-bility for individual animals might be expected to differ fromthat of the right side for example the animal might be morelikely to approach the boat from one side than from the otherDolphins are also well known to have heterogeneous behavior(Wilson Reid Grellier Thompson and Hammond 2004)

Thus to incorporate heterogeneity as well as observationfrom two sides survival and capture probabilities are definedas logit(φdj ) = α + βj + γd and logit(πdjs) = δs + εjs + ζds where γd denotes a survival effect ζds denotes a recapture ef-fect for animal d s isin [LR] for the left or right side andlogit(x) = log(x(1 minus x)) The logit function is the standardchoice for binary data

Note that the heterogeneity effects are not used in the popu-lation and observation models because both of these are basedon both the nicked and unnicked populations and members ofthe latter will not have an individual effect parameter Althoughwe could adopt a random-effects model to ascribe individualeffects to unnicked animals this would imply that we believethe heterogeneity of survival and recapture to be the same inboth the nicked and unnicked populations which may not bethe case

The overall probability has components for animals in twosituations There are nJ animals observed at the final time andsome nJ lowast animals last observed at some earlier time and thatmight have died or simply are not seen Each individual historyprobability is conditional on the first time that the animal isseen j1

d Thus the likelihood associated with the histories of n = nJ +

nJ lowast dolphins is given by

prod

sisin[LR]

nprod

d=1

Jprod

j=1

L(hdjs)

=nJprod

d=1

Jprod

j=j1d +1

φdjminus1I (πdjLπdjRhdjs)

timesnJ +nJlowastprod

d=nJ +1

hlprod

j=j1d +1

φdjminus1I (πdjLπdjRhdjs)

(

1 minus φdhl

+Jsum

j=hl+1

φdj

jprod

k=hl+1

φdkminus1(1 minus πdkL)(1 minus πdkR)

)

(1)

in which φdj denotes the survival probability φdj = 1 minus φdj

when j lt J and φdj = 1 when j = J πdjs denotes the recap-ture probability from animal d observed in time j from sides isin [LR] j1

d and hl denote the first and last times that individ-ual d is seen and I (πdjLπdjRhdjs) = (1 minus πdjL)(1 minus πdjR)

when hdjs = 0 I (πdjLπdjRhdjs) = πdjL(1 minus πdjR) whenhdjs = 1 and I (πdjLπdjRhdjs) = (1 minus πdjL)πdjR whenhdjs = 2 I (πdjLπdjRhdjs) = πdjLπdjR when hdjs = 3 Theφdj and πdjs parameters are defined only for 1 le j lt J minus1 and2 le j lt J The values of nJ j1

d and hl are determined fromthe capture histories

952 Journal of the American Statistical Association September 2008

We refer to this model as the recapture model It yieldsthe probability P(hdjs |φdj γd πdjLπdjR ζdL ζdR) This isa CormackndashJollyndashSeber model which has been discussed else-where (Seber 2002 Lebreton Burnham Clobert and Anderson1992 Link and Barker 2005)

This completes the model specification To summarize wedefine a population model that describes the recruitment (ωj )and survival (θj ) of animals in the nicked population (Nj ) Therelative sizes of the nicked population (Nj ) and total population(Wj ) are described by the inflation model which makes use ofobserved counts of nicked and unnicked animals Using the cap-ture probability (ρjs ) and the observed number of nicked ani-mals we estimate the nicked population Finally we estimatethe heterogeneous capture (πdjs ) and survival (φdj ) probabili-ties from the capture histories Because this is a Bayesian analy-sis we also specify priors to the following parameters S0 B0ωj W0 N0 α βj γd δs εjs and ζds We describe the priorsin detail in the next section

4 ANALYSIS

In this section we describe the statistical methodology re-quired to analyze the model described in the previous sectionWe adopt a Bayesian approach and describe the prior specifica-tion and simulation techniques required here In this modelingprocess we use a Bayesian approach to allow for uncertaintyin measurement and parameters used The Bayesian approachallows for simultaneous consideration of all data and modelsallowing for the proper propagation of uncertainty throughoutthe model Identifiability is also less of a concern in a Bayesiananalysis in which prior information is supplied (Gelfand andSahu 1999) Moreover the Bayesian approach has an advan-tage in that MCMC methods can be used which greatly simpli-fies the computation compared with the corresponding classi-cal tools The remainder of this section discusses the process ofprior specification and provides details of the implementation

41 Priors

The first stage in the Bayesian analysis is to elicit priors foreach of the model parameters We do so here by grouping theparameters according to the component of the model in whichthey appear The priors used and their associated parameters aresummarized in what follows

The Population Model The model specifies the distribu-tion of parameters Sj and Bj in terms of Sjminus1 and Bjminus1 forj ge 1 Therefore we need a prior for S0 and B0 Becausethey only appear together we set N0 = S0 + B0 and assigna Poisson prior N0 sim Po(λN) Its mean takes a gamma priorλN sim Gam(76 01) with parameters selected to obtain a meanfor the nicked population of 76 as estimated by Wilson et al(1999) and a large variance so that it becomes uninformativecompared with the range of reasonable population values

Because the recruitment rates of nicked dolphins cannot bedirectly estimated from the photo-identification data set thepriors are parameterized with results from a separate analy-sis of randomly selected photographs with at least 2 years ofdata Because it is thought that the rate of nicking might begreater for older animals due to increased social interactionsestimates are estimated from photographs of individuals that

are first identified as known-age calves Identification is doneusing all attributes such as fin shape and the presence of le-sions Photographs of seven of these animals are available forestimating the age at which these dolphins are first seen to havea nick in the dorsal fin We consider these cases a random se-lection from the unnicked population so that the mean num-ber of nicks per year gives an approximate probability of nick-ing per year We set the prior for ωj to be reasonably vagueωj sim Beta(02381) Although rough these assumptions givea crude estimate for the parameter which is then updated bythe data

The Inflation Model The initial population size is assigneda negative-binomial prior W0 minusN0 sim NegBin(N0 υ0) The ini-tial inflation factor υ0 is assigned a beta prior υ0 sim Beta(11)which is equivalent to a noninformative uniform prior

The Recapture Model The recapture model survival pa-rameters have independent normal priors N(0 185) and thecapture parameters have independent normal priors N(0 15)These priors span the range (01) on the logit scale Largervariances are not used because these would obtain a U-shapeddistribution on the logit scale

42 Implementation

Once priors are assigned the full joint distribution can bedetermined as the product of the likelihood with the associatedprior distributions The resulting posterior is complex and high-dimensional and so inference is obtained in the form of poste-rior means and variances obtained through MCMC simulationWe choose to use an implementation in which each parameter isupdated in turn using either a MetropolisndashHastings or a Gibbsupdate depending on the form of the associated posterior con-ditional distributions The updating strategy is detailed in theAppendix All simulation software is written in FORTRAN 77The model was run for 500000 iterations and a 50 burn-in was used Sensitivity studies and standard diagnostic tech-niques were used (Brooks and Roberts 1998) to assess modelvalidity

43 Simulation Study

To test the MCMC code and to demonstrate the utility of theframework developed earlier we construct two artificial pop-ulations one with homogeneous capture and survival proba-bilities and the other with heterogeneous capture and survivalprobabilities These populations are constructed using the fol-lowing procedure An initial population of animals (n = 130)is sampled and a proportion of this initial population is ran-domly marked as nicked At each time period in the simulationanimals are randomly selected to die some of the unnicked be-come nicked and a proportion of the remaining animals arerandomly selected to be observed For each population threesimulations are run with differing priors for the ωj and λN para-meters The priors are such that the three cases represent vaguepriors precise priors and very precise priors but in each casethe means differ from the actual population values One possi-ble outcome is that the precise prior simulations track the priormeans more closely than the vague simulations

The simulations are illustrated in Table 1 which shows theactual total abundances the medians of the posteriors for Wj

Corkrey et al Bayesian Recapture Population Model 953

Table 1 Actual and estimated median total abundances for two artificial populations with 95 HPDI and posterior p values fromthree simulations on each with differing priors

A B C

Population Year Wj Wj p value Wj p value Wj p value

1 1990 124 119(99139) 28 112(97130) 07 107(90124) 021991 120 125(105146) 67 118(95134) 39 114(97135) 251992 114 116(96135) 54 110(93127) 30 106(92123) 171993 109 113(95134) 63 108(92124) 41 105(90121) 261994 103 111(93130) 81 106(88124) 63 103(86118) 491995 97 105(89125) 86 102(87118) 74 100(85114) 621996 95 98(84115) 66 95(81110) 48 93(81107) 381997 91 99(83114) 85 96(80109) 76 95(80108) 651998 89 94(79110) 73 91(77105) 56 90(77103) 521999 85 92(77108) 78 89(75103) 69 87(75103) 632000 76 82(6598) 76 79(6696) 65 79(6594) 592001 70 79(6299) 84 77(5895) 79 76(6095) 752002 68 76(5593) 78 75(5691) 75 73(5892) 70

2 1990 124 135(100178) 71 121(96159) 42 116(90149) 261991 122 123(90158) 51 112(87140) 21 108(84134) 121992 119 114(87148) 35 105(84130) 10 101(78124) 061993 111 106(82138) 34 99(75121) 11 95(74117) 061994 106 102(76130) 33 93(75118) 12 91(68115) 071995 100 96(70122) 38 91(74114) 16 88(67108) 081996 93 93(66119) 48 87(69109) 26 85(65105) 161997 87 91(67115) 60 85(68106) 37 82(62102) 281998 77 79(59101) 56 76(6295) 39 72(5791) 281999 74 84(60108) 80 80(62101) 75 77(5897) 622000 72 78(5799) 67 74(5894) 58 72(5591) 462001 67 71(4992) 62 69(4891) 57 66(4688) 432002 64 68(4191) 59 65(4492) 51 63(4388) 44

NOTE Population 1 has homogeneous capture probability (75) and survival probability (95) while population 2 has heterogeneous capture probability (mean 55 range 05ndash98) andsurvival probability (mean 92 range 68ndash99) The simulations have different priors for ωj and λN Simulation run A has vague priors [ω sim Beta(029412368) λN sim Gam(76 01)]B has more precise priors [ω sim Beta(023710) λN sim Gam(761)] and C has very precise priors [ω sim Beta(929939155) λN sim Gam(7600100)] Their means are the same ineach case (023 and 76) but each differs from the actual population values (06 and 106)

their 95 highest posterior density intervals (HPDI) and a p

value The p values are calculated by counting the proportionof iterations in which the Wj exceeded the true value Examina-tion of the results indicate that the median posterior estimatesgenerally agree with the actual values The agreement is sat-isfactory whether or not there is heterogeneity assumed Thep values are in the range 02ndash88 suggesting that the modelworks correctly The posterior p values for the recapture modelcoefficients are given in Table 2 The p values for α δL andδR are summarized and those for the remaining parameters aresummarized as the proportion of p values in each that lie insidethe range 01ndash99 Almost all p values are within the range 01ndash99 the only exceptions are the capture coefficients for a smallnumber of dolphins

In general the total population estimates for the homoge-neous case are slightly greater and the 95 HPDI is narrowerthen the heterogeneous case but all intervals contain the popu-lation values The estimates also shrink slightly for the mod-els with more precise priors However the p values in eachcase suggest an adequate fit even though some priors are de-liberately set to incorrect values In the case of the recapturecoefficients there is no apparent difference between the homo-geneous and heterogeneous models or with any trend in theprecision of the priors

5 RESULTS

We obtain estimates of heterogeneous capture and survivalcoefficients variation of capture and survival over time over-all survival and capture probabilities the abundance and thetrend in abundance The survival probabilities and capture co-efficients from the left and right sides are shown in Figure 2 andTable 3 We can obtain an overall capture probability for the twosides by calculating exp(δ)(1+exp(δ)) The mean overall cap-

Table 2 Summary of posterior p values for survival and capturecoefficients for two artificial populations from three simulations

on each with differing priors

Population α βj γd δL εjL ζdL δR εjR ζdR

1 16 100 100 66 100 98 60 100 10010 100 100 59 100 98 53 100 10008 100 100 59 100 98 48 100 100

2 34 100 100 17 100 97 50 100 9822 100 100 15 100 97 37 100 9923 100 100 12 100 97 39 100 99

NOTE The populations and simulations are described in Table 1 The p values are calcu-lated as the proportion of iterations greater than the actual population value The p valuesfor α δL and δR are shown Because the remaining parameters are vectors of coefficientsthose p values are summarized as the proportion of p values in each that are found insidethe range 01ndash99

954 Journal of the American Statistical Association September 2008

(a)

(b) (c)

Figure 2 (a) Variation of mean capture coefficients over time forthe left side εL (minusminus) and right side εR (- - bull - - bull) The two hor-izontal lines indicate the global mean capture coefficients δL (mdashmdash)and δR (minus minus minus) On the left vertical axis are the mean left-side het-erogeneity capture coefficients ζL and on the right vertical axis arethe mean right-side heterogeneity capture coefficients ζR Standardserrors for each coefficient are shown as vertical bars (b) Variation ofmean survival probability over time (c) Histogram of mean heteroge-neous survival probabilities

ture probability for the left side is 48 (95 HPDI = 29 68)and for the right side it is 58 (95 HPDI = 39 77) Althoughthis is a small difference the probability of the right capturecoefficient exceeding the left is 81 Examination of Figure 2(a)shows that the variation of capture coefficients over time is largecompared with the overall coefficients and that the left (εL) andright (εR) coefficients tend to track each other The probabilityof the right capture coefficient exceeding the left varies overa wide range 07ndash96 This range results from some years inwhich the probability tends strongly to the left side (1995 and2000 probability of observing the left side 95) or right side(1998 1999 2001 probability of observing the right side 91)Finally the range of the heterogeneous coefficients is compara-ble to that of the time-varying coefficients

The mean heterogeneous survival coefficient is not corre-lated with either of the mean capture heterogeneous coeffi-cients (right r = 009 left r = 022) However the meanleft and right heterogeneous capture coefficients are correlated(r = 714) indicating that animals that are more likely to beseen from one side also are more likely to be seen from theother side However the heterogeneity capture coefficient of theright side (mean 56) exceeded that of the left side (mean 23)with 85 of animals having higher right coefficients than left

Table 3 Left-side and right-side capture probabilities and survivalprobabilities by year

Year Left capture Right capture Survival

1990 1(11)

1991 54(40 68) 60(47 75) 93(84 99)

1992 74(59 86) 75(60 86) 98(94100)

1993 31(20 44) 46(33 61) 95(88100)

1994 36(24 47) 53(35 66) 95(86100)

1995 42(27 55) 71(56 88) 94(86100)

1996 30(17 42) 48(34 63) 93(82100)

1997 23(14 36) 37(23 51) 88(75 98)

1998 39(24 53) 31(20 45) 94(84100)

1999 45(31 63) 40(25 54) 93(82100)

2000 24(14 36) 57(40 72) 96(88100)

2001 87(78 94) 82(73 91) 96(89100)

2002 88(80 96) 89(80 96)

NOTE For each the mean and the 95 HPDI limits are given There is concordancebetween the left-side and right-side results The capture probabilities are more variableand reach minimums in 1997 and 2000 for the left side and in 1998 for the right sidebut both reach maximums in 2002 Survival reaches a maximum in 1992 declines to aminimum in 1997 and then rises again in later years

coefficients Because this is observed after adjustments for theoverall and time-varying components it suggests an effect dueto behavior or sampling protocol

The variation in the capture probability over time is likelyto result from a combination of factors including variation indolphin ranging patterns (see Wilson et al 2004) weather con-ditions and other logistical factors that influence sampling indifferent areas Most notably sampling protocols were changedin 2000 specifically to increase the probability of capture Thischange has proven to have been successful in the latter part ofthe study while estimates of the survival probabilities and to-tal population size remain unaffected Second another study(Wilson et al 2004) has shown that the ranging pattern of thepopulation changed in recent years with some animals spend-ing more time in the southern part of the population range alongthe east coast of Scotland south of the Moray Firth to St An-drews and the Firth of Forth The effect of this would be toreduce the capture probability in the core study area in the in-ner Moray Firth and to potentially increase heterogeneity incapture probabilities The more southerly areas also have beensampled but the data are more sparse and cover only a smallnumber of years Once these histories have become more exten-sive we expect that the capture probabilities will become lessvariable Unlike other models our estimates are also correctedfor individual heterogeneity Variation in capture probabilitiesalso might have arisen from heterogeneous effort in space Al-though this was not explicitly modeled we could have stratifiedthe capture histories by area by including separate capture co-efficients for each area

The overall mean survival estimate through the study periodis 93 (standard deviation 029 95 HPDI = 861 979) Thisis similar to an earlier maximum likelihood estimate of sur-vival for the cohort of nicked individuals from this populationfirst observed in 1990 (Saunders-Reed Hammond Grellier andThompson 1999) But our model estimate is slightly lower thansurvival estimates from other bottlenose populations of 962 plusmn76 in Sarasota Bay Florida (Wells and Scott 1990) 952 plusmn

Corkrey et al Bayesian Recapture Population Model 955

150 in Doubtful Sound New Zealand (Haase 2000) and 95ndash99 in the Sado estuary Portugal (Gaspar 2003) However thismay be because our study included subadults while the otherstudies reported adult survival Variation in survival probabil-ity occurs over time [Fig 2(b) Table 3] It is of interest thatsurvival reached a minimum in 1997 but later returned to lev-els comparable to earlier years The proportion of iterations inwhich the survival for 1997 are above the mean survival forthe remaining years (mean φ = 95 95 HPDI = 85 100)is only 08 The apparent low survival in 1997 may be due tomigration because this is not distinguishable from survival inthe model It is possible that an emigration event occurred in1997 and that those individuals have not yet returned Consid-erable heterogeneity in survival (range 77ndash97) is observedas shown in Figure 2(c) The survival probabilities are not dis-tributed smoothly suggesting multimodality The animals witha low survival coefficient may have died or migrated In thecase of migration this result would indicate that some animalsless consistently reside in the core areas compared with oth-ers

Individuals with higher survival coefficients are frequentlythose seen earliest in the study There is little information avail-able on individuals seen only more recently and consequentlytheir coefficients are close to 0 Some survival coefficients aremore negative and these correspond to animals that have notbeen seen for some time As time goes on and in the absenceof a confirmed death or sighting these coefficients can be ex-pected to become more negative However some animals thatare observed only on a few occasions have negative coefficientsThe estimation of individual survival coefficients means that es-timation of other coefficients in the model are adjusted for vari-able survival The estimation of individual capture probabilitiesallows the results to be combined with other individual covari-ates such as sex or body size if these data become availableSimilarly where individuals differ in their home ranges indi-vidual capture or survival probabilities could be related to localenvironmental conditions within their range to prey availabil-ity or to water quality

Estimates of the total population size from bilateral analy-sis of the left and right photo-identification series and estimatesfrom unilateral analyses of the left and right sides are shown inFigure 3(a) These are similar both to the maximum likelihoodestimate of 129 (95 confidence interval = 110ndash174) for datacollected in 1992 (Wilson et al 1999) and to a Bayesian mul-tisite mark recapture estimate of the nicked population of 85(95 probability interval = 76ndash263) for data collected in 2001(Durban et al 2005) The credible intervals of the abundanceestimates are generally narrower in the bilateral analysis (meanwidth 886) than in left-side analysis (mean width 1373) andright-side analysis (mean width 937) Figure 3(b) shows theprior and mean posterior recruitment probabilities ωj It is in-teresting that higher probabilities occur in 1996 and 2001 Thismight be considered to variations in capture probability butwhereas the capture probability is high in 2001 it remains lowin 1996 [see Fig 2(a)] Instead calculating the probabilities ofthe posterior exceeding the prior recruitment probabilities wefind they are below 5 (range 04ndash44) for all years except 2001in which it is 85 The higher recruitment probability in 2001may be due to an improved protocol in that year that led to the

(a)

(b)

Figure 3 (a) Mean total abundance estimates from a bilateral analy-sis with 95 HPDI Also shown are estimates from separate analysesof left (minus minus minus) and right (middot middot middot) sides along with associated 95 HPDIrepresented by symbols left + right Minimum numbers of ani-mals known to be alive are indicated by diamonds (b) Mean posteriorrecruitment probabilities ωj with 95 HPDI The prior value is rep-resented as a dotted line

identification of previously unrecognized nicked animals Weconsider the rise in 1996 to have insufficient evidence to sug-gest an increase in recruitment in that year

Earlier demographic modeling of this population has pre-dicted that the population is in decline (Saunders-Reed et al1999) To examine this we calculate the statistic [prodJ

j=2 Wj

(Wjminus1)]1(Jminus1) Values of this statistic gt1 indicate an increas-ing population whereas values lt1 indicates a decreasing pop-ulation The mean value is 989 (95 HPDI = 958 102)and the probability of it being lt10 is 77 This suggests thatthe population may be decreasing but that the evidence fora decline is weak By doing this we are able to use the fulltime series of photo identification data to obtain simultaneousposterior estimates of the Wj to compare the probability of apopulation increase or decrease In agreement with the earlierpredictive modeling these data indicate greater probability of adecline than of an increase However because it is likely thatthe population is expanding its range (Wilson et al 2004) thedecline may be confounded with temporary emigration

956 Journal of the American Statistical Association September 2008

6 DISCUSSION

Changes in the size of wildlife populations are often assessedby fitting trend models to independent abundance estimatesbut the power of these techniques to detect trends often canbe low (Gerrodette 1987 Taylor Martinez Gerrodette Bar-low and Hrovat 2007) Here we use a capturendashrecapture modelthat incorporates an underlying population model and an ob-servation model We provide a probability that a population isin decline or increasing Using photoidentification data fromthis bottlenose dolphin population we generate the first em-pirical estimate of trends in abundance for any of the popula-tions of small cetaceans inhabiting European waters Becausethe methodological changes in sampling likely are absorbedby the time-varying capture coefficient the remaining variationdue to individual capture tendencies might reasonably be de-scribed by the individual capture coefficient This then allowsexploration of capture and survival relating to individuals ratherthan to populations Information on individual survival could beimportant for addressing some management issues for exam-ple individuals differing in their spatial distribution

By decomposing the capture probabilities into overall time-varying and heterogeneous terms we find that the time-varyingcontribution and heterogeneous variation are comparable inmagnitude and their variation is greater than that of the overallterm This information can be used in the design of monitoringprograms by suggesting how effort should be distributed Thedifference in overall left and right capture probabilities mightbe interpretable as an artifact of the sampling techniques or ofbehavioral effects However after allowing for this the prob-ability of observing one side also varied over time with someyears strongly favoring one side or the other This might be ex-plained by variation in protocol between years such as changesin personnel The relative sizes of these effects may be of usein evaluating consistency of protocol over time Finally afterallowing for the overall difference and time variation the het-erogeneous capture coefficients exhibit a bias to the one sidepossibly due to behavior or protocol These findings demon-strate that bilateral decomposition of capture probabilities canprovide useful information

We decompose the survival probability similarly and we findit to be unusually low in 1997 suggesting a possible multimodalheterogeneous survival probability that may be the result of mi-gration or survival variation This type of decomposition allowsexploration of such effects in more detail In particular the mul-timodality of heterogeneous survival suggests that the popula-tion comprises individuals with different strategies some vis-iting the core study area more or less often Ongoing work isattempting to quantify individual heterogeneity such as in ex-posure to sewage contaminated waters and boat-based tourismTherefore this technique could be used to determine whetherexposure to different anthropogenic stressors is related to sur-vival

Our model also makes use of priors such as those on ωj thatare not informed by the recapture data set but make use of in-formation from other sources An alternative approach mightbe to specify these as informative priors based on expert opin-ion or tuned to obtain satisfactory results Our application il-lustrates how disparate data sources may be combined into a

single analysis Although the data source for the prior recruit-ment probability is small and problematic it is the only oneavailable Even when some priors are defined in the simulationstudy to incorrect values the estimates of parameters of interestremain valid The posterior probability also adds to the state ofknowledge of this parameter

Previous use of capturendashrecapture models with cetacean pho-toidentification data highlights the potential violation of modelassumptions due to heterogeneity in capture andor survivalprobabilities This model explicitly accounts for heterogene-ity and also provides simultaneous estimates of individual sur-vival and capture probabilities This can be used both to ex-plore the biological basis of heterogeneity and to improve sam-pling programs The individual survival coefficients also can beused to explore aspects of biology where correlates are avail-able for example the relationship of survival to age sex orother variables may be of interest Abundance under the as-sumption of heterogeneous survival has been estimated by LeeHuang and Ou (2004) whereas individual and time-varyingsurvival estimates have been obtained using band-recoverymodels (Grosbois and Thompson 2005) Other studies have ex-amined survival by group for example Franklin Andersonand Burnham (2002) obtained separate estimates of group-levelsurvival for males and females using stratification in a bandingstudy Although we lack such variables as age and sex theo-retically it would be possible to examine individual survivalby sex either as a post hoc comparison or through their inte-gration into the model itself For example survival and cap-ture could be modeled as logit(φ) = α + βj + γd + τ andlogit(π) = δ + εj + ζd + χ where τ corresponds to stage-specific (eg young immature reproductive female senes-cent) survival and χ corresponds to capture probabilities forvarious categories (eg young reproductive adult)

Other cetacean studies have examined heterogeneity White-head (2001b) obtained an estimate of the coefficient of variation(CV) for identification from a randomization test of successiveidentifications in whale track data (The CV is also a measure ofcapture heterogeneity) They found that the CV was significantin one of the two years and that probably depended on the sizesof the animals being observed This is not comparable to thedolphin observations in which heterogeneity is more likely toresult from differences in animal behavior In the present studywe use a likelihood in which heterogeneity of survival and cap-ture has been incorporated by a logit function This approachwas described previously (Pledger Pollock and Norris 2003Pledger and Schwarz 2002) within a non-Bayesian contextPledger et al (2003) classified animals into an unknown set oflatent classes each of which could have a separate survival andcapture probability A simplification allows for time-varyingand animal-level heterogeneity This approach is similar to oursbut we also integrate a dynamic population model and an ob-servation model into the estimation process Goodman (2004)also implemented a population model in a Bayesian context thatcombines capturendashrecapture data with carcass-recovery databut assumes the absence of heterogeneity in capture probabil-ities Durban et al (2005) estimated abundance for the samepopulation as us but used a multisite Bayesian log-linear modeland model averaging (King and Brooks 2001) across the pos-sible models with varying combinations of interactions Our

Corkrey et al Bayesian Recapture Population Model 957

model could be extended to include multiple sites and yearsby using a capturendashrecapture history that includes site informa-tion and a yearndashsite term in the capture probability expressionBy doing this we suggest that geographic areas with more datacan be used to strengthen estimates from poorly sampled areasThis also may account for some individual variation currentlyidentified as individual capture heterogeneity because some in-dividuals may be more closely associated with some areas thanothers and may attenuate a possible confounding of migrationwith survival

We use a separate data set and a negative-binomial model toinflate the estimate of the proportion of nicked animals in thesamples Another study (Da Silva Rodrigues Leite and Mi-lan 2003) obtained an abundance estimate for bowhead whalesusing an inflation factor derived from the numbers of good andbad photographs of marked and unmarked whales Our analysisavoids the lack of independence between good photographs anddistinctively marked animals in the photographs that might in-validate the estimate of variance in the inflation factor Otherstudies avoid this issue by including only separately catego-rized good-quality photographs (eg Wilson et al 1999 ReadUrian Wilson and Waples 2003) However independent es-timates of inflation when available such as in our study arepreferred

We do not know of other cetacean studies that combinephoto-identification series of the left and right sides but ananalogous approach is that of log-linear capturendashrecapturemodeling (King and Brooks 2001) in which animals are identi-fied from separate lists Some photo-identification studies mightuse information from both sides in the process of identifyinganimals (Joyce and Dorsey 1990) A more robust approach isto perform the identification on each side separately We haveshown how these separate data can then be combined

A major advantage to our modeling approach is its extensibil-ity Within the Bayesian framework models can be comparedby such methods as reversible-jump Markov chain Monte Carlotechniques (Green 1995) Model extensions include extendingthe population model considering an individual-level approachThis would allow incorporation of the heterogeneity into thispart of the model and also modeling of the unobserved popu-lation More detailed modeling of heterogeneity could be ex-plored by including hyperpriors for the variances of the recap-ture model terms For convenience we analyze the data on anannual basis Additional information would be available if theanalyses were conducted on a finer time scale such as monthsor by individual trip This would allow the model to be extendedto include seasonal effects For example dolphins are observedmore frequently in summer months than in winter months andthis tendency could be included in an extended model by chang-ing the capture probability to be logit(π) = δt + εj + ζd whereδt t isin [1 12] is a monthly capture coefficient This maybe of use in modeling fine-scale movements The response vari-able itself also can be generalized Although we are careful toonly include data from high-quality photographs and nicked an-imals there is still the possibility of some identification errorsparticularly for more subtly marked individuals A possible ex-tension of the model could allow identification to be assigneddifferent levels of certainty for example by incorporating anintermediate state between nonrecapture and recapture that we

term ldquopossible recapturerdquo This could be done by replacing thelogit function logit(ν) = log(ν(1 minus ν)) where ν correspondsto the probability of recapture by a proportional odds function(McCullagh 1980) This technique has previously found appli-cation in a study of photoidentification used to model photo-graphic quality scores and individual distinctiveness of hump-back whale tail flukes (Friday Smith Stevick and Allen 2000)In this approach the probability of ldquopossible recapturerdquo is givenby ν1 = π1 the probability of certain recapture is given byν2 = π1 + π2 and includes the possible recapture category andthe probability of no capture is given by ν0 = 1 minus π1 + π2There are two proportional odds functions corresponding topossible recapture and certain recapture which are logit(ν1) =log(π1(π2 +π3)) and logit(ν2) = log((π1 +π2)π3) We thencould extend the recapture model using logit(ν1) = α+βj +γk1

and logit(ν2) = α + βj + γk2 This proportional odds approachmay be particularly useful in situations where automatic match-ing algorithms (Saacutenchez-Mariacuten 2000) provide an estimate ofthe likelihood of a match which could then be incorporatedinto abundance estimate models

APPENDIX DERIVATION OF POSTERIORS

Here we show how we derive the posterior conditional distributionsfor the model parameters and describe how each is updated within theMCMC algorithm The full joint distribution has the form shown in(A1) In (A1) the first two lines correspond to the population modelthe third line corresponds to the inflation model the fourth line cor-responds to the observation model the fifth line corresponds to therecapture model and the remaining lines correspond to priors

π = P(B1|ω1N0W0)P (S1|αN0)

(Sec 31)

timesSprod

j=2

P(Bj |Sjminus1Bjminus1Wjminus1ωj )

times P(Sj |Sjminus1Bjminus1 αβjminus1)

(Sec 31)

timesSprod

j=1

P(Wj |Sj Bj υj )

Tjprod

i=1

P(tnji |υj twji )

(Sec 32)

timesSprod

j=1

P(nLj|δL εjLSj Bj )P (nRj

|δR εjRSj Bj )

(Sec 33)

timesSprod

j=1

P(Hdj |αβj γd δL εjL ζdL δR εjR ζdR)

(Sec 34)

times P(N0|λN)P (W0|N0 υ0)P (α δL δRλN υ0)

(Sec 41)

timesSprod

j=1

P(βj εjL εjRωj υj )

nprod

d=1

P(γd)P (ζdL)P (ζdR)

(Sec 41) (A1)

958 Journal of the American Statistical Association September 2008

For each parameter we obtain the corresponding posterior condi-tional distribution by extracting all terms containing that parameterfrom the full joint distribution When the conditional distributions havea standard form we use the Gibbs sampler (Brooks 1998) to update theparameters otherwise we use MetropolisndashHastings updates (Chib andGreenberg 1995)

Within a MetropolisndashHastings update we propose a new valuefor a parameter x and denote the proposed value by xprime To decidewhether to accept the proposed value we calculate an acceptance ra-tio α = min(1A) where A is the acceptance ratio The acceptanceratio is given by A = π(xprime)q(x)π(x)q(xprime) where π(x) is the poste-rior probability of x and q(x) is the probability of the proposed valueWe accept the new value xprime with probability α otherwise we leavethe value of x unchanged (See Chib and Greenberg 1995 for an exam-ple)

Normal proposal distributions are used for continuous parameterscentered around the current parameter value Discrete uniform pro-posals are used for discrete variables each centered around the currentparameter values The discrete proposals also require constraints to en-sure that the proposed values are valid For example a proposed valuefor N0 must be less than the current value of W0 We provide a detaileddescription of each update next

Population Model

The population model has parameters ωj λ Bj Sj and N0 andare updated using Gibbs or MetropolisndashHastings samplers The condi-tional posterior distributions for ωj and λN can be derived as

ωj |Bj Sjminus1Bjminus1

sim Beta(Bj + 0238Wjminus1 minus Sjminus1 minus Bjminus1 minus Bj + 1)

and

λN |N0 sim Gam(76 + N0101)

These are individually updated using the Gibbs sampler during eachiteration of the Markov chain

The parameters Bj Sj and N0 are updated by the MetropolisndashHastings algorithm using acceptance ratios calculated as described ear-lier We use uniform proposals centered on the current parameter valueand limited to the range of possible values

For the Sj rsquos we use three alternative proposals

Sprimej=1 sim Unif

(max

(0 S1 minus 5 S2 minus B1max(n1Ln1R) minus N1

2 minus B1)

min(S1 + 5N0W1 minus B1W1 minus B1 minus B2

W1 minus B1 minus 1))

Sprime2lejltJ sim Unif

(max

(0 Sj minus 5 Sj+1 minus Bj max(njLnjR) minus Bj

2 minus Bj

)

min(Sj + 5 Sjminus1 + Bjminus1Wj minus Bj Wj minus Bj minus Bj+1

Wj minus Bj minus 1))

and

Sprimej=J sim Unif

(max

(0 SJ minus 5max(nJLnJR) minus BJ 2 minus NJ

)

min(SJ minus 5 Sjminus1 + Bjminus1Wj minus Bj Wj minus Bj minus 1))

For Bj we use three alternative proposals

B prime1 sim Unif

(max

(0B1 minus 15 S2 minus S1max(n1Ln1R) minus S1

2 minus S1)

min(B1 + 15W0 minus N0W1 minus S1W1 minus S1 minus B2

W1 minus S1 minus 1))

B prime2lejltJ sim Unif

(max

(0Bj minus 15 Sj+1 minus Sj max(njLnjR) minus Sj

2 minus Sj

)

min(Bj + 15Wjminus1 minus Sjminus1 minus Bjminus1Wj minus Sj

Wj minus Sj minus Bj+1Wj minus Sj minus 1))

and

B primeJ sim Unif

(max

(0BJ minus 15max(nJLnJR) minus SJ 2 minus SJ

)

min(BJ + 15WJminus1 minus SJminus1 minus BJminus1

WJ minus SJ WJ minus SJ minus 1))

For N0 we use

N prime0 sim Unif

(max(5N0 minus 15 S1)min(N0 + 15W0 minus B1W0 minus 1)

)

Inflation Model

The inflation model parameters Wj and υj are updated usingMetropolisndashHastings and Gibbs updates The conditional posterior dis-tributions for υj can be derived as

υ0|N0W0 sim Beta(N0 + 1W0 minus N0 + 1)

and

υjgt0|Sj Bj Wj tnji twji sim Beta

(

Sj + Bj +T (j)sum

i=1

tnji + 1

Wj minus Sj minus Bj +T (j)sum

i=1

twji minus tnji + 1

)

These are updated separately using the Gibbs sampler on each itera-tion

The remaining parameter Wj has a nonstandard conditional pos-terior distribution We use larger updates than in the population modelbecause the initial total population is expected to be larger in size

W prime0 sim Unif

(max(0W0 minus 15N0 + 1N0 + B1)W0 + 15

)

W prime1lejltJ sim Unif

(max(0Wj minus 15 Sj + Bj + 1 Sj + Bj + Bj+1)

Wj + 15)

and

W primeJ sim Unif

(max(0WJ minus 15 SJ + BJ + 1)WJ + 15

)

Recapture Model

For the recapture model we have parameters α βj γd δL εjLζdL δR εjR and ζdR These all have nonstandard conditional pos-terior distributions and we use the MetropolisndashHastings algorithm toupdate the parameters on each iteration We use individual normal dis-tribution proposals centered on the current parameter values and withvariance equal to the prior variance for example

αprime sim N(α185)

[Received June 2005 Revised May 2007]

Corkrey et al Bayesian Recapture Population Model 959

REFERENCES

Brooks S P (1998) ldquoMarkov Chain Monte Carlo Method and Its ApplicationrdquoThe Statistician 47 69ndash100

Brooks S P and Roberts G O (1998) ldquoConvergence Assessment Techniquesfor Markov Chain Monte Carlordquo Statistics and Computing 8 319ndash335

Buckland S T (1982) ldquoA CapturendashRecapture Survival Analysisrdquo Journal ofAnimal Ecology 51 833ndash847

(1990) ldquoEstimation of Survival Rates From Sightings of IndividuallyIdentifiable Whalesrdquo in Individual Recognition of Cetaceans Use of Photo-Identification and Other Techniques to Estimate Population Parameters edsP S Hammond S A Mizroch and G P Donovan Cambridge UK Inter-national Whaling Commission pp 149ndash153

Burnham K and Overton W (1979) ldquoRobust Estimation of Population SizeWhen Capture Probabilities Vary Among Animalsrdquo Ecology 60 927ndash936

Carothers A (1979) ldquoQuantifying Unequal Catchability and Its Effect on Sur-vival Estimates in an Actual Populationrdquo Journal of Animal Ecology 48863ndash869

Chao A (1987) ldquoEstimating the Population Size for CapturendashRecapture DataWith Unequal Catchabilityrdquo Biometrics 43 783ndash791

Chao A Lee S-M and Jeng S-L (1992) ldquoEstimating Population Size forCapturendashRecapture Data When Capture Probabilities Vary by Time and Indi-vidual Animalrdquo Biometrics 48 201ndash216

Chib S and Greenberg E (1995) ldquoUnderstanding the MetropolisndashHastingsAlgorithmrdquo The American Statistician 49 327ndash335

Cormack R (1972) ldquoThe Logic of CapturendashRecapture Experimentsrdquo Biomet-rics 28 337ndash343

Coull B A and Agresti A (1999) ldquoThe Use of Mixed Logit Models to Re-flect Heterogeneity in CapturendashRecapture Studiesrdquo Biometrics 55 294ndash301

Da Silva C Q Rodrigues J Leite J G and Milan L A (2003) ldquoBayesianEstimation of the Size of a Closed Population Using Photo-ID With Part ofthe Population Uncatchablerdquo Communications in Statistics 32 677ndash696

Dorazio R M and Royle J A (2003) ldquoMixture Models for Estimating theSize of a Closed Population When Capture Rates Vary Among IndividualsrdquoBiometrics 59 351ndash364

Durban J Elston D Ellifrit D Dickson E Hammon P and Thompson P(2005) ldquoMultisite Mark-Recapture for Cetaceans Population Estimates WithBayesian Model Averagingrdquo Marine Mammal Science 21 80ndash92

Fienberg S E Johnson M S and Junker B W (1999) ldquoClassical Multi-level and Bayesian Approaches to Population Size Estimation Using MultipleListsrdquo Journal of the Royal Statistical Society Ser A 162 383ndash405

Forcada J and Aguilar A (2000) ldquoUse of Photographic Identification inCapturendashRecapture Studies of Mediterranean Monk Sealsrdquo Marine MammalScience 16 767ndash793

Franklin A B Anderson D R and Burnham K P (2002) ldquoEstimation ofLong-Term Trends and Variation in Avian Survival Probabilities Using Ran-dom Effects Modelsrdquo Journal of Applied Statistics 29 267ndash287

Friday N Smith T D Stevick P T and Allen J (2000) ldquoMeasure-ment of Photographic Quality and Individual Distinctiveness for the Photo-graphic Identification of Humpback Whales Megaptera novaeangliaerdquo Ma-rine Mammal Science 16 355ndash374

Gaspar R (2003) ldquoStatus of the Resident Bottlenose Dolphin Population inthe Sado Estuary Past Present and Futurerdquo unpublished doctoral dissertationUniversity of St Andrews School of Biology

Gelfand A E and Sahu S K (1999) ldquoIdentifiability Improper Priors andGibbs Sampling for Generalized Linear Modelsrdquo Journal of the AmericanStatistical Association 94 247ndash253

Gerrodette T (1987) ldquoA Power Analysis for Detecting Trendsrdquo Ecology 681364ndash1372

Goodman D (2004) ldquoMethods for Joint Inference From Multiple Data Sourcesfor Improved Estimates of Population Size and Survival Ratesrdquo Marine Mam-mal Science 20 401ndash423

Green P J (1995) ldquoReversible-Jump Markov Change Monte Carlo BayesianModel Determinationrdquo Biometrika 82 711ndash732

Grosbois V and Thompson P M (2005) ldquoNorth Atlantic Climate VariationInfluences Survival in Adult Fulmarsrdquo Oikos 109 273ndash290

Haase P (2000) ldquoSocial Organisation Behaviour and Population Parametersof Bottlenose Dolphins in Doubtful Sound Fiordlandrdquo unpublished mastersdissertation University of Otago Dept of Marine Science

Hammond P (1986) ldquoEstimating the Size of Naturally Marked Whale Pop-ulations Using CapturendashRecapture Techniquesrdquo Report of the InternationalWhaling Commission Special Issue 8 253ndash282

(1990) ldquoHeterogeneity in the Gulf of Maine Estimating HumpbackWhale Population Size When Capture Probabilities Are not Equalrdquo in In-dividual Recognition of Cetaceans Use of Photo-Identification and OtherTechniques to Estimate Population Parameters eds P S Hammond S A

Mizroch and G P Donovan Cambridge UK International Whaling Com-mission pp 135ndash153

Huggins R (1989) ldquoOn the Statistical Analysis of Capture Experimentsrdquo Bio-metrika 76 133ndash140

Joyce G G and Dorsey E M (1990) ldquoA Feasibility Study on the Useof Photo-Identification Techniques for Southern Hemisphere Minke WhaleStock Assessmentrdquo in Individual Recognition of Cetaceans Use of Photo-Identification and Other Techniques to Estimate Population Parameters edsP S Hammond S A Mizroch and G P Donovan Cambridge UK Inter-national Whaling Commission pp 419ndash423

King R and Brooks S (2001) ldquoOn the Bayesian Analysis of PopulationSizerdquo Biometrika 88 317ndash336

Lebreton J-D Burnham K P Clobert J and Anderson S R (1992)ldquoModeling Survival and Testing Biological Hypotheses Using Marked An-imals A Unified Approach With Case Studiesrdquo Ecological Monographs 6267ndash118

Lee S-M Huang L-H H and Ou S-T (2004) ldquoBand Recovery Model In-ference With Heterogeoneous Survival Ratesrdquo Statistica Sinica 14 513ndash531

Link W A and Barker R J (2005) ldquoModeling Association Among De-mographic Parameters in Analysis of Open Population CapturendashRecaptureDatardquo Biometrics 61 46ndash54

Mann J and Smuts B (1999) ldquoBehavioral Development in Wild BottlenoseDolphin Newborns (Tursiops sp)rdquo Behaviour 136 529ndash566

McCullagh P (1980) ldquoRegression Models for Ordinal Datardquo Journal of theRoyal Statistical Society Ser B 42 109ndash142

Otis D L Burnham K P White G C and Anderson D R (1978) ldquoStatis-tical Inference From Capture Data on Closed Animal Populationsrdquo WildlifeMonographs 62 135

Pledger S (2000) ldquoUnified Maximum Likelihood Estimates for ClosedCapturendashRecapture Models Using Mixturesrdquo Biometrics 56 434ndash442

Pledger S and Schwarz C J (2002) ldquoModelling Heterogeneity of Survivalin Band-Recovery Data Using Mixturesrdquo Journal of Applied Statistics 29315ndash327

Pledger S Pollock K H and Norris J L (2003) ldquoOpen CapturendashRecaptureModels With Heterogeneity I CormackndashJollyndashSeber Modelrdquo Ecology 59786ndash794

Pollock K H Nichols J D Brownie C and Hines J E (1990) ldquoStatisticalInference for CapturendashRecapture Samplingrdquo Wildlife Monographs 107 1ndash97

Read A Urian K W Wilson B and Waples D (2003) ldquoAbundance andStock Identity of Bottlenose Dolphins in the Bayes Sounds and Estuaries ofNorth Carolinardquo Marine Mammal Science 19 59ndash73

Saacutenchez-Mariacuten F J (2000) ldquoAutomatic Recognition of Biological ShapesWith and Without Representations of Shaperdquo Artificial Intelligence in Medi-cine 18 173ndash186

Saunders-Reed C A Hammond P S Grellier K and Thompson P M(1999) ldquoDevelopment of a Population Model for Bottlenose Dolphinsrdquo Re-search Survey and Monitoring Report 156 Scottish Natural Heritage

Scottish Natural Heritage (1995) Natura 2000 A Guide to the 1992 EC Habi-tats Directive in Scotlandrsquos Marine Environment Perth UK Scottish Nat-ural Heritage

Seber G (2002) The Estimation of Animal Abundance Caldwell NJ Black-burn Press

Stevick P T Palsboslashll P J Smith T D Bravington M V and Ham-mond P S (2001) ldquoErrors in Identification Using Natural Markings RatesSources and Effects on CapturendashRecapture Estimates of Abundancerdquo Cana-dian Journal of Fisheries and Aquatic Science 58 1861ndash1870

Taylor B L Martinez M Gerrodette T Barlow J and Hrovat Y N (2007)ldquoLessons From Monitoring Trends in Abundance of Marine Mammalsrdquo Ma-rine Mammal Science 23 157ndash175

Wells R S and Scott M D (1990) ldquoEstimating Bottlenose Dolphin Popu-lation Parameters From Individual Identification and CapturendashRelease Tech-niquesrdquo in Individual Recognition of Cetaceans Use of Photo-Identificationand Other Techniques to Estimate Population Parameters eds P S Ham-mond S A Mizroch and G P Donovan Cambridge UK InternationalWhaling Commission pp 407ndash415

Whitehead H (2001a) ldquoAnalysis of Animal Movement Using Opportunis-tic Individual Identifications Application to Sperm Whalesrdquo Ecology 821417ndash1432

(2001b) ldquoDirect Estimation of Within-Group Heterogeneity in Photo-Identification of Sperm Whalesrdquo Marine Mammal Science 17 718ndash728

Williams B K Nichols J D and Conroy M J (2001) Analysis and Man-agement of Animal Populations San Diego Academic Press

Wilson B Hammond P S and Thompson P M (1999) ldquoEstimating Size andAssessing Trends in a Coastal Bottlenose Dolphin Populationrdquo EcologicalApplications 9 288ndash300

Wilson B Reid R J Grellier K Thompson P M and Hammond P S(2004) ldquoConsidering the Temporal When Managing the Spatial A Popula-

Corkrey et al Bayesian Recapture Population Model 949

to estimate the proportion of the sampled population having dis-tinctive marks (Seber 2002) or by double sampling (Pollock etal 1990)

To address the aforementioned issues we construct a modelincorporating heterogeneity in capture and survival that makesuse of the available data consisting of left-side and right-sidephotographic series Individuals are regarded as captured af-ter they have been identified by a specific type of dorsal finmarking called ldquonicksrdquo that are visible in these photographsBecause these nicks appear on animals only as they maturewe include a nicking-recruitment process in the model How-ever we lack knowledge of the probability of having nicks andthe rate of acquiring nicks Rather than adopt vague priors forthese probabilities we include separate data sources that pro-vide these quantities We also make use of the survival andcapture information from the recapture model to obtain an esti-mate of the population size Because some priors remain vaguewe assess the effect of prior specification using simulated dataWe apply the method to photo-identification data from a nat-urally marked bottlenose dolphin (Tursiops truncatus) popula-tion from the Moray Firth Scotland This is the only residentcoastal population of bottlenose dolphins in the North Sea anda Special Area of Conservation has been established to pro-tect these animals in response to the 1992 EC Habitats Direc-tive (Council Directive 9243EEC Scottish Natural Heritage1995) The UK government needs information on trends inthe abundance of this population to report on the success of thisconservation program to the European Union

In this article we develop a general framework for estimat-ing the total dolphin population size by embedding a standardcapturendashrecapture model within a state-space model that de-scribes the evolution of population size over time We begin inSection 2 by describing the data available In Section 3 we dis-cuss implementation of the model In Section 4 we discuss theBayesian approach including the use of Markov chain MonteCarlo (MCMC) techniques the specification of priors and asimulation study In Section 5 we provide the results of ouranalysis and discuss the implications for the Moray Firth bot-tlenose dolphin (Tursiops truncatus) population We concludewith discussion of the wider implications of our research andthe utility of our approach for other wild animal populations

2 DATA COLLECTION AND FORMAT

Between 1990 and 2002 boat-based surveys were con-ducted both in the inner Moray Firth and along adjacent coastsFull details of survey protocols have been provided by (Wil-son Thompson and Hammond 1997 Wilson Hammond andThompson 1999) The number of trips made varied amongyears and areas Most trips were made between May and Sep-tember with each trip encountering one or more groups of dol-phins Because the data collection effort in winter was low andavailable only for the early years of the study these data areexcluded from the analysis Between 1990 and 1999 trips fol-lowed a fixed route (Wilson et al 1997) but from 2000 onwardopportunistic surveys were conducted in an attempt to increasethe number of dolphin groups sighted

When bottlenose dolphins were encountered photographswere taken of their dorsal fins using an single-lens reflux cam-era with a telephoto lens (Wilson et al 1999) Both 35-mm

transparency and digital cameras were used during the studywith all photos subjected to the same strict grading of photo-graphic quality (described in Wilson et al 1999) Only high-quality pictures were used to minimize errors in identification(Stevick Palsboslashll Smith Bravington and Hammond 2001Forcada and Aguilar 2000)

Individual bottlenose dolphins are identifiable by markingsalong the whole of the dorsal fin (see Fig 1) Such fin markshave been shown to be of varying persistence and reliability(Wilson et al 1999) therefore we restricted our data set onthose dolphins with nicks to the posterior edge of the dorsalfin which have been shown to be relatively permanent andto accumulate over time The accrual of individual nicks overtime changes the dorsal edge profile but reliable identificationcan still be obtained if the rest of the dorsal edge remains un-affected Estimated nicked proportions are determined from aseparate data set comprising the number of animals encounteredin each trip and the number of those that were nicked

On each occasion animals could be photographed from theleft side the right side or both sides Individuals photographedfrom one side can be identified at later times from photographsof either side because the dorsal trailing edge is visible fromeither side This means that differences in left and right captureprobability can be related to animal behavior and boat differ-ences and left and right capture probabilities are of potentialuse in sampling programs A total of 13 individuals are onlyknown from the left side and another 13 are only known fromthe right side These individuals are included in the calculationsince there is no evidence that they are not members of the pop-ulation

Each animal is assigned a unique identification code and arecord made of the date and location it was observed The ob-servations of a single animal are represented as a capture historyconsisting of a series of 0rsquos and 1rsquos where 1 indicates that theanimal was observed in the corresponding time period and 0 in-dicates the animal was not observed The entire data set consistsof a matrix in which each row corresponds to a capture historyfor a single animal and each column represents one calendaryear Separate history matrices are constructed as described ear-lier for photoidentification series of the left and right sides Inaddition a bilateral history is constructed from both the left andright side series in which 0 1 2 and 3 indicate that the animalwas unobserved or observed from the left side right side orboth sides on each occasion We denote the capture status foranimal d at time j as hdj hdj isin [0123]

3 MODEL CONSTRUCTION

The overall model comprises a series of submodels that re-late different elements of the observed (and indeed unobserved)data to common population parameters We begin with a de-scription of the component of the model that divides the pop-ulation into two distinct classes those animals that are nickedand thus identifiable and those that are not nicked Using a stan-dard capturendashrecapture model the size of the nicked populationcan be determined and from this the total population size canbe determined by estimating the proportion of nicked animals

950 Journal of the American Statistical Association September 2008

Figure 1 Example of dorsal fin showing nicks to the trailing edge Insets show the changes in the dorsal fin for this animal over time withthe earliest photograph on the left and the latest on the right The background image corresponds to the third inset from the right Apparentchanges in shape of the inset images are the result of differences in camera perspective The effect of these nicks is to produce a serrated edgethe whole of which is used to confirm the identity of the animal rather than just a selected few nicks Other markings such as skin lesions arenot considered reliable enough to be used in this study Minor changes in some nicks may occur over time but as long as other nicks on thedorsal edge remain intact identification can proceed

31 Modeling the Nicked Population

We begin with the assumption that young animals are ini-tially unnicked but that as they mature they will accumu-late nicks as they increasingly engage in social interactionsThis is a reasonable assumption in light of the social develop-ment of juvenile bottlenose dolphins (Mann and Smuts 1999Haase 2000) The animals can be considered as being recruitedfrom the unnicked population to the nicked population Once inthe nicked population individuals become observable and can-not return to the unmarked population

We denote the number of dolphins recruited from the un-nicked population to the nicked population at time 1 le j le J

as Bj and the number of nicked animals that survived betweentimes j minus 1 and j as Sj Thus the nicked population size attime j is given by Sj + Bj that is the current population com-prises new recruits and surviving former members The nickedpopulation is denoted by Nj = Sj + Bj

We assume that previously nicked animals survive from timej minus1 to time j with probability θj = exp(α+βj )(1+exp(α+βj )) where α denotes a global survival tendency and βj de-notes a time-varying survival component Thus the number ofsurviving animals Sj is given a binomial distribution so thatSj sim Bin(Sjminus1 + Bjminus1 θjminus1)

We also assume that unnicked animals acquire nicks withprobability ωj so that the number of newly nicked animals has

a binomial distribution Bj sim Bin(Wjminus1 minus Njminus1 ωj ) where attime j Wj denotes the total population and Nj denotes thenicked population Given this definition ωj is the probabilityof acquiring nicks and also surviving We believe that the num-ber of newly nicked animals is relatively small however

We refer to this component of the model as the popula-tion model It provides the probability distributions P(Sj |Sjminus1

Bjminus1 θjminus1) and P(Bj |Wjminus1Njminus1ωj )

32 Modeling the Total Population

In this case study the total population size and its variationover time is the primary statistic of interest but the total pop-ulation comprises both nicked and unnicked animals We canhowever estimate the size of the total population from the sizeof the nicked population using data on the proportion of ani-mals in each group that have nicked dorsal fins This is possiblebecause although unnicked individuals cannot be reliably iden-tified over periods of months or years they are recognizableover a period of days (see Wilson et al 1999) This allows usto count the number of unnicked individuals captured on high-quality photographs in any one encounter and to estimate theproportion of nicked and unnicked animals in the group to pro-vide an inflation factor

Then the unnicked part of the total population is given byWj minus Nj If the probability of an individual having nicks is

Corkrey et al Bayesian Recapture Population Model 951

given by υj then the size of the unnicked population at timej can be modeled as a negative binomial distribution condi-tional on the size of the nicked population so that Wj minus Nj simNegBin(Nj υj ) This approach allows the use of an indepen-dent data set to inform the υj rather than relying on the re-capture data to estimate this parameter The negative binomialdistribution is preferred over a binomial distribution becausethe Nj rsquos already have been assigned a distribution in the popu-lation model

In each time period j suppose that there are Tj trips eachobserving twji animals for i = 1 Tj of which tnji are nickedNumerous trips are undertaken each year and the numbernicked is separately estimated on each encounter A binomialdistribution then can be used to describe the number of nickedanimals seen per trip so that tnji sim Bin(twji υj ) Thus the ob-servations tnji and twji allow us to estimate υj and thus the totalpopulation size given an estimate of the nicked population Werefer to this as the inflation model It provides the probabilitydistributions P(Wj |Sj Bj υj ) and P(tnji |twji υj )

33 The Observation Model

In any time period only a portion of the nicked populationis seen Examination of the data suggests that the left and rightcapture probabilities differ at the population level The prob-ability that an individual animal would be seen from the leftor right side s isin [LR] is given by ρjs = exp(δs + εjs)(1 +exp(δs + εjs)) where δs denotes a global capture tendency andεjs denotes a time-varying capture component

Then the number of nicked animals seen from side s njs fol-lows a binomial distribution njs sim Bin(Nj ρjs) where njL =sum

d hdj isin [13] and njR = sumd hdj isin [23] We refer to this

model as the observation model It provides the probability dis-tributions P(njL|Nj ρjL) and P(njR|Nj ρjR)

We consider the njL and njR as independent counts A dol-phin seen in one year from one side contributes to the prob-ability of observing that side and one seen from both sidescontributes to both probabilities The introduction of captureprobabilities for each side allows the combination of the twophotographic series that otherwise would have been analyzedseparately because they would be correlated It is also of inter-est to estimate these probabilities to assess the capture methodsin use We include the term εjs to allow for observed differ-ences between years in observing all dolphins from one sideversus the other These differences may have arisen from dif-ferences in boat work

34 The Recapture Model

The population and observation models require estimates ofthe parameters α βj δL εjL δR and εjR corresponding tothe survival and capture probabilities These probabilities canbe obtained from the capture history data To do this we cal-culate the likelihood of the entire capture history matrix whichis the product of the probabilities associated with each animalrsquosindividual history L(hdj )

To combine the photo-identification series from the left andright sides into a single bilateral analysis the model allows dif-ferent capture probabilities when animals are seen from eachside The number of animals seen also can differ between the

left and right sides Other parameters do not depend on the sidefrom which an animal is observed for example the populationsize and survival probability of an animal should not depend onwhich side is observed However the left-side capture proba-bility for individual animals might be expected to differ fromthat of the right side for example the animal might be morelikely to approach the boat from one side than from the otherDolphins are also well known to have heterogeneous behavior(Wilson Reid Grellier Thompson and Hammond 2004)

Thus to incorporate heterogeneity as well as observationfrom two sides survival and capture probabilities are definedas logit(φdj ) = α + βj + γd and logit(πdjs) = δs + εjs + ζds where γd denotes a survival effect ζds denotes a recapture ef-fect for animal d s isin [LR] for the left or right side andlogit(x) = log(x(1 minus x)) The logit function is the standardchoice for binary data

Note that the heterogeneity effects are not used in the popu-lation and observation models because both of these are basedon both the nicked and unnicked populations and members ofthe latter will not have an individual effect parameter Althoughwe could adopt a random-effects model to ascribe individualeffects to unnicked animals this would imply that we believethe heterogeneity of survival and recapture to be the same inboth the nicked and unnicked populations which may not bethe case

The overall probability has components for animals in twosituations There are nJ animals observed at the final time andsome nJ lowast animals last observed at some earlier time and thatmight have died or simply are not seen Each individual historyprobability is conditional on the first time that the animal isseen j1

d Thus the likelihood associated with the histories of n = nJ +

nJ lowast dolphins is given by

prod

sisin[LR]

nprod

d=1

Jprod

j=1

L(hdjs)

=nJprod

d=1

Jprod

j=j1d +1

φdjminus1I (πdjLπdjRhdjs)

timesnJ +nJlowastprod

d=nJ +1

hlprod

j=j1d +1

φdjminus1I (πdjLπdjRhdjs)

(

1 minus φdhl

+Jsum

j=hl+1

φdj

jprod

k=hl+1

φdkminus1(1 minus πdkL)(1 minus πdkR)

)

(1)

in which φdj denotes the survival probability φdj = 1 minus φdj

when j lt J and φdj = 1 when j = J πdjs denotes the recap-ture probability from animal d observed in time j from sides isin [LR] j1

d and hl denote the first and last times that individ-ual d is seen and I (πdjLπdjRhdjs) = (1 minus πdjL)(1 minus πdjR)

when hdjs = 0 I (πdjLπdjRhdjs) = πdjL(1 minus πdjR) whenhdjs = 1 and I (πdjLπdjRhdjs) = (1 minus πdjL)πdjR whenhdjs = 2 I (πdjLπdjRhdjs) = πdjLπdjR when hdjs = 3 Theφdj and πdjs parameters are defined only for 1 le j lt J minus1 and2 le j lt J The values of nJ j1

d and hl are determined fromthe capture histories

952 Journal of the American Statistical Association September 2008

We refer to this model as the recapture model It yieldsthe probability P(hdjs |φdj γd πdjLπdjR ζdL ζdR) This isa CormackndashJollyndashSeber model which has been discussed else-where (Seber 2002 Lebreton Burnham Clobert and Anderson1992 Link and Barker 2005)

This completes the model specification To summarize wedefine a population model that describes the recruitment (ωj )and survival (θj ) of animals in the nicked population (Nj ) Therelative sizes of the nicked population (Nj ) and total population(Wj ) are described by the inflation model which makes use ofobserved counts of nicked and unnicked animals Using the cap-ture probability (ρjs ) and the observed number of nicked ani-mals we estimate the nicked population Finally we estimatethe heterogeneous capture (πdjs ) and survival (φdj ) probabili-ties from the capture histories Because this is a Bayesian analy-sis we also specify priors to the following parameters S0 B0ωj W0 N0 α βj γd δs εjs and ζds We describe the priorsin detail in the next section

4 ANALYSIS

In this section we describe the statistical methodology re-quired to analyze the model described in the previous sectionWe adopt a Bayesian approach and describe the prior specifica-tion and simulation techniques required here In this modelingprocess we use a Bayesian approach to allow for uncertaintyin measurement and parameters used The Bayesian approachallows for simultaneous consideration of all data and modelsallowing for the proper propagation of uncertainty throughoutthe model Identifiability is also less of a concern in a Bayesiananalysis in which prior information is supplied (Gelfand andSahu 1999) Moreover the Bayesian approach has an advan-tage in that MCMC methods can be used which greatly simpli-fies the computation compared with the corresponding classi-cal tools The remainder of this section discusses the process ofprior specification and provides details of the implementation

41 Priors

The first stage in the Bayesian analysis is to elicit priors foreach of the model parameters We do so here by grouping theparameters according to the component of the model in whichthey appear The priors used and their associated parameters aresummarized in what follows

The Population Model The model specifies the distribu-tion of parameters Sj and Bj in terms of Sjminus1 and Bjminus1 forj ge 1 Therefore we need a prior for S0 and B0 Becausethey only appear together we set N0 = S0 + B0 and assigna Poisson prior N0 sim Po(λN) Its mean takes a gamma priorλN sim Gam(76 01) with parameters selected to obtain a meanfor the nicked population of 76 as estimated by Wilson et al(1999) and a large variance so that it becomes uninformativecompared with the range of reasonable population values

Because the recruitment rates of nicked dolphins cannot bedirectly estimated from the photo-identification data set thepriors are parameterized with results from a separate analy-sis of randomly selected photographs with at least 2 years ofdata Because it is thought that the rate of nicking might begreater for older animals due to increased social interactionsestimates are estimated from photographs of individuals that

are first identified as known-age calves Identification is doneusing all attributes such as fin shape and the presence of le-sions Photographs of seven of these animals are available forestimating the age at which these dolphins are first seen to havea nick in the dorsal fin We consider these cases a random se-lection from the unnicked population so that the mean num-ber of nicks per year gives an approximate probability of nick-ing per year We set the prior for ωj to be reasonably vagueωj sim Beta(02381) Although rough these assumptions givea crude estimate for the parameter which is then updated bythe data

The Inflation Model The initial population size is assigneda negative-binomial prior W0 minusN0 sim NegBin(N0 υ0) The ini-tial inflation factor υ0 is assigned a beta prior υ0 sim Beta(11)which is equivalent to a noninformative uniform prior

The Recapture Model The recapture model survival pa-rameters have independent normal priors N(0 185) and thecapture parameters have independent normal priors N(0 15)These priors span the range (01) on the logit scale Largervariances are not used because these would obtain a U-shapeddistribution on the logit scale

42 Implementation

Once priors are assigned the full joint distribution can bedetermined as the product of the likelihood with the associatedprior distributions The resulting posterior is complex and high-dimensional and so inference is obtained in the form of poste-rior means and variances obtained through MCMC simulationWe choose to use an implementation in which each parameter isupdated in turn using either a MetropolisndashHastings or a Gibbsupdate depending on the form of the associated posterior con-ditional distributions The updating strategy is detailed in theAppendix All simulation software is written in FORTRAN 77The model was run for 500000 iterations and a 50 burn-in was used Sensitivity studies and standard diagnostic tech-niques were used (Brooks and Roberts 1998) to assess modelvalidity

43 Simulation Study

To test the MCMC code and to demonstrate the utility of theframework developed earlier we construct two artificial pop-ulations one with homogeneous capture and survival proba-bilities and the other with heterogeneous capture and survivalprobabilities These populations are constructed using the fol-lowing procedure An initial population of animals (n = 130)is sampled and a proportion of this initial population is ran-domly marked as nicked At each time period in the simulationanimals are randomly selected to die some of the unnicked be-come nicked and a proportion of the remaining animals arerandomly selected to be observed For each population threesimulations are run with differing priors for the ωj and λN para-meters The priors are such that the three cases represent vaguepriors precise priors and very precise priors but in each casethe means differ from the actual population values One possi-ble outcome is that the precise prior simulations track the priormeans more closely than the vague simulations

The simulations are illustrated in Table 1 which shows theactual total abundances the medians of the posteriors for Wj

Corkrey et al Bayesian Recapture Population Model 953

Table 1 Actual and estimated median total abundances for two artificial populations with 95 HPDI and posterior p values fromthree simulations on each with differing priors

A B C

Population Year Wj Wj p value Wj p value Wj p value

1 1990 124 119(99139) 28 112(97130) 07 107(90124) 021991 120 125(105146) 67 118(95134) 39 114(97135) 251992 114 116(96135) 54 110(93127) 30 106(92123) 171993 109 113(95134) 63 108(92124) 41 105(90121) 261994 103 111(93130) 81 106(88124) 63 103(86118) 491995 97 105(89125) 86 102(87118) 74 100(85114) 621996 95 98(84115) 66 95(81110) 48 93(81107) 381997 91 99(83114) 85 96(80109) 76 95(80108) 651998 89 94(79110) 73 91(77105) 56 90(77103) 521999 85 92(77108) 78 89(75103) 69 87(75103) 632000 76 82(6598) 76 79(6696) 65 79(6594) 592001 70 79(6299) 84 77(5895) 79 76(6095) 752002 68 76(5593) 78 75(5691) 75 73(5892) 70

2 1990 124 135(100178) 71 121(96159) 42 116(90149) 261991 122 123(90158) 51 112(87140) 21 108(84134) 121992 119 114(87148) 35 105(84130) 10 101(78124) 061993 111 106(82138) 34 99(75121) 11 95(74117) 061994 106 102(76130) 33 93(75118) 12 91(68115) 071995 100 96(70122) 38 91(74114) 16 88(67108) 081996 93 93(66119) 48 87(69109) 26 85(65105) 161997 87 91(67115) 60 85(68106) 37 82(62102) 281998 77 79(59101) 56 76(6295) 39 72(5791) 281999 74 84(60108) 80 80(62101) 75 77(5897) 622000 72 78(5799) 67 74(5894) 58 72(5591) 462001 67 71(4992) 62 69(4891) 57 66(4688) 432002 64 68(4191) 59 65(4492) 51 63(4388) 44

NOTE Population 1 has homogeneous capture probability (75) and survival probability (95) while population 2 has heterogeneous capture probability (mean 55 range 05ndash98) andsurvival probability (mean 92 range 68ndash99) The simulations have different priors for ωj and λN Simulation run A has vague priors [ω sim Beta(029412368) λN sim Gam(76 01)]B has more precise priors [ω sim Beta(023710) λN sim Gam(761)] and C has very precise priors [ω sim Beta(929939155) λN sim Gam(7600100)] Their means are the same ineach case (023 and 76) but each differs from the actual population values (06 and 106)

their 95 highest posterior density intervals (HPDI) and a p

value The p values are calculated by counting the proportionof iterations in which the Wj exceeded the true value Examina-tion of the results indicate that the median posterior estimatesgenerally agree with the actual values The agreement is sat-isfactory whether or not there is heterogeneity assumed Thep values are in the range 02ndash88 suggesting that the modelworks correctly The posterior p values for the recapture modelcoefficients are given in Table 2 The p values for α δL andδR are summarized and those for the remaining parameters aresummarized as the proportion of p values in each that lie insidethe range 01ndash99 Almost all p values are within the range 01ndash99 the only exceptions are the capture coefficients for a smallnumber of dolphins

In general the total population estimates for the homoge-neous case are slightly greater and the 95 HPDI is narrowerthen the heterogeneous case but all intervals contain the popu-lation values The estimates also shrink slightly for the mod-els with more precise priors However the p values in eachcase suggest an adequate fit even though some priors are de-liberately set to incorrect values In the case of the recapturecoefficients there is no apparent difference between the homo-geneous and heterogeneous models or with any trend in theprecision of the priors

5 RESULTS

We obtain estimates of heterogeneous capture and survivalcoefficients variation of capture and survival over time over-all survival and capture probabilities the abundance and thetrend in abundance The survival probabilities and capture co-efficients from the left and right sides are shown in Figure 2 andTable 3 We can obtain an overall capture probability for the twosides by calculating exp(δ)(1+exp(δ)) The mean overall cap-

Table 2 Summary of posterior p values for survival and capturecoefficients for two artificial populations from three simulations

on each with differing priors

Population α βj γd δL εjL ζdL δR εjR ζdR

1 16 100 100 66 100 98 60 100 10010 100 100 59 100 98 53 100 10008 100 100 59 100 98 48 100 100

2 34 100 100 17 100 97 50 100 9822 100 100 15 100 97 37 100 9923 100 100 12 100 97 39 100 99

NOTE The populations and simulations are described in Table 1 The p values are calcu-lated as the proportion of iterations greater than the actual population value The p valuesfor α δL and δR are shown Because the remaining parameters are vectors of coefficientsthose p values are summarized as the proportion of p values in each that are found insidethe range 01ndash99

954 Journal of the American Statistical Association September 2008

(a)

(b) (c)

Figure 2 (a) Variation of mean capture coefficients over time forthe left side εL (minusminus) and right side εR (- - bull - - bull) The two hor-izontal lines indicate the global mean capture coefficients δL (mdashmdash)and δR (minus minus minus) On the left vertical axis are the mean left-side het-erogeneity capture coefficients ζL and on the right vertical axis arethe mean right-side heterogeneity capture coefficients ζR Standardserrors for each coefficient are shown as vertical bars (b) Variation ofmean survival probability over time (c) Histogram of mean heteroge-neous survival probabilities

ture probability for the left side is 48 (95 HPDI = 29 68)and for the right side it is 58 (95 HPDI = 39 77) Althoughthis is a small difference the probability of the right capturecoefficient exceeding the left is 81 Examination of Figure 2(a)shows that the variation of capture coefficients over time is largecompared with the overall coefficients and that the left (εL) andright (εR) coefficients tend to track each other The probabilityof the right capture coefficient exceeding the left varies overa wide range 07ndash96 This range results from some years inwhich the probability tends strongly to the left side (1995 and2000 probability of observing the left side 95) or right side(1998 1999 2001 probability of observing the right side 91)Finally the range of the heterogeneous coefficients is compara-ble to that of the time-varying coefficients

The mean heterogeneous survival coefficient is not corre-lated with either of the mean capture heterogeneous coeffi-cients (right r = 009 left r = 022) However the meanleft and right heterogeneous capture coefficients are correlated(r = 714) indicating that animals that are more likely to beseen from one side also are more likely to be seen from theother side However the heterogeneity capture coefficient of theright side (mean 56) exceeded that of the left side (mean 23)with 85 of animals having higher right coefficients than left

Table 3 Left-side and right-side capture probabilities and survivalprobabilities by year

Year Left capture Right capture Survival

1990 1(11)

1991 54(40 68) 60(47 75) 93(84 99)

1992 74(59 86) 75(60 86) 98(94100)

1993 31(20 44) 46(33 61) 95(88100)

1994 36(24 47) 53(35 66) 95(86100)

1995 42(27 55) 71(56 88) 94(86100)

1996 30(17 42) 48(34 63) 93(82100)

1997 23(14 36) 37(23 51) 88(75 98)

1998 39(24 53) 31(20 45) 94(84100)

1999 45(31 63) 40(25 54) 93(82100)

2000 24(14 36) 57(40 72) 96(88100)

2001 87(78 94) 82(73 91) 96(89100)

2002 88(80 96) 89(80 96)

NOTE For each the mean and the 95 HPDI limits are given There is concordancebetween the left-side and right-side results The capture probabilities are more variableand reach minimums in 1997 and 2000 for the left side and in 1998 for the right sidebut both reach maximums in 2002 Survival reaches a maximum in 1992 declines to aminimum in 1997 and then rises again in later years

coefficients Because this is observed after adjustments for theoverall and time-varying components it suggests an effect dueto behavior or sampling protocol

The variation in the capture probability over time is likelyto result from a combination of factors including variation indolphin ranging patterns (see Wilson et al 2004) weather con-ditions and other logistical factors that influence sampling indifferent areas Most notably sampling protocols were changedin 2000 specifically to increase the probability of capture Thischange has proven to have been successful in the latter part ofthe study while estimates of the survival probabilities and to-tal population size remain unaffected Second another study(Wilson et al 2004) has shown that the ranging pattern of thepopulation changed in recent years with some animals spend-ing more time in the southern part of the population range alongthe east coast of Scotland south of the Moray Firth to St An-drews and the Firth of Forth The effect of this would be toreduce the capture probability in the core study area in the in-ner Moray Firth and to potentially increase heterogeneity incapture probabilities The more southerly areas also have beensampled but the data are more sparse and cover only a smallnumber of years Once these histories have become more exten-sive we expect that the capture probabilities will become lessvariable Unlike other models our estimates are also correctedfor individual heterogeneity Variation in capture probabilitiesalso might have arisen from heterogeneous effort in space Al-though this was not explicitly modeled we could have stratifiedthe capture histories by area by including separate capture co-efficients for each area

The overall mean survival estimate through the study periodis 93 (standard deviation 029 95 HPDI = 861 979) Thisis similar to an earlier maximum likelihood estimate of sur-vival for the cohort of nicked individuals from this populationfirst observed in 1990 (Saunders-Reed Hammond Grellier andThompson 1999) But our model estimate is slightly lower thansurvival estimates from other bottlenose populations of 962 plusmn76 in Sarasota Bay Florida (Wells and Scott 1990) 952 plusmn

Corkrey et al Bayesian Recapture Population Model 955

150 in Doubtful Sound New Zealand (Haase 2000) and 95ndash99 in the Sado estuary Portugal (Gaspar 2003) However thismay be because our study included subadults while the otherstudies reported adult survival Variation in survival probabil-ity occurs over time [Fig 2(b) Table 3] It is of interest thatsurvival reached a minimum in 1997 but later returned to lev-els comparable to earlier years The proportion of iterations inwhich the survival for 1997 are above the mean survival forthe remaining years (mean φ = 95 95 HPDI = 85 100)is only 08 The apparent low survival in 1997 may be due tomigration because this is not distinguishable from survival inthe model It is possible that an emigration event occurred in1997 and that those individuals have not yet returned Consid-erable heterogeneity in survival (range 77ndash97) is observedas shown in Figure 2(c) The survival probabilities are not dis-tributed smoothly suggesting multimodality The animals witha low survival coefficient may have died or migrated In thecase of migration this result would indicate that some animalsless consistently reside in the core areas compared with oth-ers

Individuals with higher survival coefficients are frequentlythose seen earliest in the study There is little information avail-able on individuals seen only more recently and consequentlytheir coefficients are close to 0 Some survival coefficients aremore negative and these correspond to animals that have notbeen seen for some time As time goes on and in the absenceof a confirmed death or sighting these coefficients can be ex-pected to become more negative However some animals thatare observed only on a few occasions have negative coefficientsThe estimation of individual survival coefficients means that es-timation of other coefficients in the model are adjusted for vari-able survival The estimation of individual capture probabilitiesallows the results to be combined with other individual covari-ates such as sex or body size if these data become availableSimilarly where individuals differ in their home ranges indi-vidual capture or survival probabilities could be related to localenvironmental conditions within their range to prey availabil-ity or to water quality

Estimates of the total population size from bilateral analy-sis of the left and right photo-identification series and estimatesfrom unilateral analyses of the left and right sides are shown inFigure 3(a) These are similar both to the maximum likelihoodestimate of 129 (95 confidence interval = 110ndash174) for datacollected in 1992 (Wilson et al 1999) and to a Bayesian mul-tisite mark recapture estimate of the nicked population of 85(95 probability interval = 76ndash263) for data collected in 2001(Durban et al 2005) The credible intervals of the abundanceestimates are generally narrower in the bilateral analysis (meanwidth 886) than in left-side analysis (mean width 1373) andright-side analysis (mean width 937) Figure 3(b) shows theprior and mean posterior recruitment probabilities ωj It is in-teresting that higher probabilities occur in 1996 and 2001 Thismight be considered to variations in capture probability butwhereas the capture probability is high in 2001 it remains lowin 1996 [see Fig 2(a)] Instead calculating the probabilities ofthe posterior exceeding the prior recruitment probabilities wefind they are below 5 (range 04ndash44) for all years except 2001in which it is 85 The higher recruitment probability in 2001may be due to an improved protocol in that year that led to the

(a)

(b)

Figure 3 (a) Mean total abundance estimates from a bilateral analy-sis with 95 HPDI Also shown are estimates from separate analysesof left (minus minus minus) and right (middot middot middot) sides along with associated 95 HPDIrepresented by symbols left + right Minimum numbers of ani-mals known to be alive are indicated by diamonds (b) Mean posteriorrecruitment probabilities ωj with 95 HPDI The prior value is rep-resented as a dotted line

identification of previously unrecognized nicked animals Weconsider the rise in 1996 to have insufficient evidence to sug-gest an increase in recruitment in that year

Earlier demographic modeling of this population has pre-dicted that the population is in decline (Saunders-Reed et al1999) To examine this we calculate the statistic [prodJ

j=2 Wj

(Wjminus1)]1(Jminus1) Values of this statistic gt1 indicate an increas-ing population whereas values lt1 indicates a decreasing pop-ulation The mean value is 989 (95 HPDI = 958 102)and the probability of it being lt10 is 77 This suggests thatthe population may be decreasing but that the evidence fora decline is weak By doing this we are able to use the fulltime series of photo identification data to obtain simultaneousposterior estimates of the Wj to compare the probability of apopulation increase or decrease In agreement with the earlierpredictive modeling these data indicate greater probability of adecline than of an increase However because it is likely thatthe population is expanding its range (Wilson et al 2004) thedecline may be confounded with temporary emigration

956 Journal of the American Statistical Association September 2008

6 DISCUSSION

Changes in the size of wildlife populations are often assessedby fitting trend models to independent abundance estimatesbut the power of these techniques to detect trends often canbe low (Gerrodette 1987 Taylor Martinez Gerrodette Bar-low and Hrovat 2007) Here we use a capturendashrecapture modelthat incorporates an underlying population model and an ob-servation model We provide a probability that a population isin decline or increasing Using photoidentification data fromthis bottlenose dolphin population we generate the first em-pirical estimate of trends in abundance for any of the popula-tions of small cetaceans inhabiting European waters Becausethe methodological changes in sampling likely are absorbedby the time-varying capture coefficient the remaining variationdue to individual capture tendencies might reasonably be de-scribed by the individual capture coefficient This then allowsexploration of capture and survival relating to individuals ratherthan to populations Information on individual survival could beimportant for addressing some management issues for exam-ple individuals differing in their spatial distribution

By decomposing the capture probabilities into overall time-varying and heterogeneous terms we find that the time-varyingcontribution and heterogeneous variation are comparable inmagnitude and their variation is greater than that of the overallterm This information can be used in the design of monitoringprograms by suggesting how effort should be distributed Thedifference in overall left and right capture probabilities mightbe interpretable as an artifact of the sampling techniques or ofbehavioral effects However after allowing for this the prob-ability of observing one side also varied over time with someyears strongly favoring one side or the other This might be ex-plained by variation in protocol between years such as changesin personnel The relative sizes of these effects may be of usein evaluating consistency of protocol over time Finally afterallowing for the overall difference and time variation the het-erogeneous capture coefficients exhibit a bias to the one sidepossibly due to behavior or protocol These findings demon-strate that bilateral decomposition of capture probabilities canprovide useful information

We decompose the survival probability similarly and we findit to be unusually low in 1997 suggesting a possible multimodalheterogeneous survival probability that may be the result of mi-gration or survival variation This type of decomposition allowsexploration of such effects in more detail In particular the mul-timodality of heterogeneous survival suggests that the popula-tion comprises individuals with different strategies some vis-iting the core study area more or less often Ongoing work isattempting to quantify individual heterogeneity such as in ex-posure to sewage contaminated waters and boat-based tourismTherefore this technique could be used to determine whetherexposure to different anthropogenic stressors is related to sur-vival

Our model also makes use of priors such as those on ωj thatare not informed by the recapture data set but make use of in-formation from other sources An alternative approach mightbe to specify these as informative priors based on expert opin-ion or tuned to obtain satisfactory results Our application il-lustrates how disparate data sources may be combined into a

single analysis Although the data source for the prior recruit-ment probability is small and problematic it is the only oneavailable Even when some priors are defined in the simulationstudy to incorrect values the estimates of parameters of interestremain valid The posterior probability also adds to the state ofknowledge of this parameter

Previous use of capturendashrecapture models with cetacean pho-toidentification data highlights the potential violation of modelassumptions due to heterogeneity in capture andor survivalprobabilities This model explicitly accounts for heterogene-ity and also provides simultaneous estimates of individual sur-vival and capture probabilities This can be used both to ex-plore the biological basis of heterogeneity and to improve sam-pling programs The individual survival coefficients also can beused to explore aspects of biology where correlates are avail-able for example the relationship of survival to age sex orother variables may be of interest Abundance under the as-sumption of heterogeneous survival has been estimated by LeeHuang and Ou (2004) whereas individual and time-varyingsurvival estimates have been obtained using band-recoverymodels (Grosbois and Thompson 2005) Other studies have ex-amined survival by group for example Franklin Andersonand Burnham (2002) obtained separate estimates of group-levelsurvival for males and females using stratification in a bandingstudy Although we lack such variables as age and sex theo-retically it would be possible to examine individual survivalby sex either as a post hoc comparison or through their inte-gration into the model itself For example survival and cap-ture could be modeled as logit(φ) = α + βj + γd + τ andlogit(π) = δ + εj + ζd + χ where τ corresponds to stage-specific (eg young immature reproductive female senes-cent) survival and χ corresponds to capture probabilities forvarious categories (eg young reproductive adult)

Other cetacean studies have examined heterogeneity White-head (2001b) obtained an estimate of the coefficient of variation(CV) for identification from a randomization test of successiveidentifications in whale track data (The CV is also a measure ofcapture heterogeneity) They found that the CV was significantin one of the two years and that probably depended on the sizesof the animals being observed This is not comparable to thedolphin observations in which heterogeneity is more likely toresult from differences in animal behavior In the present studywe use a likelihood in which heterogeneity of survival and cap-ture has been incorporated by a logit function This approachwas described previously (Pledger Pollock and Norris 2003Pledger and Schwarz 2002) within a non-Bayesian contextPledger et al (2003) classified animals into an unknown set oflatent classes each of which could have a separate survival andcapture probability A simplification allows for time-varyingand animal-level heterogeneity This approach is similar to oursbut we also integrate a dynamic population model and an ob-servation model into the estimation process Goodman (2004)also implemented a population model in a Bayesian context thatcombines capturendashrecapture data with carcass-recovery databut assumes the absence of heterogeneity in capture probabil-ities Durban et al (2005) estimated abundance for the samepopulation as us but used a multisite Bayesian log-linear modeland model averaging (King and Brooks 2001) across the pos-sible models with varying combinations of interactions Our

Corkrey et al Bayesian Recapture Population Model 957

model could be extended to include multiple sites and yearsby using a capturendashrecapture history that includes site informa-tion and a yearndashsite term in the capture probability expressionBy doing this we suggest that geographic areas with more datacan be used to strengthen estimates from poorly sampled areasThis also may account for some individual variation currentlyidentified as individual capture heterogeneity because some in-dividuals may be more closely associated with some areas thanothers and may attenuate a possible confounding of migrationwith survival

We use a separate data set and a negative-binomial model toinflate the estimate of the proportion of nicked animals in thesamples Another study (Da Silva Rodrigues Leite and Mi-lan 2003) obtained an abundance estimate for bowhead whalesusing an inflation factor derived from the numbers of good andbad photographs of marked and unmarked whales Our analysisavoids the lack of independence between good photographs anddistinctively marked animals in the photographs that might in-validate the estimate of variance in the inflation factor Otherstudies avoid this issue by including only separately catego-rized good-quality photographs (eg Wilson et al 1999 ReadUrian Wilson and Waples 2003) However independent es-timates of inflation when available such as in our study arepreferred

We do not know of other cetacean studies that combinephoto-identification series of the left and right sides but ananalogous approach is that of log-linear capturendashrecapturemodeling (King and Brooks 2001) in which animals are identi-fied from separate lists Some photo-identification studies mightuse information from both sides in the process of identifyinganimals (Joyce and Dorsey 1990) A more robust approach isto perform the identification on each side separately We haveshown how these separate data can then be combined

A major advantage to our modeling approach is its extensibil-ity Within the Bayesian framework models can be comparedby such methods as reversible-jump Markov chain Monte Carlotechniques (Green 1995) Model extensions include extendingthe population model considering an individual-level approachThis would allow incorporation of the heterogeneity into thispart of the model and also modeling of the unobserved popu-lation More detailed modeling of heterogeneity could be ex-plored by including hyperpriors for the variances of the recap-ture model terms For convenience we analyze the data on anannual basis Additional information would be available if theanalyses were conducted on a finer time scale such as monthsor by individual trip This would allow the model to be extendedto include seasonal effects For example dolphins are observedmore frequently in summer months than in winter months andthis tendency could be included in an extended model by chang-ing the capture probability to be logit(π) = δt + εj + ζd whereδt t isin [1 12] is a monthly capture coefficient This maybe of use in modeling fine-scale movements The response vari-able itself also can be generalized Although we are careful toonly include data from high-quality photographs and nicked an-imals there is still the possibility of some identification errorsparticularly for more subtly marked individuals A possible ex-tension of the model could allow identification to be assigneddifferent levels of certainty for example by incorporating anintermediate state between nonrecapture and recapture that we

term ldquopossible recapturerdquo This could be done by replacing thelogit function logit(ν) = log(ν(1 minus ν)) where ν correspondsto the probability of recapture by a proportional odds function(McCullagh 1980) This technique has previously found appli-cation in a study of photoidentification used to model photo-graphic quality scores and individual distinctiveness of hump-back whale tail flukes (Friday Smith Stevick and Allen 2000)In this approach the probability of ldquopossible recapturerdquo is givenby ν1 = π1 the probability of certain recapture is given byν2 = π1 + π2 and includes the possible recapture category andthe probability of no capture is given by ν0 = 1 minus π1 + π2There are two proportional odds functions corresponding topossible recapture and certain recapture which are logit(ν1) =log(π1(π2 +π3)) and logit(ν2) = log((π1 +π2)π3) We thencould extend the recapture model using logit(ν1) = α+βj +γk1

and logit(ν2) = α + βj + γk2 This proportional odds approachmay be particularly useful in situations where automatic match-ing algorithms (Saacutenchez-Mariacuten 2000) provide an estimate ofthe likelihood of a match which could then be incorporatedinto abundance estimate models

APPENDIX DERIVATION OF POSTERIORS

Here we show how we derive the posterior conditional distributionsfor the model parameters and describe how each is updated within theMCMC algorithm The full joint distribution has the form shown in(A1) In (A1) the first two lines correspond to the population modelthe third line corresponds to the inflation model the fourth line cor-responds to the observation model the fifth line corresponds to therecapture model and the remaining lines correspond to priors

π = P(B1|ω1N0W0)P (S1|αN0)

(Sec 31)

timesSprod

j=2

P(Bj |Sjminus1Bjminus1Wjminus1ωj )

times P(Sj |Sjminus1Bjminus1 αβjminus1)

(Sec 31)

timesSprod

j=1

P(Wj |Sj Bj υj )

Tjprod

i=1

P(tnji |υj twji )

(Sec 32)

timesSprod

j=1

P(nLj|δL εjLSj Bj )P (nRj

|δR εjRSj Bj )

(Sec 33)

timesSprod

j=1

P(Hdj |αβj γd δL εjL ζdL δR εjR ζdR)

(Sec 34)

times P(N0|λN)P (W0|N0 υ0)P (α δL δRλN υ0)

(Sec 41)

timesSprod

j=1

P(βj εjL εjRωj υj )

nprod

d=1

P(γd)P (ζdL)P (ζdR)

(Sec 41) (A1)

958 Journal of the American Statistical Association September 2008

For each parameter we obtain the corresponding posterior condi-tional distribution by extracting all terms containing that parameterfrom the full joint distribution When the conditional distributions havea standard form we use the Gibbs sampler (Brooks 1998) to update theparameters otherwise we use MetropolisndashHastings updates (Chib andGreenberg 1995)

Within a MetropolisndashHastings update we propose a new valuefor a parameter x and denote the proposed value by xprime To decidewhether to accept the proposed value we calculate an acceptance ra-tio α = min(1A) where A is the acceptance ratio The acceptanceratio is given by A = π(xprime)q(x)π(x)q(xprime) where π(x) is the poste-rior probability of x and q(x) is the probability of the proposed valueWe accept the new value xprime with probability α otherwise we leavethe value of x unchanged (See Chib and Greenberg 1995 for an exam-ple)

Normal proposal distributions are used for continuous parameterscentered around the current parameter value Discrete uniform pro-posals are used for discrete variables each centered around the currentparameter values The discrete proposals also require constraints to en-sure that the proposed values are valid For example a proposed valuefor N0 must be less than the current value of W0 We provide a detaileddescription of each update next

Population Model

The population model has parameters ωj λ Bj Sj and N0 andare updated using Gibbs or MetropolisndashHastings samplers The condi-tional posterior distributions for ωj and λN can be derived as

ωj |Bj Sjminus1Bjminus1

sim Beta(Bj + 0238Wjminus1 minus Sjminus1 minus Bjminus1 minus Bj + 1)

and

λN |N0 sim Gam(76 + N0101)

These are individually updated using the Gibbs sampler during eachiteration of the Markov chain

The parameters Bj Sj and N0 are updated by the MetropolisndashHastings algorithm using acceptance ratios calculated as described ear-lier We use uniform proposals centered on the current parameter valueand limited to the range of possible values

For the Sj rsquos we use three alternative proposals

Sprimej=1 sim Unif

(max

(0 S1 minus 5 S2 minus B1max(n1Ln1R) minus N1

2 minus B1)

min(S1 + 5N0W1 minus B1W1 minus B1 minus B2

W1 minus B1 minus 1))

Sprime2lejltJ sim Unif

(max

(0 Sj minus 5 Sj+1 minus Bj max(njLnjR) minus Bj

2 minus Bj

)

min(Sj + 5 Sjminus1 + Bjminus1Wj minus Bj Wj minus Bj minus Bj+1

Wj minus Bj minus 1))

and

Sprimej=J sim Unif

(max

(0 SJ minus 5max(nJLnJR) minus BJ 2 minus NJ

)

min(SJ minus 5 Sjminus1 + Bjminus1Wj minus Bj Wj minus Bj minus 1))

For Bj we use three alternative proposals

B prime1 sim Unif

(max

(0B1 minus 15 S2 minus S1max(n1Ln1R) minus S1

2 minus S1)

min(B1 + 15W0 minus N0W1 minus S1W1 minus S1 minus B2

W1 minus S1 minus 1))

B prime2lejltJ sim Unif

(max

(0Bj minus 15 Sj+1 minus Sj max(njLnjR) minus Sj

2 minus Sj

)

min(Bj + 15Wjminus1 minus Sjminus1 minus Bjminus1Wj minus Sj

Wj minus Sj minus Bj+1Wj minus Sj minus 1))

and

B primeJ sim Unif

(max

(0BJ minus 15max(nJLnJR) minus SJ 2 minus SJ

)

min(BJ + 15WJminus1 minus SJminus1 minus BJminus1

WJ minus SJ WJ minus SJ minus 1))

For N0 we use

N prime0 sim Unif

(max(5N0 minus 15 S1)min(N0 + 15W0 minus B1W0 minus 1)

)

Inflation Model

The inflation model parameters Wj and υj are updated usingMetropolisndashHastings and Gibbs updates The conditional posterior dis-tributions for υj can be derived as

υ0|N0W0 sim Beta(N0 + 1W0 minus N0 + 1)

and

υjgt0|Sj Bj Wj tnji twji sim Beta

(

Sj + Bj +T (j)sum

i=1

tnji + 1

Wj minus Sj minus Bj +T (j)sum

i=1

twji minus tnji + 1

)

These are updated separately using the Gibbs sampler on each itera-tion

The remaining parameter Wj has a nonstandard conditional pos-terior distribution We use larger updates than in the population modelbecause the initial total population is expected to be larger in size

W prime0 sim Unif

(max(0W0 minus 15N0 + 1N0 + B1)W0 + 15

)

W prime1lejltJ sim Unif

(max(0Wj minus 15 Sj + Bj + 1 Sj + Bj + Bj+1)

Wj + 15)

and

W primeJ sim Unif

(max(0WJ minus 15 SJ + BJ + 1)WJ + 15

)

Recapture Model

For the recapture model we have parameters α βj γd δL εjLζdL δR εjR and ζdR These all have nonstandard conditional pos-terior distributions and we use the MetropolisndashHastings algorithm toupdate the parameters on each iteration We use individual normal dis-tribution proposals centered on the current parameter values and withvariance equal to the prior variance for example

αprime sim N(α185)

[Received June 2005 Revised May 2007]

Corkrey et al Bayesian Recapture Population Model 959

REFERENCES

Brooks S P (1998) ldquoMarkov Chain Monte Carlo Method and Its ApplicationrdquoThe Statistician 47 69ndash100

Brooks S P and Roberts G O (1998) ldquoConvergence Assessment Techniquesfor Markov Chain Monte Carlordquo Statistics and Computing 8 319ndash335

Buckland S T (1982) ldquoA CapturendashRecapture Survival Analysisrdquo Journal ofAnimal Ecology 51 833ndash847

(1990) ldquoEstimation of Survival Rates From Sightings of IndividuallyIdentifiable Whalesrdquo in Individual Recognition of Cetaceans Use of Photo-Identification and Other Techniques to Estimate Population Parameters edsP S Hammond S A Mizroch and G P Donovan Cambridge UK Inter-national Whaling Commission pp 149ndash153

Burnham K and Overton W (1979) ldquoRobust Estimation of Population SizeWhen Capture Probabilities Vary Among Animalsrdquo Ecology 60 927ndash936

Carothers A (1979) ldquoQuantifying Unequal Catchability and Its Effect on Sur-vival Estimates in an Actual Populationrdquo Journal of Animal Ecology 48863ndash869

Chao A (1987) ldquoEstimating the Population Size for CapturendashRecapture DataWith Unequal Catchabilityrdquo Biometrics 43 783ndash791

Chao A Lee S-M and Jeng S-L (1992) ldquoEstimating Population Size forCapturendashRecapture Data When Capture Probabilities Vary by Time and Indi-vidual Animalrdquo Biometrics 48 201ndash216

Chib S and Greenberg E (1995) ldquoUnderstanding the MetropolisndashHastingsAlgorithmrdquo The American Statistician 49 327ndash335

Cormack R (1972) ldquoThe Logic of CapturendashRecapture Experimentsrdquo Biomet-rics 28 337ndash343

Coull B A and Agresti A (1999) ldquoThe Use of Mixed Logit Models to Re-flect Heterogeneity in CapturendashRecapture Studiesrdquo Biometrics 55 294ndash301

Da Silva C Q Rodrigues J Leite J G and Milan L A (2003) ldquoBayesianEstimation of the Size of a Closed Population Using Photo-ID With Part ofthe Population Uncatchablerdquo Communications in Statistics 32 677ndash696

Dorazio R M and Royle J A (2003) ldquoMixture Models for Estimating theSize of a Closed Population When Capture Rates Vary Among IndividualsrdquoBiometrics 59 351ndash364

Durban J Elston D Ellifrit D Dickson E Hammon P and Thompson P(2005) ldquoMultisite Mark-Recapture for Cetaceans Population Estimates WithBayesian Model Averagingrdquo Marine Mammal Science 21 80ndash92

Fienberg S E Johnson M S and Junker B W (1999) ldquoClassical Multi-level and Bayesian Approaches to Population Size Estimation Using MultipleListsrdquo Journal of the Royal Statistical Society Ser A 162 383ndash405

Forcada J and Aguilar A (2000) ldquoUse of Photographic Identification inCapturendashRecapture Studies of Mediterranean Monk Sealsrdquo Marine MammalScience 16 767ndash793

Franklin A B Anderson D R and Burnham K P (2002) ldquoEstimation ofLong-Term Trends and Variation in Avian Survival Probabilities Using Ran-dom Effects Modelsrdquo Journal of Applied Statistics 29 267ndash287

Friday N Smith T D Stevick P T and Allen J (2000) ldquoMeasure-ment of Photographic Quality and Individual Distinctiveness for the Photo-graphic Identification of Humpback Whales Megaptera novaeangliaerdquo Ma-rine Mammal Science 16 355ndash374

Gaspar R (2003) ldquoStatus of the Resident Bottlenose Dolphin Population inthe Sado Estuary Past Present and Futurerdquo unpublished doctoral dissertationUniversity of St Andrews School of Biology

Gelfand A E and Sahu S K (1999) ldquoIdentifiability Improper Priors andGibbs Sampling for Generalized Linear Modelsrdquo Journal of the AmericanStatistical Association 94 247ndash253

Gerrodette T (1987) ldquoA Power Analysis for Detecting Trendsrdquo Ecology 681364ndash1372

Goodman D (2004) ldquoMethods for Joint Inference From Multiple Data Sourcesfor Improved Estimates of Population Size and Survival Ratesrdquo Marine Mam-mal Science 20 401ndash423

Green P J (1995) ldquoReversible-Jump Markov Change Monte Carlo BayesianModel Determinationrdquo Biometrika 82 711ndash732

Grosbois V and Thompson P M (2005) ldquoNorth Atlantic Climate VariationInfluences Survival in Adult Fulmarsrdquo Oikos 109 273ndash290

Haase P (2000) ldquoSocial Organisation Behaviour and Population Parametersof Bottlenose Dolphins in Doubtful Sound Fiordlandrdquo unpublished mastersdissertation University of Otago Dept of Marine Science

Hammond P (1986) ldquoEstimating the Size of Naturally Marked Whale Pop-ulations Using CapturendashRecapture Techniquesrdquo Report of the InternationalWhaling Commission Special Issue 8 253ndash282

(1990) ldquoHeterogeneity in the Gulf of Maine Estimating HumpbackWhale Population Size When Capture Probabilities Are not Equalrdquo in In-dividual Recognition of Cetaceans Use of Photo-Identification and OtherTechniques to Estimate Population Parameters eds P S Hammond S A

Mizroch and G P Donovan Cambridge UK International Whaling Com-mission pp 135ndash153

Huggins R (1989) ldquoOn the Statistical Analysis of Capture Experimentsrdquo Bio-metrika 76 133ndash140

Joyce G G and Dorsey E M (1990) ldquoA Feasibility Study on the Useof Photo-Identification Techniques for Southern Hemisphere Minke WhaleStock Assessmentrdquo in Individual Recognition of Cetaceans Use of Photo-Identification and Other Techniques to Estimate Population Parameters edsP S Hammond S A Mizroch and G P Donovan Cambridge UK Inter-national Whaling Commission pp 419ndash423

King R and Brooks S (2001) ldquoOn the Bayesian Analysis of PopulationSizerdquo Biometrika 88 317ndash336

Lebreton J-D Burnham K P Clobert J and Anderson S R (1992)ldquoModeling Survival and Testing Biological Hypotheses Using Marked An-imals A Unified Approach With Case Studiesrdquo Ecological Monographs 6267ndash118

Lee S-M Huang L-H H and Ou S-T (2004) ldquoBand Recovery Model In-ference With Heterogeoneous Survival Ratesrdquo Statistica Sinica 14 513ndash531

Link W A and Barker R J (2005) ldquoModeling Association Among De-mographic Parameters in Analysis of Open Population CapturendashRecaptureDatardquo Biometrics 61 46ndash54

Mann J and Smuts B (1999) ldquoBehavioral Development in Wild BottlenoseDolphin Newborns (Tursiops sp)rdquo Behaviour 136 529ndash566

McCullagh P (1980) ldquoRegression Models for Ordinal Datardquo Journal of theRoyal Statistical Society Ser B 42 109ndash142

Otis D L Burnham K P White G C and Anderson D R (1978) ldquoStatis-tical Inference From Capture Data on Closed Animal Populationsrdquo WildlifeMonographs 62 135

Pledger S (2000) ldquoUnified Maximum Likelihood Estimates for ClosedCapturendashRecapture Models Using Mixturesrdquo Biometrics 56 434ndash442

Pledger S and Schwarz C J (2002) ldquoModelling Heterogeneity of Survivalin Band-Recovery Data Using Mixturesrdquo Journal of Applied Statistics 29315ndash327

Pledger S Pollock K H and Norris J L (2003) ldquoOpen CapturendashRecaptureModels With Heterogeneity I CormackndashJollyndashSeber Modelrdquo Ecology 59786ndash794

Pollock K H Nichols J D Brownie C and Hines J E (1990) ldquoStatisticalInference for CapturendashRecapture Samplingrdquo Wildlife Monographs 107 1ndash97

Read A Urian K W Wilson B and Waples D (2003) ldquoAbundance andStock Identity of Bottlenose Dolphins in the Bayes Sounds and Estuaries ofNorth Carolinardquo Marine Mammal Science 19 59ndash73

Saacutenchez-Mariacuten F J (2000) ldquoAutomatic Recognition of Biological ShapesWith and Without Representations of Shaperdquo Artificial Intelligence in Medi-cine 18 173ndash186

Saunders-Reed C A Hammond P S Grellier K and Thompson P M(1999) ldquoDevelopment of a Population Model for Bottlenose Dolphinsrdquo Re-search Survey and Monitoring Report 156 Scottish Natural Heritage

Scottish Natural Heritage (1995) Natura 2000 A Guide to the 1992 EC Habi-tats Directive in Scotlandrsquos Marine Environment Perth UK Scottish Nat-ural Heritage

Seber G (2002) The Estimation of Animal Abundance Caldwell NJ Black-burn Press

Stevick P T Palsboslashll P J Smith T D Bravington M V and Ham-mond P S (2001) ldquoErrors in Identification Using Natural Markings RatesSources and Effects on CapturendashRecapture Estimates of Abundancerdquo Cana-dian Journal of Fisheries and Aquatic Science 58 1861ndash1870

Taylor B L Martinez M Gerrodette T Barlow J and Hrovat Y N (2007)ldquoLessons From Monitoring Trends in Abundance of Marine Mammalsrdquo Ma-rine Mammal Science 23 157ndash175

Wells R S and Scott M D (1990) ldquoEstimating Bottlenose Dolphin Popu-lation Parameters From Individual Identification and CapturendashRelease Tech-niquesrdquo in Individual Recognition of Cetaceans Use of Photo-Identificationand Other Techniques to Estimate Population Parameters eds P S Ham-mond S A Mizroch and G P Donovan Cambridge UK InternationalWhaling Commission pp 407ndash415

Whitehead H (2001a) ldquoAnalysis of Animal Movement Using Opportunis-tic Individual Identifications Application to Sperm Whalesrdquo Ecology 821417ndash1432

(2001b) ldquoDirect Estimation of Within-Group Heterogeneity in Photo-Identification of Sperm Whalesrdquo Marine Mammal Science 17 718ndash728

Williams B K Nichols J D and Conroy M J (2001) Analysis and Man-agement of Animal Populations San Diego Academic Press

Wilson B Hammond P S and Thompson P M (1999) ldquoEstimating Size andAssessing Trends in a Coastal Bottlenose Dolphin Populationrdquo EcologicalApplications 9 288ndash300

Wilson B Reid R J Grellier K Thompson P M and Hammond P S(2004) ldquoConsidering the Temporal When Managing the Spatial A Popula-

950 Journal of the American Statistical Association September 2008

Figure 1 Example of dorsal fin showing nicks to the trailing edge Insets show the changes in the dorsal fin for this animal over time withthe earliest photograph on the left and the latest on the right The background image corresponds to the third inset from the right Apparentchanges in shape of the inset images are the result of differences in camera perspective The effect of these nicks is to produce a serrated edgethe whole of which is used to confirm the identity of the animal rather than just a selected few nicks Other markings such as skin lesions arenot considered reliable enough to be used in this study Minor changes in some nicks may occur over time but as long as other nicks on thedorsal edge remain intact identification can proceed

31 Modeling the Nicked Population

We begin with the assumption that young animals are ini-tially unnicked but that as they mature they will accumu-late nicks as they increasingly engage in social interactionsThis is a reasonable assumption in light of the social develop-ment of juvenile bottlenose dolphins (Mann and Smuts 1999Haase 2000) The animals can be considered as being recruitedfrom the unnicked population to the nicked population Once inthe nicked population individuals become observable and can-not return to the unmarked population

We denote the number of dolphins recruited from the un-nicked population to the nicked population at time 1 le j le J

as Bj and the number of nicked animals that survived betweentimes j minus 1 and j as Sj Thus the nicked population size attime j is given by Sj + Bj that is the current population com-prises new recruits and surviving former members The nickedpopulation is denoted by Nj = Sj + Bj

We assume that previously nicked animals survive from timej minus1 to time j with probability θj = exp(α+βj )(1+exp(α+βj )) where α denotes a global survival tendency and βj de-notes a time-varying survival component Thus the number ofsurviving animals Sj is given a binomial distribution so thatSj sim Bin(Sjminus1 + Bjminus1 θjminus1)

We also assume that unnicked animals acquire nicks withprobability ωj so that the number of newly nicked animals has

a binomial distribution Bj sim Bin(Wjminus1 minus Njminus1 ωj ) where attime j Wj denotes the total population and Nj denotes thenicked population Given this definition ωj is the probabilityof acquiring nicks and also surviving We believe that the num-ber of newly nicked animals is relatively small however

We refer to this component of the model as the popula-tion model It provides the probability distributions P(Sj |Sjminus1

Bjminus1 θjminus1) and P(Bj |Wjminus1Njminus1ωj )

32 Modeling the Total Population

In this case study the total population size and its variationover time is the primary statistic of interest but the total pop-ulation comprises both nicked and unnicked animals We canhowever estimate the size of the total population from the sizeof the nicked population using data on the proportion of ani-mals in each group that have nicked dorsal fins This is possiblebecause although unnicked individuals cannot be reliably iden-tified over periods of months or years they are recognizableover a period of days (see Wilson et al 1999) This allows usto count the number of unnicked individuals captured on high-quality photographs in any one encounter and to estimate theproportion of nicked and unnicked animals in the group to pro-vide an inflation factor

Then the unnicked part of the total population is given byWj minus Nj If the probability of an individual having nicks is

Corkrey et al Bayesian Recapture Population Model 951

given by υj then the size of the unnicked population at timej can be modeled as a negative binomial distribution condi-tional on the size of the nicked population so that Wj minus Nj simNegBin(Nj υj ) This approach allows the use of an indepen-dent data set to inform the υj rather than relying on the re-capture data to estimate this parameter The negative binomialdistribution is preferred over a binomial distribution becausethe Nj rsquos already have been assigned a distribution in the popu-lation model

In each time period j suppose that there are Tj trips eachobserving twji animals for i = 1 Tj of which tnji are nickedNumerous trips are undertaken each year and the numbernicked is separately estimated on each encounter A binomialdistribution then can be used to describe the number of nickedanimals seen per trip so that tnji sim Bin(twji υj ) Thus the ob-servations tnji and twji allow us to estimate υj and thus the totalpopulation size given an estimate of the nicked population Werefer to this as the inflation model It provides the probabilitydistributions P(Wj |Sj Bj υj ) and P(tnji |twji υj )

33 The Observation Model

In any time period only a portion of the nicked populationis seen Examination of the data suggests that the left and rightcapture probabilities differ at the population level The prob-ability that an individual animal would be seen from the leftor right side s isin [LR] is given by ρjs = exp(δs + εjs)(1 +exp(δs + εjs)) where δs denotes a global capture tendency andεjs denotes a time-varying capture component

Then the number of nicked animals seen from side s njs fol-lows a binomial distribution njs sim Bin(Nj ρjs) where njL =sum

d hdj isin [13] and njR = sumd hdj isin [23] We refer to this

model as the observation model It provides the probability dis-tributions P(njL|Nj ρjL) and P(njR|Nj ρjR)

We consider the njL and njR as independent counts A dol-phin seen in one year from one side contributes to the prob-ability of observing that side and one seen from both sidescontributes to both probabilities The introduction of captureprobabilities for each side allows the combination of the twophotographic series that otherwise would have been analyzedseparately because they would be correlated It is also of inter-est to estimate these probabilities to assess the capture methodsin use We include the term εjs to allow for observed differ-ences between years in observing all dolphins from one sideversus the other These differences may have arisen from dif-ferences in boat work

34 The Recapture Model

The population and observation models require estimates ofthe parameters α βj δL εjL δR and εjR corresponding tothe survival and capture probabilities These probabilities canbe obtained from the capture history data To do this we cal-culate the likelihood of the entire capture history matrix whichis the product of the probabilities associated with each animalrsquosindividual history L(hdj )

To combine the photo-identification series from the left andright sides into a single bilateral analysis the model allows dif-ferent capture probabilities when animals are seen from eachside The number of animals seen also can differ between the

left and right sides Other parameters do not depend on the sidefrom which an animal is observed for example the populationsize and survival probability of an animal should not depend onwhich side is observed However the left-side capture proba-bility for individual animals might be expected to differ fromthat of the right side for example the animal might be morelikely to approach the boat from one side than from the otherDolphins are also well known to have heterogeneous behavior(Wilson Reid Grellier Thompson and Hammond 2004)

Thus to incorporate heterogeneity as well as observationfrom two sides survival and capture probabilities are definedas logit(φdj ) = α + βj + γd and logit(πdjs) = δs + εjs + ζds where γd denotes a survival effect ζds denotes a recapture ef-fect for animal d s isin [LR] for the left or right side andlogit(x) = log(x(1 minus x)) The logit function is the standardchoice for binary data

Note that the heterogeneity effects are not used in the popu-lation and observation models because both of these are basedon both the nicked and unnicked populations and members ofthe latter will not have an individual effect parameter Althoughwe could adopt a random-effects model to ascribe individualeffects to unnicked animals this would imply that we believethe heterogeneity of survival and recapture to be the same inboth the nicked and unnicked populations which may not bethe case

The overall probability has components for animals in twosituations There are nJ animals observed at the final time andsome nJ lowast animals last observed at some earlier time and thatmight have died or simply are not seen Each individual historyprobability is conditional on the first time that the animal isseen j1

d Thus the likelihood associated with the histories of n = nJ +

nJ lowast dolphins is given by

prod

sisin[LR]

nprod

d=1

Jprod

j=1

L(hdjs)

=nJprod

d=1

Jprod

j=j1d +1

φdjminus1I (πdjLπdjRhdjs)

timesnJ +nJlowastprod

d=nJ +1

hlprod

j=j1d +1

φdjminus1I (πdjLπdjRhdjs)

(

1 minus φdhl

+Jsum

j=hl+1

φdj

jprod

k=hl+1

φdkminus1(1 minus πdkL)(1 minus πdkR)

)

(1)

in which φdj denotes the survival probability φdj = 1 minus φdj

when j lt J and φdj = 1 when j = J πdjs denotes the recap-ture probability from animal d observed in time j from sides isin [LR] j1

d and hl denote the first and last times that individ-ual d is seen and I (πdjLπdjRhdjs) = (1 minus πdjL)(1 minus πdjR)

when hdjs = 0 I (πdjLπdjRhdjs) = πdjL(1 minus πdjR) whenhdjs = 1 and I (πdjLπdjRhdjs) = (1 minus πdjL)πdjR whenhdjs = 2 I (πdjLπdjRhdjs) = πdjLπdjR when hdjs = 3 Theφdj and πdjs parameters are defined only for 1 le j lt J minus1 and2 le j lt J The values of nJ j1

d and hl are determined fromthe capture histories

952 Journal of the American Statistical Association September 2008

We refer to this model as the recapture model It yieldsthe probability P(hdjs |φdj γd πdjLπdjR ζdL ζdR) This isa CormackndashJollyndashSeber model which has been discussed else-where (Seber 2002 Lebreton Burnham Clobert and Anderson1992 Link and Barker 2005)

This completes the model specification To summarize wedefine a population model that describes the recruitment (ωj )and survival (θj ) of animals in the nicked population (Nj ) Therelative sizes of the nicked population (Nj ) and total population(Wj ) are described by the inflation model which makes use ofobserved counts of nicked and unnicked animals Using the cap-ture probability (ρjs ) and the observed number of nicked ani-mals we estimate the nicked population Finally we estimatethe heterogeneous capture (πdjs ) and survival (φdj ) probabili-ties from the capture histories Because this is a Bayesian analy-sis we also specify priors to the following parameters S0 B0ωj W0 N0 α βj γd δs εjs and ζds We describe the priorsin detail in the next section

4 ANALYSIS

In this section we describe the statistical methodology re-quired to analyze the model described in the previous sectionWe adopt a Bayesian approach and describe the prior specifica-tion and simulation techniques required here In this modelingprocess we use a Bayesian approach to allow for uncertaintyin measurement and parameters used The Bayesian approachallows for simultaneous consideration of all data and modelsallowing for the proper propagation of uncertainty throughoutthe model Identifiability is also less of a concern in a Bayesiananalysis in which prior information is supplied (Gelfand andSahu 1999) Moreover the Bayesian approach has an advan-tage in that MCMC methods can be used which greatly simpli-fies the computation compared with the corresponding classi-cal tools The remainder of this section discusses the process ofprior specification and provides details of the implementation

41 Priors

The first stage in the Bayesian analysis is to elicit priors foreach of the model parameters We do so here by grouping theparameters according to the component of the model in whichthey appear The priors used and their associated parameters aresummarized in what follows

The Population Model The model specifies the distribu-tion of parameters Sj and Bj in terms of Sjminus1 and Bjminus1 forj ge 1 Therefore we need a prior for S0 and B0 Becausethey only appear together we set N0 = S0 + B0 and assigna Poisson prior N0 sim Po(λN) Its mean takes a gamma priorλN sim Gam(76 01) with parameters selected to obtain a meanfor the nicked population of 76 as estimated by Wilson et al(1999) and a large variance so that it becomes uninformativecompared with the range of reasonable population values

Because the recruitment rates of nicked dolphins cannot bedirectly estimated from the photo-identification data set thepriors are parameterized with results from a separate analy-sis of randomly selected photographs with at least 2 years ofdata Because it is thought that the rate of nicking might begreater for older animals due to increased social interactionsestimates are estimated from photographs of individuals that

are first identified as known-age calves Identification is doneusing all attributes such as fin shape and the presence of le-sions Photographs of seven of these animals are available forestimating the age at which these dolphins are first seen to havea nick in the dorsal fin We consider these cases a random se-lection from the unnicked population so that the mean num-ber of nicks per year gives an approximate probability of nick-ing per year We set the prior for ωj to be reasonably vagueωj sim Beta(02381) Although rough these assumptions givea crude estimate for the parameter which is then updated bythe data

The Inflation Model The initial population size is assigneda negative-binomial prior W0 minusN0 sim NegBin(N0 υ0) The ini-tial inflation factor υ0 is assigned a beta prior υ0 sim Beta(11)which is equivalent to a noninformative uniform prior

The Recapture Model The recapture model survival pa-rameters have independent normal priors N(0 185) and thecapture parameters have independent normal priors N(0 15)These priors span the range (01) on the logit scale Largervariances are not used because these would obtain a U-shapeddistribution on the logit scale

42 Implementation

Once priors are assigned the full joint distribution can bedetermined as the product of the likelihood with the associatedprior distributions The resulting posterior is complex and high-dimensional and so inference is obtained in the form of poste-rior means and variances obtained through MCMC simulationWe choose to use an implementation in which each parameter isupdated in turn using either a MetropolisndashHastings or a Gibbsupdate depending on the form of the associated posterior con-ditional distributions The updating strategy is detailed in theAppendix All simulation software is written in FORTRAN 77The model was run for 500000 iterations and a 50 burn-in was used Sensitivity studies and standard diagnostic tech-niques were used (Brooks and Roberts 1998) to assess modelvalidity

43 Simulation Study

To test the MCMC code and to demonstrate the utility of theframework developed earlier we construct two artificial pop-ulations one with homogeneous capture and survival proba-bilities and the other with heterogeneous capture and survivalprobabilities These populations are constructed using the fol-lowing procedure An initial population of animals (n = 130)is sampled and a proportion of this initial population is ran-domly marked as nicked At each time period in the simulationanimals are randomly selected to die some of the unnicked be-come nicked and a proportion of the remaining animals arerandomly selected to be observed For each population threesimulations are run with differing priors for the ωj and λN para-meters The priors are such that the three cases represent vaguepriors precise priors and very precise priors but in each casethe means differ from the actual population values One possi-ble outcome is that the precise prior simulations track the priormeans more closely than the vague simulations

The simulations are illustrated in Table 1 which shows theactual total abundances the medians of the posteriors for Wj

Corkrey et al Bayesian Recapture Population Model 953

Table 1 Actual and estimated median total abundances for two artificial populations with 95 HPDI and posterior p values fromthree simulations on each with differing priors

A B C

Population Year Wj Wj p value Wj p value Wj p value

1 1990 124 119(99139) 28 112(97130) 07 107(90124) 021991 120 125(105146) 67 118(95134) 39 114(97135) 251992 114 116(96135) 54 110(93127) 30 106(92123) 171993 109 113(95134) 63 108(92124) 41 105(90121) 261994 103 111(93130) 81 106(88124) 63 103(86118) 491995 97 105(89125) 86 102(87118) 74 100(85114) 621996 95 98(84115) 66 95(81110) 48 93(81107) 381997 91 99(83114) 85 96(80109) 76 95(80108) 651998 89 94(79110) 73 91(77105) 56 90(77103) 521999 85 92(77108) 78 89(75103) 69 87(75103) 632000 76 82(6598) 76 79(6696) 65 79(6594) 592001 70 79(6299) 84 77(5895) 79 76(6095) 752002 68 76(5593) 78 75(5691) 75 73(5892) 70

2 1990 124 135(100178) 71 121(96159) 42 116(90149) 261991 122 123(90158) 51 112(87140) 21 108(84134) 121992 119 114(87148) 35 105(84130) 10 101(78124) 061993 111 106(82138) 34 99(75121) 11 95(74117) 061994 106 102(76130) 33 93(75118) 12 91(68115) 071995 100 96(70122) 38 91(74114) 16 88(67108) 081996 93 93(66119) 48 87(69109) 26 85(65105) 161997 87 91(67115) 60 85(68106) 37 82(62102) 281998 77 79(59101) 56 76(6295) 39 72(5791) 281999 74 84(60108) 80 80(62101) 75 77(5897) 622000 72 78(5799) 67 74(5894) 58 72(5591) 462001 67 71(4992) 62 69(4891) 57 66(4688) 432002 64 68(4191) 59 65(4492) 51 63(4388) 44

NOTE Population 1 has homogeneous capture probability (75) and survival probability (95) while population 2 has heterogeneous capture probability (mean 55 range 05ndash98) andsurvival probability (mean 92 range 68ndash99) The simulations have different priors for ωj and λN Simulation run A has vague priors [ω sim Beta(029412368) λN sim Gam(76 01)]B has more precise priors [ω sim Beta(023710) λN sim Gam(761)] and C has very precise priors [ω sim Beta(929939155) λN sim Gam(7600100)] Their means are the same ineach case (023 and 76) but each differs from the actual population values (06 and 106)

their 95 highest posterior density intervals (HPDI) and a p

value The p values are calculated by counting the proportionof iterations in which the Wj exceeded the true value Examina-tion of the results indicate that the median posterior estimatesgenerally agree with the actual values The agreement is sat-isfactory whether or not there is heterogeneity assumed Thep values are in the range 02ndash88 suggesting that the modelworks correctly The posterior p values for the recapture modelcoefficients are given in Table 2 The p values for α δL andδR are summarized and those for the remaining parameters aresummarized as the proportion of p values in each that lie insidethe range 01ndash99 Almost all p values are within the range 01ndash99 the only exceptions are the capture coefficients for a smallnumber of dolphins

In general the total population estimates for the homoge-neous case are slightly greater and the 95 HPDI is narrowerthen the heterogeneous case but all intervals contain the popu-lation values The estimates also shrink slightly for the mod-els with more precise priors However the p values in eachcase suggest an adequate fit even though some priors are de-liberately set to incorrect values In the case of the recapturecoefficients there is no apparent difference between the homo-geneous and heterogeneous models or with any trend in theprecision of the priors

5 RESULTS

We obtain estimates of heterogeneous capture and survivalcoefficients variation of capture and survival over time over-all survival and capture probabilities the abundance and thetrend in abundance The survival probabilities and capture co-efficients from the left and right sides are shown in Figure 2 andTable 3 We can obtain an overall capture probability for the twosides by calculating exp(δ)(1+exp(δ)) The mean overall cap-

Table 2 Summary of posterior p values for survival and capturecoefficients for two artificial populations from three simulations

on each with differing priors

Population α βj γd δL εjL ζdL δR εjR ζdR

1 16 100 100 66 100 98 60 100 10010 100 100 59 100 98 53 100 10008 100 100 59 100 98 48 100 100

2 34 100 100 17 100 97 50 100 9822 100 100 15 100 97 37 100 9923 100 100 12 100 97 39 100 99

NOTE The populations and simulations are described in Table 1 The p values are calcu-lated as the proportion of iterations greater than the actual population value The p valuesfor α δL and δR are shown Because the remaining parameters are vectors of coefficientsthose p values are summarized as the proportion of p values in each that are found insidethe range 01ndash99

954 Journal of the American Statistical Association September 2008

(a)

(b) (c)

Figure 2 (a) Variation of mean capture coefficients over time forthe left side εL (minusminus) and right side εR (- - bull - - bull) The two hor-izontal lines indicate the global mean capture coefficients δL (mdashmdash)and δR (minus minus minus) On the left vertical axis are the mean left-side het-erogeneity capture coefficients ζL and on the right vertical axis arethe mean right-side heterogeneity capture coefficients ζR Standardserrors for each coefficient are shown as vertical bars (b) Variation ofmean survival probability over time (c) Histogram of mean heteroge-neous survival probabilities

ture probability for the left side is 48 (95 HPDI = 29 68)and for the right side it is 58 (95 HPDI = 39 77) Althoughthis is a small difference the probability of the right capturecoefficient exceeding the left is 81 Examination of Figure 2(a)shows that the variation of capture coefficients over time is largecompared with the overall coefficients and that the left (εL) andright (εR) coefficients tend to track each other The probabilityof the right capture coefficient exceeding the left varies overa wide range 07ndash96 This range results from some years inwhich the probability tends strongly to the left side (1995 and2000 probability of observing the left side 95) or right side(1998 1999 2001 probability of observing the right side 91)Finally the range of the heterogeneous coefficients is compara-ble to that of the time-varying coefficients

The mean heterogeneous survival coefficient is not corre-lated with either of the mean capture heterogeneous coeffi-cients (right r = 009 left r = 022) However the meanleft and right heterogeneous capture coefficients are correlated(r = 714) indicating that animals that are more likely to beseen from one side also are more likely to be seen from theother side However the heterogeneity capture coefficient of theright side (mean 56) exceeded that of the left side (mean 23)with 85 of animals having higher right coefficients than left

Table 3 Left-side and right-side capture probabilities and survivalprobabilities by year

Year Left capture Right capture Survival

1990 1(11)

1991 54(40 68) 60(47 75) 93(84 99)

1992 74(59 86) 75(60 86) 98(94100)

1993 31(20 44) 46(33 61) 95(88100)

1994 36(24 47) 53(35 66) 95(86100)

1995 42(27 55) 71(56 88) 94(86100)

1996 30(17 42) 48(34 63) 93(82100)

1997 23(14 36) 37(23 51) 88(75 98)

1998 39(24 53) 31(20 45) 94(84100)

1999 45(31 63) 40(25 54) 93(82100)

2000 24(14 36) 57(40 72) 96(88100)

2001 87(78 94) 82(73 91) 96(89100)

2002 88(80 96) 89(80 96)

NOTE For each the mean and the 95 HPDI limits are given There is concordancebetween the left-side and right-side results The capture probabilities are more variableand reach minimums in 1997 and 2000 for the left side and in 1998 for the right sidebut both reach maximums in 2002 Survival reaches a maximum in 1992 declines to aminimum in 1997 and then rises again in later years

coefficients Because this is observed after adjustments for theoverall and time-varying components it suggests an effect dueto behavior or sampling protocol

The variation in the capture probability over time is likelyto result from a combination of factors including variation indolphin ranging patterns (see Wilson et al 2004) weather con-ditions and other logistical factors that influence sampling indifferent areas Most notably sampling protocols were changedin 2000 specifically to increase the probability of capture Thischange has proven to have been successful in the latter part ofthe study while estimates of the survival probabilities and to-tal population size remain unaffected Second another study(Wilson et al 2004) has shown that the ranging pattern of thepopulation changed in recent years with some animals spend-ing more time in the southern part of the population range alongthe east coast of Scotland south of the Moray Firth to St An-drews and the Firth of Forth The effect of this would be toreduce the capture probability in the core study area in the in-ner Moray Firth and to potentially increase heterogeneity incapture probabilities The more southerly areas also have beensampled but the data are more sparse and cover only a smallnumber of years Once these histories have become more exten-sive we expect that the capture probabilities will become lessvariable Unlike other models our estimates are also correctedfor individual heterogeneity Variation in capture probabilitiesalso might have arisen from heterogeneous effort in space Al-though this was not explicitly modeled we could have stratifiedthe capture histories by area by including separate capture co-efficients for each area

The overall mean survival estimate through the study periodis 93 (standard deviation 029 95 HPDI = 861 979) Thisis similar to an earlier maximum likelihood estimate of sur-vival for the cohort of nicked individuals from this populationfirst observed in 1990 (Saunders-Reed Hammond Grellier andThompson 1999) But our model estimate is slightly lower thansurvival estimates from other bottlenose populations of 962 plusmn76 in Sarasota Bay Florida (Wells and Scott 1990) 952 plusmn

Corkrey et al Bayesian Recapture Population Model 955

150 in Doubtful Sound New Zealand (Haase 2000) and 95ndash99 in the Sado estuary Portugal (Gaspar 2003) However thismay be because our study included subadults while the otherstudies reported adult survival Variation in survival probabil-ity occurs over time [Fig 2(b) Table 3] It is of interest thatsurvival reached a minimum in 1997 but later returned to lev-els comparable to earlier years The proportion of iterations inwhich the survival for 1997 are above the mean survival forthe remaining years (mean φ = 95 95 HPDI = 85 100)is only 08 The apparent low survival in 1997 may be due tomigration because this is not distinguishable from survival inthe model It is possible that an emigration event occurred in1997 and that those individuals have not yet returned Consid-erable heterogeneity in survival (range 77ndash97) is observedas shown in Figure 2(c) The survival probabilities are not dis-tributed smoothly suggesting multimodality The animals witha low survival coefficient may have died or migrated In thecase of migration this result would indicate that some animalsless consistently reside in the core areas compared with oth-ers

Individuals with higher survival coefficients are frequentlythose seen earliest in the study There is little information avail-able on individuals seen only more recently and consequentlytheir coefficients are close to 0 Some survival coefficients aremore negative and these correspond to animals that have notbeen seen for some time As time goes on and in the absenceof a confirmed death or sighting these coefficients can be ex-pected to become more negative However some animals thatare observed only on a few occasions have negative coefficientsThe estimation of individual survival coefficients means that es-timation of other coefficients in the model are adjusted for vari-able survival The estimation of individual capture probabilitiesallows the results to be combined with other individual covari-ates such as sex or body size if these data become availableSimilarly where individuals differ in their home ranges indi-vidual capture or survival probabilities could be related to localenvironmental conditions within their range to prey availabil-ity or to water quality

Estimates of the total population size from bilateral analy-sis of the left and right photo-identification series and estimatesfrom unilateral analyses of the left and right sides are shown inFigure 3(a) These are similar both to the maximum likelihoodestimate of 129 (95 confidence interval = 110ndash174) for datacollected in 1992 (Wilson et al 1999) and to a Bayesian mul-tisite mark recapture estimate of the nicked population of 85(95 probability interval = 76ndash263) for data collected in 2001(Durban et al 2005) The credible intervals of the abundanceestimates are generally narrower in the bilateral analysis (meanwidth 886) than in left-side analysis (mean width 1373) andright-side analysis (mean width 937) Figure 3(b) shows theprior and mean posterior recruitment probabilities ωj It is in-teresting that higher probabilities occur in 1996 and 2001 Thismight be considered to variations in capture probability butwhereas the capture probability is high in 2001 it remains lowin 1996 [see Fig 2(a)] Instead calculating the probabilities ofthe posterior exceeding the prior recruitment probabilities wefind they are below 5 (range 04ndash44) for all years except 2001in which it is 85 The higher recruitment probability in 2001may be due to an improved protocol in that year that led to the

(a)

(b)

Figure 3 (a) Mean total abundance estimates from a bilateral analy-sis with 95 HPDI Also shown are estimates from separate analysesof left (minus minus minus) and right (middot middot middot) sides along with associated 95 HPDIrepresented by symbols left + right Minimum numbers of ani-mals known to be alive are indicated by diamonds (b) Mean posteriorrecruitment probabilities ωj with 95 HPDI The prior value is rep-resented as a dotted line

identification of previously unrecognized nicked animals Weconsider the rise in 1996 to have insufficient evidence to sug-gest an increase in recruitment in that year

Earlier demographic modeling of this population has pre-dicted that the population is in decline (Saunders-Reed et al1999) To examine this we calculate the statistic [prodJ

j=2 Wj

(Wjminus1)]1(Jminus1) Values of this statistic gt1 indicate an increas-ing population whereas values lt1 indicates a decreasing pop-ulation The mean value is 989 (95 HPDI = 958 102)and the probability of it being lt10 is 77 This suggests thatthe population may be decreasing but that the evidence fora decline is weak By doing this we are able to use the fulltime series of photo identification data to obtain simultaneousposterior estimates of the Wj to compare the probability of apopulation increase or decrease In agreement with the earlierpredictive modeling these data indicate greater probability of adecline than of an increase However because it is likely thatthe population is expanding its range (Wilson et al 2004) thedecline may be confounded with temporary emigration

956 Journal of the American Statistical Association September 2008

6 DISCUSSION

Changes in the size of wildlife populations are often assessedby fitting trend models to independent abundance estimatesbut the power of these techniques to detect trends often canbe low (Gerrodette 1987 Taylor Martinez Gerrodette Bar-low and Hrovat 2007) Here we use a capturendashrecapture modelthat incorporates an underlying population model and an ob-servation model We provide a probability that a population isin decline or increasing Using photoidentification data fromthis bottlenose dolphin population we generate the first em-pirical estimate of trends in abundance for any of the popula-tions of small cetaceans inhabiting European waters Becausethe methodological changes in sampling likely are absorbedby the time-varying capture coefficient the remaining variationdue to individual capture tendencies might reasonably be de-scribed by the individual capture coefficient This then allowsexploration of capture and survival relating to individuals ratherthan to populations Information on individual survival could beimportant for addressing some management issues for exam-ple individuals differing in their spatial distribution

By decomposing the capture probabilities into overall time-varying and heterogeneous terms we find that the time-varyingcontribution and heterogeneous variation are comparable inmagnitude and their variation is greater than that of the overallterm This information can be used in the design of monitoringprograms by suggesting how effort should be distributed Thedifference in overall left and right capture probabilities mightbe interpretable as an artifact of the sampling techniques or ofbehavioral effects However after allowing for this the prob-ability of observing one side also varied over time with someyears strongly favoring one side or the other This might be ex-plained by variation in protocol between years such as changesin personnel The relative sizes of these effects may be of usein evaluating consistency of protocol over time Finally afterallowing for the overall difference and time variation the het-erogeneous capture coefficients exhibit a bias to the one sidepossibly due to behavior or protocol These findings demon-strate that bilateral decomposition of capture probabilities canprovide useful information

We decompose the survival probability similarly and we findit to be unusually low in 1997 suggesting a possible multimodalheterogeneous survival probability that may be the result of mi-gration or survival variation This type of decomposition allowsexploration of such effects in more detail In particular the mul-timodality of heterogeneous survival suggests that the popula-tion comprises individuals with different strategies some vis-iting the core study area more or less often Ongoing work isattempting to quantify individual heterogeneity such as in ex-posure to sewage contaminated waters and boat-based tourismTherefore this technique could be used to determine whetherexposure to different anthropogenic stressors is related to sur-vival

Our model also makes use of priors such as those on ωj thatare not informed by the recapture data set but make use of in-formation from other sources An alternative approach mightbe to specify these as informative priors based on expert opin-ion or tuned to obtain satisfactory results Our application il-lustrates how disparate data sources may be combined into a

single analysis Although the data source for the prior recruit-ment probability is small and problematic it is the only oneavailable Even when some priors are defined in the simulationstudy to incorrect values the estimates of parameters of interestremain valid The posterior probability also adds to the state ofknowledge of this parameter

Previous use of capturendashrecapture models with cetacean pho-toidentification data highlights the potential violation of modelassumptions due to heterogeneity in capture andor survivalprobabilities This model explicitly accounts for heterogene-ity and also provides simultaneous estimates of individual sur-vival and capture probabilities This can be used both to ex-plore the biological basis of heterogeneity and to improve sam-pling programs The individual survival coefficients also can beused to explore aspects of biology where correlates are avail-able for example the relationship of survival to age sex orother variables may be of interest Abundance under the as-sumption of heterogeneous survival has been estimated by LeeHuang and Ou (2004) whereas individual and time-varyingsurvival estimates have been obtained using band-recoverymodels (Grosbois and Thompson 2005) Other studies have ex-amined survival by group for example Franklin Andersonand Burnham (2002) obtained separate estimates of group-levelsurvival for males and females using stratification in a bandingstudy Although we lack such variables as age and sex theo-retically it would be possible to examine individual survivalby sex either as a post hoc comparison or through their inte-gration into the model itself For example survival and cap-ture could be modeled as logit(φ) = α + βj + γd + τ andlogit(π) = δ + εj + ζd + χ where τ corresponds to stage-specific (eg young immature reproductive female senes-cent) survival and χ corresponds to capture probabilities forvarious categories (eg young reproductive adult)

Other cetacean studies have examined heterogeneity White-head (2001b) obtained an estimate of the coefficient of variation(CV) for identification from a randomization test of successiveidentifications in whale track data (The CV is also a measure ofcapture heterogeneity) They found that the CV was significantin one of the two years and that probably depended on the sizesof the animals being observed This is not comparable to thedolphin observations in which heterogeneity is more likely toresult from differences in animal behavior In the present studywe use a likelihood in which heterogeneity of survival and cap-ture has been incorporated by a logit function This approachwas described previously (Pledger Pollock and Norris 2003Pledger and Schwarz 2002) within a non-Bayesian contextPledger et al (2003) classified animals into an unknown set oflatent classes each of which could have a separate survival andcapture probability A simplification allows for time-varyingand animal-level heterogeneity This approach is similar to oursbut we also integrate a dynamic population model and an ob-servation model into the estimation process Goodman (2004)also implemented a population model in a Bayesian context thatcombines capturendashrecapture data with carcass-recovery databut assumes the absence of heterogeneity in capture probabil-ities Durban et al (2005) estimated abundance for the samepopulation as us but used a multisite Bayesian log-linear modeland model averaging (King and Brooks 2001) across the pos-sible models with varying combinations of interactions Our

Corkrey et al Bayesian Recapture Population Model 957

model could be extended to include multiple sites and yearsby using a capturendashrecapture history that includes site informa-tion and a yearndashsite term in the capture probability expressionBy doing this we suggest that geographic areas with more datacan be used to strengthen estimates from poorly sampled areasThis also may account for some individual variation currentlyidentified as individual capture heterogeneity because some in-dividuals may be more closely associated with some areas thanothers and may attenuate a possible confounding of migrationwith survival

We use a separate data set and a negative-binomial model toinflate the estimate of the proportion of nicked animals in thesamples Another study (Da Silva Rodrigues Leite and Mi-lan 2003) obtained an abundance estimate for bowhead whalesusing an inflation factor derived from the numbers of good andbad photographs of marked and unmarked whales Our analysisavoids the lack of independence between good photographs anddistinctively marked animals in the photographs that might in-validate the estimate of variance in the inflation factor Otherstudies avoid this issue by including only separately catego-rized good-quality photographs (eg Wilson et al 1999 ReadUrian Wilson and Waples 2003) However independent es-timates of inflation when available such as in our study arepreferred

We do not know of other cetacean studies that combinephoto-identification series of the left and right sides but ananalogous approach is that of log-linear capturendashrecapturemodeling (King and Brooks 2001) in which animals are identi-fied from separate lists Some photo-identification studies mightuse information from both sides in the process of identifyinganimals (Joyce and Dorsey 1990) A more robust approach isto perform the identification on each side separately We haveshown how these separate data can then be combined

A major advantage to our modeling approach is its extensibil-ity Within the Bayesian framework models can be comparedby such methods as reversible-jump Markov chain Monte Carlotechniques (Green 1995) Model extensions include extendingthe population model considering an individual-level approachThis would allow incorporation of the heterogeneity into thispart of the model and also modeling of the unobserved popu-lation More detailed modeling of heterogeneity could be ex-plored by including hyperpriors for the variances of the recap-ture model terms For convenience we analyze the data on anannual basis Additional information would be available if theanalyses were conducted on a finer time scale such as monthsor by individual trip This would allow the model to be extendedto include seasonal effects For example dolphins are observedmore frequently in summer months than in winter months andthis tendency could be included in an extended model by chang-ing the capture probability to be logit(π) = δt + εj + ζd whereδt t isin [1 12] is a monthly capture coefficient This maybe of use in modeling fine-scale movements The response vari-able itself also can be generalized Although we are careful toonly include data from high-quality photographs and nicked an-imals there is still the possibility of some identification errorsparticularly for more subtly marked individuals A possible ex-tension of the model could allow identification to be assigneddifferent levels of certainty for example by incorporating anintermediate state between nonrecapture and recapture that we

term ldquopossible recapturerdquo This could be done by replacing thelogit function logit(ν) = log(ν(1 minus ν)) where ν correspondsto the probability of recapture by a proportional odds function(McCullagh 1980) This technique has previously found appli-cation in a study of photoidentification used to model photo-graphic quality scores and individual distinctiveness of hump-back whale tail flukes (Friday Smith Stevick and Allen 2000)In this approach the probability of ldquopossible recapturerdquo is givenby ν1 = π1 the probability of certain recapture is given byν2 = π1 + π2 and includes the possible recapture category andthe probability of no capture is given by ν0 = 1 minus π1 + π2There are two proportional odds functions corresponding topossible recapture and certain recapture which are logit(ν1) =log(π1(π2 +π3)) and logit(ν2) = log((π1 +π2)π3) We thencould extend the recapture model using logit(ν1) = α+βj +γk1

and logit(ν2) = α + βj + γk2 This proportional odds approachmay be particularly useful in situations where automatic match-ing algorithms (Saacutenchez-Mariacuten 2000) provide an estimate ofthe likelihood of a match which could then be incorporatedinto abundance estimate models

APPENDIX DERIVATION OF POSTERIORS

Here we show how we derive the posterior conditional distributionsfor the model parameters and describe how each is updated within theMCMC algorithm The full joint distribution has the form shown in(A1) In (A1) the first two lines correspond to the population modelthe third line corresponds to the inflation model the fourth line cor-responds to the observation model the fifth line corresponds to therecapture model and the remaining lines correspond to priors

π = P(B1|ω1N0W0)P (S1|αN0)

(Sec 31)

timesSprod

j=2

P(Bj |Sjminus1Bjminus1Wjminus1ωj )

times P(Sj |Sjminus1Bjminus1 αβjminus1)

(Sec 31)

timesSprod

j=1

P(Wj |Sj Bj υj )

Tjprod

i=1

P(tnji |υj twji )

(Sec 32)

timesSprod

j=1

P(nLj|δL εjLSj Bj )P (nRj

|δR εjRSj Bj )

(Sec 33)

timesSprod

j=1

P(Hdj |αβj γd δL εjL ζdL δR εjR ζdR)

(Sec 34)

times P(N0|λN)P (W0|N0 υ0)P (α δL δRλN υ0)

(Sec 41)

timesSprod

j=1

P(βj εjL εjRωj υj )

nprod

d=1

P(γd)P (ζdL)P (ζdR)

(Sec 41) (A1)

958 Journal of the American Statistical Association September 2008

For each parameter we obtain the corresponding posterior condi-tional distribution by extracting all terms containing that parameterfrom the full joint distribution When the conditional distributions havea standard form we use the Gibbs sampler (Brooks 1998) to update theparameters otherwise we use MetropolisndashHastings updates (Chib andGreenberg 1995)

Within a MetropolisndashHastings update we propose a new valuefor a parameter x and denote the proposed value by xprime To decidewhether to accept the proposed value we calculate an acceptance ra-tio α = min(1A) where A is the acceptance ratio The acceptanceratio is given by A = π(xprime)q(x)π(x)q(xprime) where π(x) is the poste-rior probability of x and q(x) is the probability of the proposed valueWe accept the new value xprime with probability α otherwise we leavethe value of x unchanged (See Chib and Greenberg 1995 for an exam-ple)

Normal proposal distributions are used for continuous parameterscentered around the current parameter value Discrete uniform pro-posals are used for discrete variables each centered around the currentparameter values The discrete proposals also require constraints to en-sure that the proposed values are valid For example a proposed valuefor N0 must be less than the current value of W0 We provide a detaileddescription of each update next

Population Model

The population model has parameters ωj λ Bj Sj and N0 andare updated using Gibbs or MetropolisndashHastings samplers The condi-tional posterior distributions for ωj and λN can be derived as

ωj |Bj Sjminus1Bjminus1

sim Beta(Bj + 0238Wjminus1 minus Sjminus1 minus Bjminus1 minus Bj + 1)

and

λN |N0 sim Gam(76 + N0101)

These are individually updated using the Gibbs sampler during eachiteration of the Markov chain

The parameters Bj Sj and N0 are updated by the MetropolisndashHastings algorithm using acceptance ratios calculated as described ear-lier We use uniform proposals centered on the current parameter valueand limited to the range of possible values

For the Sj rsquos we use three alternative proposals

Sprimej=1 sim Unif

(max

(0 S1 minus 5 S2 minus B1max(n1Ln1R) minus N1

2 minus B1)

min(S1 + 5N0W1 minus B1W1 minus B1 minus B2

W1 minus B1 minus 1))

Sprime2lejltJ sim Unif

(max

(0 Sj minus 5 Sj+1 minus Bj max(njLnjR) minus Bj

2 minus Bj

)

min(Sj + 5 Sjminus1 + Bjminus1Wj minus Bj Wj minus Bj minus Bj+1

Wj minus Bj minus 1))

and

Sprimej=J sim Unif

(max

(0 SJ minus 5max(nJLnJR) minus BJ 2 minus NJ

)

min(SJ minus 5 Sjminus1 + Bjminus1Wj minus Bj Wj minus Bj minus 1))

For Bj we use three alternative proposals

B prime1 sim Unif

(max

(0B1 minus 15 S2 minus S1max(n1Ln1R) minus S1

2 minus S1)

min(B1 + 15W0 minus N0W1 minus S1W1 minus S1 minus B2

W1 minus S1 minus 1))

B prime2lejltJ sim Unif

(max

(0Bj minus 15 Sj+1 minus Sj max(njLnjR) minus Sj

2 minus Sj

)

min(Bj + 15Wjminus1 minus Sjminus1 minus Bjminus1Wj minus Sj

Wj minus Sj minus Bj+1Wj minus Sj minus 1))

and

B primeJ sim Unif

(max

(0BJ minus 15max(nJLnJR) minus SJ 2 minus SJ

)

min(BJ + 15WJminus1 minus SJminus1 minus BJminus1

WJ minus SJ WJ minus SJ minus 1))

For N0 we use

N prime0 sim Unif

(max(5N0 minus 15 S1)min(N0 + 15W0 minus B1W0 minus 1)

)

Inflation Model

The inflation model parameters Wj and υj are updated usingMetropolisndashHastings and Gibbs updates The conditional posterior dis-tributions for υj can be derived as

υ0|N0W0 sim Beta(N0 + 1W0 minus N0 + 1)

and

υjgt0|Sj Bj Wj tnji twji sim Beta

(

Sj + Bj +T (j)sum

i=1

tnji + 1

Wj minus Sj minus Bj +T (j)sum

i=1

twji minus tnji + 1

)

These are updated separately using the Gibbs sampler on each itera-tion

The remaining parameter Wj has a nonstandard conditional pos-terior distribution We use larger updates than in the population modelbecause the initial total population is expected to be larger in size

W prime0 sim Unif

(max(0W0 minus 15N0 + 1N0 + B1)W0 + 15

)

W prime1lejltJ sim Unif

(max(0Wj minus 15 Sj + Bj + 1 Sj + Bj + Bj+1)

Wj + 15)

and

W primeJ sim Unif

(max(0WJ minus 15 SJ + BJ + 1)WJ + 15

)

Recapture Model

For the recapture model we have parameters α βj γd δL εjLζdL δR εjR and ζdR These all have nonstandard conditional pos-terior distributions and we use the MetropolisndashHastings algorithm toupdate the parameters on each iteration We use individual normal dis-tribution proposals centered on the current parameter values and withvariance equal to the prior variance for example

αprime sim N(α185)

[Received June 2005 Revised May 2007]

Corkrey et al Bayesian Recapture Population Model 959

REFERENCES

Brooks S P (1998) ldquoMarkov Chain Monte Carlo Method and Its ApplicationrdquoThe Statistician 47 69ndash100

Brooks S P and Roberts G O (1998) ldquoConvergence Assessment Techniquesfor Markov Chain Monte Carlordquo Statistics and Computing 8 319ndash335

Buckland S T (1982) ldquoA CapturendashRecapture Survival Analysisrdquo Journal ofAnimal Ecology 51 833ndash847

(1990) ldquoEstimation of Survival Rates From Sightings of IndividuallyIdentifiable Whalesrdquo in Individual Recognition of Cetaceans Use of Photo-Identification and Other Techniques to Estimate Population Parameters edsP S Hammond S A Mizroch and G P Donovan Cambridge UK Inter-national Whaling Commission pp 149ndash153

Burnham K and Overton W (1979) ldquoRobust Estimation of Population SizeWhen Capture Probabilities Vary Among Animalsrdquo Ecology 60 927ndash936

Carothers A (1979) ldquoQuantifying Unequal Catchability and Its Effect on Sur-vival Estimates in an Actual Populationrdquo Journal of Animal Ecology 48863ndash869

Chao A (1987) ldquoEstimating the Population Size for CapturendashRecapture DataWith Unequal Catchabilityrdquo Biometrics 43 783ndash791

Chao A Lee S-M and Jeng S-L (1992) ldquoEstimating Population Size forCapturendashRecapture Data When Capture Probabilities Vary by Time and Indi-vidual Animalrdquo Biometrics 48 201ndash216

Chib S and Greenberg E (1995) ldquoUnderstanding the MetropolisndashHastingsAlgorithmrdquo The American Statistician 49 327ndash335

Cormack R (1972) ldquoThe Logic of CapturendashRecapture Experimentsrdquo Biomet-rics 28 337ndash343

Coull B A and Agresti A (1999) ldquoThe Use of Mixed Logit Models to Re-flect Heterogeneity in CapturendashRecapture Studiesrdquo Biometrics 55 294ndash301

Da Silva C Q Rodrigues J Leite J G and Milan L A (2003) ldquoBayesianEstimation of the Size of a Closed Population Using Photo-ID With Part ofthe Population Uncatchablerdquo Communications in Statistics 32 677ndash696

Dorazio R M and Royle J A (2003) ldquoMixture Models for Estimating theSize of a Closed Population When Capture Rates Vary Among IndividualsrdquoBiometrics 59 351ndash364

Durban J Elston D Ellifrit D Dickson E Hammon P and Thompson P(2005) ldquoMultisite Mark-Recapture for Cetaceans Population Estimates WithBayesian Model Averagingrdquo Marine Mammal Science 21 80ndash92

Fienberg S E Johnson M S and Junker B W (1999) ldquoClassical Multi-level and Bayesian Approaches to Population Size Estimation Using MultipleListsrdquo Journal of the Royal Statistical Society Ser A 162 383ndash405

Forcada J and Aguilar A (2000) ldquoUse of Photographic Identification inCapturendashRecapture Studies of Mediterranean Monk Sealsrdquo Marine MammalScience 16 767ndash793

Franklin A B Anderson D R and Burnham K P (2002) ldquoEstimation ofLong-Term Trends and Variation in Avian Survival Probabilities Using Ran-dom Effects Modelsrdquo Journal of Applied Statistics 29 267ndash287

Friday N Smith T D Stevick P T and Allen J (2000) ldquoMeasure-ment of Photographic Quality and Individual Distinctiveness for the Photo-graphic Identification of Humpback Whales Megaptera novaeangliaerdquo Ma-rine Mammal Science 16 355ndash374

Gaspar R (2003) ldquoStatus of the Resident Bottlenose Dolphin Population inthe Sado Estuary Past Present and Futurerdquo unpublished doctoral dissertationUniversity of St Andrews School of Biology

Gelfand A E and Sahu S K (1999) ldquoIdentifiability Improper Priors andGibbs Sampling for Generalized Linear Modelsrdquo Journal of the AmericanStatistical Association 94 247ndash253

Gerrodette T (1987) ldquoA Power Analysis for Detecting Trendsrdquo Ecology 681364ndash1372

Goodman D (2004) ldquoMethods for Joint Inference From Multiple Data Sourcesfor Improved Estimates of Population Size and Survival Ratesrdquo Marine Mam-mal Science 20 401ndash423

Green P J (1995) ldquoReversible-Jump Markov Change Monte Carlo BayesianModel Determinationrdquo Biometrika 82 711ndash732

Grosbois V and Thompson P M (2005) ldquoNorth Atlantic Climate VariationInfluences Survival in Adult Fulmarsrdquo Oikos 109 273ndash290

Haase P (2000) ldquoSocial Organisation Behaviour and Population Parametersof Bottlenose Dolphins in Doubtful Sound Fiordlandrdquo unpublished mastersdissertation University of Otago Dept of Marine Science

Hammond P (1986) ldquoEstimating the Size of Naturally Marked Whale Pop-ulations Using CapturendashRecapture Techniquesrdquo Report of the InternationalWhaling Commission Special Issue 8 253ndash282

(1990) ldquoHeterogeneity in the Gulf of Maine Estimating HumpbackWhale Population Size When Capture Probabilities Are not Equalrdquo in In-dividual Recognition of Cetaceans Use of Photo-Identification and OtherTechniques to Estimate Population Parameters eds P S Hammond S A

Mizroch and G P Donovan Cambridge UK International Whaling Com-mission pp 135ndash153

Huggins R (1989) ldquoOn the Statistical Analysis of Capture Experimentsrdquo Bio-metrika 76 133ndash140

Joyce G G and Dorsey E M (1990) ldquoA Feasibility Study on the Useof Photo-Identification Techniques for Southern Hemisphere Minke WhaleStock Assessmentrdquo in Individual Recognition of Cetaceans Use of Photo-Identification and Other Techniques to Estimate Population Parameters edsP S Hammond S A Mizroch and G P Donovan Cambridge UK Inter-national Whaling Commission pp 419ndash423

King R and Brooks S (2001) ldquoOn the Bayesian Analysis of PopulationSizerdquo Biometrika 88 317ndash336

Lebreton J-D Burnham K P Clobert J and Anderson S R (1992)ldquoModeling Survival and Testing Biological Hypotheses Using Marked An-imals A Unified Approach With Case Studiesrdquo Ecological Monographs 6267ndash118

Lee S-M Huang L-H H and Ou S-T (2004) ldquoBand Recovery Model In-ference With Heterogeoneous Survival Ratesrdquo Statistica Sinica 14 513ndash531

Link W A and Barker R J (2005) ldquoModeling Association Among De-mographic Parameters in Analysis of Open Population CapturendashRecaptureDatardquo Biometrics 61 46ndash54

Mann J and Smuts B (1999) ldquoBehavioral Development in Wild BottlenoseDolphin Newborns (Tursiops sp)rdquo Behaviour 136 529ndash566

McCullagh P (1980) ldquoRegression Models for Ordinal Datardquo Journal of theRoyal Statistical Society Ser B 42 109ndash142

Otis D L Burnham K P White G C and Anderson D R (1978) ldquoStatis-tical Inference From Capture Data on Closed Animal Populationsrdquo WildlifeMonographs 62 135

Pledger S (2000) ldquoUnified Maximum Likelihood Estimates for ClosedCapturendashRecapture Models Using Mixturesrdquo Biometrics 56 434ndash442

Pledger S and Schwarz C J (2002) ldquoModelling Heterogeneity of Survivalin Band-Recovery Data Using Mixturesrdquo Journal of Applied Statistics 29315ndash327

Pledger S Pollock K H and Norris J L (2003) ldquoOpen CapturendashRecaptureModels With Heterogeneity I CormackndashJollyndashSeber Modelrdquo Ecology 59786ndash794

Pollock K H Nichols J D Brownie C and Hines J E (1990) ldquoStatisticalInference for CapturendashRecapture Samplingrdquo Wildlife Monographs 107 1ndash97

Read A Urian K W Wilson B and Waples D (2003) ldquoAbundance andStock Identity of Bottlenose Dolphins in the Bayes Sounds and Estuaries ofNorth Carolinardquo Marine Mammal Science 19 59ndash73

Saacutenchez-Mariacuten F J (2000) ldquoAutomatic Recognition of Biological ShapesWith and Without Representations of Shaperdquo Artificial Intelligence in Medi-cine 18 173ndash186

Saunders-Reed C A Hammond P S Grellier K and Thompson P M(1999) ldquoDevelopment of a Population Model for Bottlenose Dolphinsrdquo Re-search Survey and Monitoring Report 156 Scottish Natural Heritage

Scottish Natural Heritage (1995) Natura 2000 A Guide to the 1992 EC Habi-tats Directive in Scotlandrsquos Marine Environment Perth UK Scottish Nat-ural Heritage

Seber G (2002) The Estimation of Animal Abundance Caldwell NJ Black-burn Press

Stevick P T Palsboslashll P J Smith T D Bravington M V and Ham-mond P S (2001) ldquoErrors in Identification Using Natural Markings RatesSources and Effects on CapturendashRecapture Estimates of Abundancerdquo Cana-dian Journal of Fisheries and Aquatic Science 58 1861ndash1870

Taylor B L Martinez M Gerrodette T Barlow J and Hrovat Y N (2007)ldquoLessons From Monitoring Trends in Abundance of Marine Mammalsrdquo Ma-rine Mammal Science 23 157ndash175

Wells R S and Scott M D (1990) ldquoEstimating Bottlenose Dolphin Popu-lation Parameters From Individual Identification and CapturendashRelease Tech-niquesrdquo in Individual Recognition of Cetaceans Use of Photo-Identificationand Other Techniques to Estimate Population Parameters eds P S Ham-mond S A Mizroch and G P Donovan Cambridge UK InternationalWhaling Commission pp 407ndash415

Whitehead H (2001a) ldquoAnalysis of Animal Movement Using Opportunis-tic Individual Identifications Application to Sperm Whalesrdquo Ecology 821417ndash1432

(2001b) ldquoDirect Estimation of Within-Group Heterogeneity in Photo-Identification of Sperm Whalesrdquo Marine Mammal Science 17 718ndash728

Williams B K Nichols J D and Conroy M J (2001) Analysis and Man-agement of Animal Populations San Diego Academic Press

Wilson B Hammond P S and Thompson P M (1999) ldquoEstimating Size andAssessing Trends in a Coastal Bottlenose Dolphin Populationrdquo EcologicalApplications 9 288ndash300

Wilson B Reid R J Grellier K Thompson P M and Hammond P S(2004) ldquoConsidering the Temporal When Managing the Spatial A Popula-

Corkrey et al Bayesian Recapture Population Model 951

given by υj then the size of the unnicked population at timej can be modeled as a negative binomial distribution condi-tional on the size of the nicked population so that Wj minus Nj simNegBin(Nj υj ) This approach allows the use of an indepen-dent data set to inform the υj rather than relying on the re-capture data to estimate this parameter The negative binomialdistribution is preferred over a binomial distribution becausethe Nj rsquos already have been assigned a distribution in the popu-lation model

In each time period j suppose that there are Tj trips eachobserving twji animals for i = 1 Tj of which tnji are nickedNumerous trips are undertaken each year and the numbernicked is separately estimated on each encounter A binomialdistribution then can be used to describe the number of nickedanimals seen per trip so that tnji sim Bin(twji υj ) Thus the ob-servations tnji and twji allow us to estimate υj and thus the totalpopulation size given an estimate of the nicked population Werefer to this as the inflation model It provides the probabilitydistributions P(Wj |Sj Bj υj ) and P(tnji |twji υj )

33 The Observation Model

In any time period only a portion of the nicked populationis seen Examination of the data suggests that the left and rightcapture probabilities differ at the population level The prob-ability that an individual animal would be seen from the leftor right side s isin [LR] is given by ρjs = exp(δs + εjs)(1 +exp(δs + εjs)) where δs denotes a global capture tendency andεjs denotes a time-varying capture component

Then the number of nicked animals seen from side s njs fol-lows a binomial distribution njs sim Bin(Nj ρjs) where njL =sum

d hdj isin [13] and njR = sumd hdj isin [23] We refer to this

model as the observation model It provides the probability dis-tributions P(njL|Nj ρjL) and P(njR|Nj ρjR)

We consider the njL and njR as independent counts A dol-phin seen in one year from one side contributes to the prob-ability of observing that side and one seen from both sidescontributes to both probabilities The introduction of captureprobabilities for each side allows the combination of the twophotographic series that otherwise would have been analyzedseparately because they would be correlated It is also of inter-est to estimate these probabilities to assess the capture methodsin use We include the term εjs to allow for observed differ-ences between years in observing all dolphins from one sideversus the other These differences may have arisen from dif-ferences in boat work

34 The Recapture Model

The population and observation models require estimates ofthe parameters α βj δL εjL δR and εjR corresponding tothe survival and capture probabilities These probabilities canbe obtained from the capture history data To do this we cal-culate the likelihood of the entire capture history matrix whichis the product of the probabilities associated with each animalrsquosindividual history L(hdj )

To combine the photo-identification series from the left andright sides into a single bilateral analysis the model allows dif-ferent capture probabilities when animals are seen from eachside The number of animals seen also can differ between the

left and right sides Other parameters do not depend on the sidefrom which an animal is observed for example the populationsize and survival probability of an animal should not depend onwhich side is observed However the left-side capture proba-bility for individual animals might be expected to differ fromthat of the right side for example the animal might be morelikely to approach the boat from one side than from the otherDolphins are also well known to have heterogeneous behavior(Wilson Reid Grellier Thompson and Hammond 2004)

Thus to incorporate heterogeneity as well as observationfrom two sides survival and capture probabilities are definedas logit(φdj ) = α + βj + γd and logit(πdjs) = δs + εjs + ζds where γd denotes a survival effect ζds denotes a recapture ef-fect for animal d s isin [LR] for the left or right side andlogit(x) = log(x(1 minus x)) The logit function is the standardchoice for binary data

Note that the heterogeneity effects are not used in the popu-lation and observation models because both of these are basedon both the nicked and unnicked populations and members ofthe latter will not have an individual effect parameter Althoughwe could adopt a random-effects model to ascribe individualeffects to unnicked animals this would imply that we believethe heterogeneity of survival and recapture to be the same inboth the nicked and unnicked populations which may not bethe case

The overall probability has components for animals in twosituations There are nJ animals observed at the final time andsome nJ lowast animals last observed at some earlier time and thatmight have died or simply are not seen Each individual historyprobability is conditional on the first time that the animal isseen j1

d Thus the likelihood associated with the histories of n = nJ +

nJ lowast dolphins is given by

prod

sisin[LR]

nprod

d=1

Jprod

j=1

L(hdjs)

=nJprod

d=1

Jprod

j=j1d +1

φdjminus1I (πdjLπdjRhdjs)

timesnJ +nJlowastprod

d=nJ +1

hlprod

j=j1d +1

φdjminus1I (πdjLπdjRhdjs)

(

1 minus φdhl

+Jsum

j=hl+1

φdj

jprod

k=hl+1

φdkminus1(1 minus πdkL)(1 minus πdkR)

)

(1)

in which φdj denotes the survival probability φdj = 1 minus φdj

when j lt J and φdj = 1 when j = J πdjs denotes the recap-ture probability from animal d observed in time j from sides isin [LR] j1

d and hl denote the first and last times that individ-ual d is seen and I (πdjLπdjRhdjs) = (1 minus πdjL)(1 minus πdjR)

when hdjs = 0 I (πdjLπdjRhdjs) = πdjL(1 minus πdjR) whenhdjs = 1 and I (πdjLπdjRhdjs) = (1 minus πdjL)πdjR whenhdjs = 2 I (πdjLπdjRhdjs) = πdjLπdjR when hdjs = 3 Theφdj and πdjs parameters are defined only for 1 le j lt J minus1 and2 le j lt J The values of nJ j1

d and hl are determined fromthe capture histories

952 Journal of the American Statistical Association September 2008

We refer to this model as the recapture model It yieldsthe probability P(hdjs |φdj γd πdjLπdjR ζdL ζdR) This isa CormackndashJollyndashSeber model which has been discussed else-where (Seber 2002 Lebreton Burnham Clobert and Anderson1992 Link and Barker 2005)

This completes the model specification To summarize wedefine a population model that describes the recruitment (ωj )and survival (θj ) of animals in the nicked population (Nj ) Therelative sizes of the nicked population (Nj ) and total population(Wj ) are described by the inflation model which makes use ofobserved counts of nicked and unnicked animals Using the cap-ture probability (ρjs ) and the observed number of nicked ani-mals we estimate the nicked population Finally we estimatethe heterogeneous capture (πdjs ) and survival (φdj ) probabili-ties from the capture histories Because this is a Bayesian analy-sis we also specify priors to the following parameters S0 B0ωj W0 N0 α βj γd δs εjs and ζds We describe the priorsin detail in the next section

4 ANALYSIS

In this section we describe the statistical methodology re-quired to analyze the model described in the previous sectionWe adopt a Bayesian approach and describe the prior specifica-tion and simulation techniques required here In this modelingprocess we use a Bayesian approach to allow for uncertaintyin measurement and parameters used The Bayesian approachallows for simultaneous consideration of all data and modelsallowing for the proper propagation of uncertainty throughoutthe model Identifiability is also less of a concern in a Bayesiananalysis in which prior information is supplied (Gelfand andSahu 1999) Moreover the Bayesian approach has an advan-tage in that MCMC methods can be used which greatly simpli-fies the computation compared with the corresponding classi-cal tools The remainder of this section discusses the process ofprior specification and provides details of the implementation

41 Priors

The first stage in the Bayesian analysis is to elicit priors foreach of the model parameters We do so here by grouping theparameters according to the component of the model in whichthey appear The priors used and their associated parameters aresummarized in what follows

The Population Model The model specifies the distribu-tion of parameters Sj and Bj in terms of Sjminus1 and Bjminus1 forj ge 1 Therefore we need a prior for S0 and B0 Becausethey only appear together we set N0 = S0 + B0 and assigna Poisson prior N0 sim Po(λN) Its mean takes a gamma priorλN sim Gam(76 01) with parameters selected to obtain a meanfor the nicked population of 76 as estimated by Wilson et al(1999) and a large variance so that it becomes uninformativecompared with the range of reasonable population values

Because the recruitment rates of nicked dolphins cannot bedirectly estimated from the photo-identification data set thepriors are parameterized with results from a separate analy-sis of randomly selected photographs with at least 2 years ofdata Because it is thought that the rate of nicking might begreater for older animals due to increased social interactionsestimates are estimated from photographs of individuals that

are first identified as known-age calves Identification is doneusing all attributes such as fin shape and the presence of le-sions Photographs of seven of these animals are available forestimating the age at which these dolphins are first seen to havea nick in the dorsal fin We consider these cases a random se-lection from the unnicked population so that the mean num-ber of nicks per year gives an approximate probability of nick-ing per year We set the prior for ωj to be reasonably vagueωj sim Beta(02381) Although rough these assumptions givea crude estimate for the parameter which is then updated bythe data

The Inflation Model The initial population size is assigneda negative-binomial prior W0 minusN0 sim NegBin(N0 υ0) The ini-tial inflation factor υ0 is assigned a beta prior υ0 sim Beta(11)which is equivalent to a noninformative uniform prior

The Recapture Model The recapture model survival pa-rameters have independent normal priors N(0 185) and thecapture parameters have independent normal priors N(0 15)These priors span the range (01) on the logit scale Largervariances are not used because these would obtain a U-shapeddistribution on the logit scale

42 Implementation

Once priors are assigned the full joint distribution can bedetermined as the product of the likelihood with the associatedprior distributions The resulting posterior is complex and high-dimensional and so inference is obtained in the form of poste-rior means and variances obtained through MCMC simulationWe choose to use an implementation in which each parameter isupdated in turn using either a MetropolisndashHastings or a Gibbsupdate depending on the form of the associated posterior con-ditional distributions The updating strategy is detailed in theAppendix All simulation software is written in FORTRAN 77The model was run for 500000 iterations and a 50 burn-in was used Sensitivity studies and standard diagnostic tech-niques were used (Brooks and Roberts 1998) to assess modelvalidity

43 Simulation Study

To test the MCMC code and to demonstrate the utility of theframework developed earlier we construct two artificial pop-ulations one with homogeneous capture and survival proba-bilities and the other with heterogeneous capture and survivalprobabilities These populations are constructed using the fol-lowing procedure An initial population of animals (n = 130)is sampled and a proportion of this initial population is ran-domly marked as nicked At each time period in the simulationanimals are randomly selected to die some of the unnicked be-come nicked and a proportion of the remaining animals arerandomly selected to be observed For each population threesimulations are run with differing priors for the ωj and λN para-meters The priors are such that the three cases represent vaguepriors precise priors and very precise priors but in each casethe means differ from the actual population values One possi-ble outcome is that the precise prior simulations track the priormeans more closely than the vague simulations

The simulations are illustrated in Table 1 which shows theactual total abundances the medians of the posteriors for Wj

Corkrey et al Bayesian Recapture Population Model 953

Table 1 Actual and estimated median total abundances for two artificial populations with 95 HPDI and posterior p values fromthree simulations on each with differing priors

A B C

Population Year Wj Wj p value Wj p value Wj p value

1 1990 124 119(99139) 28 112(97130) 07 107(90124) 021991 120 125(105146) 67 118(95134) 39 114(97135) 251992 114 116(96135) 54 110(93127) 30 106(92123) 171993 109 113(95134) 63 108(92124) 41 105(90121) 261994 103 111(93130) 81 106(88124) 63 103(86118) 491995 97 105(89125) 86 102(87118) 74 100(85114) 621996 95 98(84115) 66 95(81110) 48 93(81107) 381997 91 99(83114) 85 96(80109) 76 95(80108) 651998 89 94(79110) 73 91(77105) 56 90(77103) 521999 85 92(77108) 78 89(75103) 69 87(75103) 632000 76 82(6598) 76 79(6696) 65 79(6594) 592001 70 79(6299) 84 77(5895) 79 76(6095) 752002 68 76(5593) 78 75(5691) 75 73(5892) 70

2 1990 124 135(100178) 71 121(96159) 42 116(90149) 261991 122 123(90158) 51 112(87140) 21 108(84134) 121992 119 114(87148) 35 105(84130) 10 101(78124) 061993 111 106(82138) 34 99(75121) 11 95(74117) 061994 106 102(76130) 33 93(75118) 12 91(68115) 071995 100 96(70122) 38 91(74114) 16 88(67108) 081996 93 93(66119) 48 87(69109) 26 85(65105) 161997 87 91(67115) 60 85(68106) 37 82(62102) 281998 77 79(59101) 56 76(6295) 39 72(5791) 281999 74 84(60108) 80 80(62101) 75 77(5897) 622000 72 78(5799) 67 74(5894) 58 72(5591) 462001 67 71(4992) 62 69(4891) 57 66(4688) 432002 64 68(4191) 59 65(4492) 51 63(4388) 44

NOTE Population 1 has homogeneous capture probability (75) and survival probability (95) while population 2 has heterogeneous capture probability (mean 55 range 05ndash98) andsurvival probability (mean 92 range 68ndash99) The simulations have different priors for ωj and λN Simulation run A has vague priors [ω sim Beta(029412368) λN sim Gam(76 01)]B has more precise priors [ω sim Beta(023710) λN sim Gam(761)] and C has very precise priors [ω sim Beta(929939155) λN sim Gam(7600100)] Their means are the same ineach case (023 and 76) but each differs from the actual population values (06 and 106)

their 95 highest posterior density intervals (HPDI) and a p

value The p values are calculated by counting the proportionof iterations in which the Wj exceeded the true value Examina-tion of the results indicate that the median posterior estimatesgenerally agree with the actual values The agreement is sat-isfactory whether or not there is heterogeneity assumed Thep values are in the range 02ndash88 suggesting that the modelworks correctly The posterior p values for the recapture modelcoefficients are given in Table 2 The p values for α δL andδR are summarized and those for the remaining parameters aresummarized as the proportion of p values in each that lie insidethe range 01ndash99 Almost all p values are within the range 01ndash99 the only exceptions are the capture coefficients for a smallnumber of dolphins

In general the total population estimates for the homoge-neous case are slightly greater and the 95 HPDI is narrowerthen the heterogeneous case but all intervals contain the popu-lation values The estimates also shrink slightly for the mod-els with more precise priors However the p values in eachcase suggest an adequate fit even though some priors are de-liberately set to incorrect values In the case of the recapturecoefficients there is no apparent difference between the homo-geneous and heterogeneous models or with any trend in theprecision of the priors

5 RESULTS

We obtain estimates of heterogeneous capture and survivalcoefficients variation of capture and survival over time over-all survival and capture probabilities the abundance and thetrend in abundance The survival probabilities and capture co-efficients from the left and right sides are shown in Figure 2 andTable 3 We can obtain an overall capture probability for the twosides by calculating exp(δ)(1+exp(δ)) The mean overall cap-

Table 2 Summary of posterior p values for survival and capturecoefficients for two artificial populations from three simulations

on each with differing priors

Population α βj γd δL εjL ζdL δR εjR ζdR

1 16 100 100 66 100 98 60 100 10010 100 100 59 100 98 53 100 10008 100 100 59 100 98 48 100 100

2 34 100 100 17 100 97 50 100 9822 100 100 15 100 97 37 100 9923 100 100 12 100 97 39 100 99

NOTE The populations and simulations are described in Table 1 The p values are calcu-lated as the proportion of iterations greater than the actual population value The p valuesfor α δL and δR are shown Because the remaining parameters are vectors of coefficientsthose p values are summarized as the proportion of p values in each that are found insidethe range 01ndash99

954 Journal of the American Statistical Association September 2008

(a)

(b) (c)

Figure 2 (a) Variation of mean capture coefficients over time forthe left side εL (minusminus) and right side εR (- - bull - - bull) The two hor-izontal lines indicate the global mean capture coefficients δL (mdashmdash)and δR (minus minus minus) On the left vertical axis are the mean left-side het-erogeneity capture coefficients ζL and on the right vertical axis arethe mean right-side heterogeneity capture coefficients ζR Standardserrors for each coefficient are shown as vertical bars (b) Variation ofmean survival probability over time (c) Histogram of mean heteroge-neous survival probabilities

ture probability for the left side is 48 (95 HPDI = 29 68)and for the right side it is 58 (95 HPDI = 39 77) Althoughthis is a small difference the probability of the right capturecoefficient exceeding the left is 81 Examination of Figure 2(a)shows that the variation of capture coefficients over time is largecompared with the overall coefficients and that the left (εL) andright (εR) coefficients tend to track each other The probabilityof the right capture coefficient exceeding the left varies overa wide range 07ndash96 This range results from some years inwhich the probability tends strongly to the left side (1995 and2000 probability of observing the left side 95) or right side(1998 1999 2001 probability of observing the right side 91)Finally the range of the heterogeneous coefficients is compara-ble to that of the time-varying coefficients

The mean heterogeneous survival coefficient is not corre-lated with either of the mean capture heterogeneous coeffi-cients (right r = 009 left r = 022) However the meanleft and right heterogeneous capture coefficients are correlated(r = 714) indicating that animals that are more likely to beseen from one side also are more likely to be seen from theother side However the heterogeneity capture coefficient of theright side (mean 56) exceeded that of the left side (mean 23)with 85 of animals having higher right coefficients than left

Table 3 Left-side and right-side capture probabilities and survivalprobabilities by year

Year Left capture Right capture Survival

1990 1(11)

1991 54(40 68) 60(47 75) 93(84 99)

1992 74(59 86) 75(60 86) 98(94100)

1993 31(20 44) 46(33 61) 95(88100)

1994 36(24 47) 53(35 66) 95(86100)

1995 42(27 55) 71(56 88) 94(86100)

1996 30(17 42) 48(34 63) 93(82100)

1997 23(14 36) 37(23 51) 88(75 98)

1998 39(24 53) 31(20 45) 94(84100)

1999 45(31 63) 40(25 54) 93(82100)

2000 24(14 36) 57(40 72) 96(88100)

2001 87(78 94) 82(73 91) 96(89100)

2002 88(80 96) 89(80 96)

NOTE For each the mean and the 95 HPDI limits are given There is concordancebetween the left-side and right-side results The capture probabilities are more variableand reach minimums in 1997 and 2000 for the left side and in 1998 for the right sidebut both reach maximums in 2002 Survival reaches a maximum in 1992 declines to aminimum in 1997 and then rises again in later years

coefficients Because this is observed after adjustments for theoverall and time-varying components it suggests an effect dueto behavior or sampling protocol

The variation in the capture probability over time is likelyto result from a combination of factors including variation indolphin ranging patterns (see Wilson et al 2004) weather con-ditions and other logistical factors that influence sampling indifferent areas Most notably sampling protocols were changedin 2000 specifically to increase the probability of capture Thischange has proven to have been successful in the latter part ofthe study while estimates of the survival probabilities and to-tal population size remain unaffected Second another study(Wilson et al 2004) has shown that the ranging pattern of thepopulation changed in recent years with some animals spend-ing more time in the southern part of the population range alongthe east coast of Scotland south of the Moray Firth to St An-drews and the Firth of Forth The effect of this would be toreduce the capture probability in the core study area in the in-ner Moray Firth and to potentially increase heterogeneity incapture probabilities The more southerly areas also have beensampled but the data are more sparse and cover only a smallnumber of years Once these histories have become more exten-sive we expect that the capture probabilities will become lessvariable Unlike other models our estimates are also correctedfor individual heterogeneity Variation in capture probabilitiesalso might have arisen from heterogeneous effort in space Al-though this was not explicitly modeled we could have stratifiedthe capture histories by area by including separate capture co-efficients for each area

The overall mean survival estimate through the study periodis 93 (standard deviation 029 95 HPDI = 861 979) Thisis similar to an earlier maximum likelihood estimate of sur-vival for the cohort of nicked individuals from this populationfirst observed in 1990 (Saunders-Reed Hammond Grellier andThompson 1999) But our model estimate is slightly lower thansurvival estimates from other bottlenose populations of 962 plusmn76 in Sarasota Bay Florida (Wells and Scott 1990) 952 plusmn

Corkrey et al Bayesian Recapture Population Model 955

150 in Doubtful Sound New Zealand (Haase 2000) and 95ndash99 in the Sado estuary Portugal (Gaspar 2003) However thismay be because our study included subadults while the otherstudies reported adult survival Variation in survival probabil-ity occurs over time [Fig 2(b) Table 3] It is of interest thatsurvival reached a minimum in 1997 but later returned to lev-els comparable to earlier years The proportion of iterations inwhich the survival for 1997 are above the mean survival forthe remaining years (mean φ = 95 95 HPDI = 85 100)is only 08 The apparent low survival in 1997 may be due tomigration because this is not distinguishable from survival inthe model It is possible that an emigration event occurred in1997 and that those individuals have not yet returned Consid-erable heterogeneity in survival (range 77ndash97) is observedas shown in Figure 2(c) The survival probabilities are not dis-tributed smoothly suggesting multimodality The animals witha low survival coefficient may have died or migrated In thecase of migration this result would indicate that some animalsless consistently reside in the core areas compared with oth-ers

Individuals with higher survival coefficients are frequentlythose seen earliest in the study There is little information avail-able on individuals seen only more recently and consequentlytheir coefficients are close to 0 Some survival coefficients aremore negative and these correspond to animals that have notbeen seen for some time As time goes on and in the absenceof a confirmed death or sighting these coefficients can be ex-pected to become more negative However some animals thatare observed only on a few occasions have negative coefficientsThe estimation of individual survival coefficients means that es-timation of other coefficients in the model are adjusted for vari-able survival The estimation of individual capture probabilitiesallows the results to be combined with other individual covari-ates such as sex or body size if these data become availableSimilarly where individuals differ in their home ranges indi-vidual capture or survival probabilities could be related to localenvironmental conditions within their range to prey availabil-ity or to water quality

Estimates of the total population size from bilateral analy-sis of the left and right photo-identification series and estimatesfrom unilateral analyses of the left and right sides are shown inFigure 3(a) These are similar both to the maximum likelihoodestimate of 129 (95 confidence interval = 110ndash174) for datacollected in 1992 (Wilson et al 1999) and to a Bayesian mul-tisite mark recapture estimate of the nicked population of 85(95 probability interval = 76ndash263) for data collected in 2001(Durban et al 2005) The credible intervals of the abundanceestimates are generally narrower in the bilateral analysis (meanwidth 886) than in left-side analysis (mean width 1373) andright-side analysis (mean width 937) Figure 3(b) shows theprior and mean posterior recruitment probabilities ωj It is in-teresting that higher probabilities occur in 1996 and 2001 Thismight be considered to variations in capture probability butwhereas the capture probability is high in 2001 it remains lowin 1996 [see Fig 2(a)] Instead calculating the probabilities ofthe posterior exceeding the prior recruitment probabilities wefind they are below 5 (range 04ndash44) for all years except 2001in which it is 85 The higher recruitment probability in 2001may be due to an improved protocol in that year that led to the

(a)

(b)

Figure 3 (a) Mean total abundance estimates from a bilateral analy-sis with 95 HPDI Also shown are estimates from separate analysesof left (minus minus minus) and right (middot middot middot) sides along with associated 95 HPDIrepresented by symbols left + right Minimum numbers of ani-mals known to be alive are indicated by diamonds (b) Mean posteriorrecruitment probabilities ωj with 95 HPDI The prior value is rep-resented as a dotted line

identification of previously unrecognized nicked animals Weconsider the rise in 1996 to have insufficient evidence to sug-gest an increase in recruitment in that year

Earlier demographic modeling of this population has pre-dicted that the population is in decline (Saunders-Reed et al1999) To examine this we calculate the statistic [prodJ

j=2 Wj

(Wjminus1)]1(Jminus1) Values of this statistic gt1 indicate an increas-ing population whereas values lt1 indicates a decreasing pop-ulation The mean value is 989 (95 HPDI = 958 102)and the probability of it being lt10 is 77 This suggests thatthe population may be decreasing but that the evidence fora decline is weak By doing this we are able to use the fulltime series of photo identification data to obtain simultaneousposterior estimates of the Wj to compare the probability of apopulation increase or decrease In agreement with the earlierpredictive modeling these data indicate greater probability of adecline than of an increase However because it is likely thatthe population is expanding its range (Wilson et al 2004) thedecline may be confounded with temporary emigration

956 Journal of the American Statistical Association September 2008

6 DISCUSSION

Changes in the size of wildlife populations are often assessedby fitting trend models to independent abundance estimatesbut the power of these techniques to detect trends often canbe low (Gerrodette 1987 Taylor Martinez Gerrodette Bar-low and Hrovat 2007) Here we use a capturendashrecapture modelthat incorporates an underlying population model and an ob-servation model We provide a probability that a population isin decline or increasing Using photoidentification data fromthis bottlenose dolphin population we generate the first em-pirical estimate of trends in abundance for any of the popula-tions of small cetaceans inhabiting European waters Becausethe methodological changes in sampling likely are absorbedby the time-varying capture coefficient the remaining variationdue to individual capture tendencies might reasonably be de-scribed by the individual capture coefficient This then allowsexploration of capture and survival relating to individuals ratherthan to populations Information on individual survival could beimportant for addressing some management issues for exam-ple individuals differing in their spatial distribution

By decomposing the capture probabilities into overall time-varying and heterogeneous terms we find that the time-varyingcontribution and heterogeneous variation are comparable inmagnitude and their variation is greater than that of the overallterm This information can be used in the design of monitoringprograms by suggesting how effort should be distributed Thedifference in overall left and right capture probabilities mightbe interpretable as an artifact of the sampling techniques or ofbehavioral effects However after allowing for this the prob-ability of observing one side also varied over time with someyears strongly favoring one side or the other This might be ex-plained by variation in protocol between years such as changesin personnel The relative sizes of these effects may be of usein evaluating consistency of protocol over time Finally afterallowing for the overall difference and time variation the het-erogeneous capture coefficients exhibit a bias to the one sidepossibly due to behavior or protocol These findings demon-strate that bilateral decomposition of capture probabilities canprovide useful information

We decompose the survival probability similarly and we findit to be unusually low in 1997 suggesting a possible multimodalheterogeneous survival probability that may be the result of mi-gration or survival variation This type of decomposition allowsexploration of such effects in more detail In particular the mul-timodality of heterogeneous survival suggests that the popula-tion comprises individuals with different strategies some vis-iting the core study area more or less often Ongoing work isattempting to quantify individual heterogeneity such as in ex-posure to sewage contaminated waters and boat-based tourismTherefore this technique could be used to determine whetherexposure to different anthropogenic stressors is related to sur-vival

Our model also makes use of priors such as those on ωj thatare not informed by the recapture data set but make use of in-formation from other sources An alternative approach mightbe to specify these as informative priors based on expert opin-ion or tuned to obtain satisfactory results Our application il-lustrates how disparate data sources may be combined into a

single analysis Although the data source for the prior recruit-ment probability is small and problematic it is the only oneavailable Even when some priors are defined in the simulationstudy to incorrect values the estimates of parameters of interestremain valid The posterior probability also adds to the state ofknowledge of this parameter

Previous use of capturendashrecapture models with cetacean pho-toidentification data highlights the potential violation of modelassumptions due to heterogeneity in capture andor survivalprobabilities This model explicitly accounts for heterogene-ity and also provides simultaneous estimates of individual sur-vival and capture probabilities This can be used both to ex-plore the biological basis of heterogeneity and to improve sam-pling programs The individual survival coefficients also can beused to explore aspects of biology where correlates are avail-able for example the relationship of survival to age sex orother variables may be of interest Abundance under the as-sumption of heterogeneous survival has been estimated by LeeHuang and Ou (2004) whereas individual and time-varyingsurvival estimates have been obtained using band-recoverymodels (Grosbois and Thompson 2005) Other studies have ex-amined survival by group for example Franklin Andersonand Burnham (2002) obtained separate estimates of group-levelsurvival for males and females using stratification in a bandingstudy Although we lack such variables as age and sex theo-retically it would be possible to examine individual survivalby sex either as a post hoc comparison or through their inte-gration into the model itself For example survival and cap-ture could be modeled as logit(φ) = α + βj + γd + τ andlogit(π) = δ + εj + ζd + χ where τ corresponds to stage-specific (eg young immature reproductive female senes-cent) survival and χ corresponds to capture probabilities forvarious categories (eg young reproductive adult)

Other cetacean studies have examined heterogeneity White-head (2001b) obtained an estimate of the coefficient of variation(CV) for identification from a randomization test of successiveidentifications in whale track data (The CV is also a measure ofcapture heterogeneity) They found that the CV was significantin one of the two years and that probably depended on the sizesof the animals being observed This is not comparable to thedolphin observations in which heterogeneity is more likely toresult from differences in animal behavior In the present studywe use a likelihood in which heterogeneity of survival and cap-ture has been incorporated by a logit function This approachwas described previously (Pledger Pollock and Norris 2003Pledger and Schwarz 2002) within a non-Bayesian contextPledger et al (2003) classified animals into an unknown set oflatent classes each of which could have a separate survival andcapture probability A simplification allows for time-varyingand animal-level heterogeneity This approach is similar to oursbut we also integrate a dynamic population model and an ob-servation model into the estimation process Goodman (2004)also implemented a population model in a Bayesian context thatcombines capturendashrecapture data with carcass-recovery databut assumes the absence of heterogeneity in capture probabil-ities Durban et al (2005) estimated abundance for the samepopulation as us but used a multisite Bayesian log-linear modeland model averaging (King and Brooks 2001) across the pos-sible models with varying combinations of interactions Our

Corkrey et al Bayesian Recapture Population Model 957

model could be extended to include multiple sites and yearsby using a capturendashrecapture history that includes site informa-tion and a yearndashsite term in the capture probability expressionBy doing this we suggest that geographic areas with more datacan be used to strengthen estimates from poorly sampled areasThis also may account for some individual variation currentlyidentified as individual capture heterogeneity because some in-dividuals may be more closely associated with some areas thanothers and may attenuate a possible confounding of migrationwith survival

We use a separate data set and a negative-binomial model toinflate the estimate of the proportion of nicked animals in thesamples Another study (Da Silva Rodrigues Leite and Mi-lan 2003) obtained an abundance estimate for bowhead whalesusing an inflation factor derived from the numbers of good andbad photographs of marked and unmarked whales Our analysisavoids the lack of independence between good photographs anddistinctively marked animals in the photographs that might in-validate the estimate of variance in the inflation factor Otherstudies avoid this issue by including only separately catego-rized good-quality photographs (eg Wilson et al 1999 ReadUrian Wilson and Waples 2003) However independent es-timates of inflation when available such as in our study arepreferred

We do not know of other cetacean studies that combinephoto-identification series of the left and right sides but ananalogous approach is that of log-linear capturendashrecapturemodeling (King and Brooks 2001) in which animals are identi-fied from separate lists Some photo-identification studies mightuse information from both sides in the process of identifyinganimals (Joyce and Dorsey 1990) A more robust approach isto perform the identification on each side separately We haveshown how these separate data can then be combined

A major advantage to our modeling approach is its extensibil-ity Within the Bayesian framework models can be comparedby such methods as reversible-jump Markov chain Monte Carlotechniques (Green 1995) Model extensions include extendingthe population model considering an individual-level approachThis would allow incorporation of the heterogeneity into thispart of the model and also modeling of the unobserved popu-lation More detailed modeling of heterogeneity could be ex-plored by including hyperpriors for the variances of the recap-ture model terms For convenience we analyze the data on anannual basis Additional information would be available if theanalyses were conducted on a finer time scale such as monthsor by individual trip This would allow the model to be extendedto include seasonal effects For example dolphins are observedmore frequently in summer months than in winter months andthis tendency could be included in an extended model by chang-ing the capture probability to be logit(π) = δt + εj + ζd whereδt t isin [1 12] is a monthly capture coefficient This maybe of use in modeling fine-scale movements The response vari-able itself also can be generalized Although we are careful toonly include data from high-quality photographs and nicked an-imals there is still the possibility of some identification errorsparticularly for more subtly marked individuals A possible ex-tension of the model could allow identification to be assigneddifferent levels of certainty for example by incorporating anintermediate state between nonrecapture and recapture that we

term ldquopossible recapturerdquo This could be done by replacing thelogit function logit(ν) = log(ν(1 minus ν)) where ν correspondsto the probability of recapture by a proportional odds function(McCullagh 1980) This technique has previously found appli-cation in a study of photoidentification used to model photo-graphic quality scores and individual distinctiveness of hump-back whale tail flukes (Friday Smith Stevick and Allen 2000)In this approach the probability of ldquopossible recapturerdquo is givenby ν1 = π1 the probability of certain recapture is given byν2 = π1 + π2 and includes the possible recapture category andthe probability of no capture is given by ν0 = 1 minus π1 + π2There are two proportional odds functions corresponding topossible recapture and certain recapture which are logit(ν1) =log(π1(π2 +π3)) and logit(ν2) = log((π1 +π2)π3) We thencould extend the recapture model using logit(ν1) = α+βj +γk1

and logit(ν2) = α + βj + γk2 This proportional odds approachmay be particularly useful in situations where automatic match-ing algorithms (Saacutenchez-Mariacuten 2000) provide an estimate ofthe likelihood of a match which could then be incorporatedinto abundance estimate models

APPENDIX DERIVATION OF POSTERIORS

Here we show how we derive the posterior conditional distributionsfor the model parameters and describe how each is updated within theMCMC algorithm The full joint distribution has the form shown in(A1) In (A1) the first two lines correspond to the population modelthe third line corresponds to the inflation model the fourth line cor-responds to the observation model the fifth line corresponds to therecapture model and the remaining lines correspond to priors

π = P(B1|ω1N0W0)P (S1|αN0)

(Sec 31)

timesSprod

j=2

P(Bj |Sjminus1Bjminus1Wjminus1ωj )

times P(Sj |Sjminus1Bjminus1 αβjminus1)

(Sec 31)

timesSprod

j=1

P(Wj |Sj Bj υj )

Tjprod

i=1

P(tnji |υj twji )

(Sec 32)

timesSprod

j=1

P(nLj|δL εjLSj Bj )P (nRj

|δR εjRSj Bj )

(Sec 33)

timesSprod

j=1

P(Hdj |αβj γd δL εjL ζdL δR εjR ζdR)

(Sec 34)

times P(N0|λN)P (W0|N0 υ0)P (α δL δRλN υ0)

(Sec 41)

timesSprod

j=1

P(βj εjL εjRωj υj )

nprod

d=1

P(γd)P (ζdL)P (ζdR)

(Sec 41) (A1)

958 Journal of the American Statistical Association September 2008

For each parameter we obtain the corresponding posterior condi-tional distribution by extracting all terms containing that parameterfrom the full joint distribution When the conditional distributions havea standard form we use the Gibbs sampler (Brooks 1998) to update theparameters otherwise we use MetropolisndashHastings updates (Chib andGreenberg 1995)

Within a MetropolisndashHastings update we propose a new valuefor a parameter x and denote the proposed value by xprime To decidewhether to accept the proposed value we calculate an acceptance ra-tio α = min(1A) where A is the acceptance ratio The acceptanceratio is given by A = π(xprime)q(x)π(x)q(xprime) where π(x) is the poste-rior probability of x and q(x) is the probability of the proposed valueWe accept the new value xprime with probability α otherwise we leavethe value of x unchanged (See Chib and Greenberg 1995 for an exam-ple)

Normal proposal distributions are used for continuous parameterscentered around the current parameter value Discrete uniform pro-posals are used for discrete variables each centered around the currentparameter values The discrete proposals also require constraints to en-sure that the proposed values are valid For example a proposed valuefor N0 must be less than the current value of W0 We provide a detaileddescription of each update next

Population Model

The population model has parameters ωj λ Bj Sj and N0 andare updated using Gibbs or MetropolisndashHastings samplers The condi-tional posterior distributions for ωj and λN can be derived as

ωj |Bj Sjminus1Bjminus1

sim Beta(Bj + 0238Wjminus1 minus Sjminus1 minus Bjminus1 minus Bj + 1)

and

λN |N0 sim Gam(76 + N0101)

These are individually updated using the Gibbs sampler during eachiteration of the Markov chain

The parameters Bj Sj and N0 are updated by the MetropolisndashHastings algorithm using acceptance ratios calculated as described ear-lier We use uniform proposals centered on the current parameter valueand limited to the range of possible values

For the Sj rsquos we use three alternative proposals

Sprimej=1 sim Unif

(max

(0 S1 minus 5 S2 minus B1max(n1Ln1R) minus N1

2 minus B1)

min(S1 + 5N0W1 minus B1W1 minus B1 minus B2

W1 minus B1 minus 1))

Sprime2lejltJ sim Unif

(max

(0 Sj minus 5 Sj+1 minus Bj max(njLnjR) minus Bj

2 minus Bj

)

min(Sj + 5 Sjminus1 + Bjminus1Wj minus Bj Wj minus Bj minus Bj+1

Wj minus Bj minus 1))

and

Sprimej=J sim Unif

(max

(0 SJ minus 5max(nJLnJR) minus BJ 2 minus NJ

)

min(SJ minus 5 Sjminus1 + Bjminus1Wj minus Bj Wj minus Bj minus 1))

For Bj we use three alternative proposals

B prime1 sim Unif

(max

(0B1 minus 15 S2 minus S1max(n1Ln1R) minus S1

2 minus S1)

min(B1 + 15W0 minus N0W1 minus S1W1 minus S1 minus B2

W1 minus S1 minus 1))

B prime2lejltJ sim Unif

(max

(0Bj minus 15 Sj+1 minus Sj max(njLnjR) minus Sj

2 minus Sj

)

min(Bj + 15Wjminus1 minus Sjminus1 minus Bjminus1Wj minus Sj

Wj minus Sj minus Bj+1Wj minus Sj minus 1))

and

B primeJ sim Unif

(max

(0BJ minus 15max(nJLnJR) minus SJ 2 minus SJ

)

min(BJ + 15WJminus1 minus SJminus1 minus BJminus1

WJ minus SJ WJ minus SJ minus 1))

For N0 we use

N prime0 sim Unif

(max(5N0 minus 15 S1)min(N0 + 15W0 minus B1W0 minus 1)

)

Inflation Model

The inflation model parameters Wj and υj are updated usingMetropolisndashHastings and Gibbs updates The conditional posterior dis-tributions for υj can be derived as

υ0|N0W0 sim Beta(N0 + 1W0 minus N0 + 1)

and

υjgt0|Sj Bj Wj tnji twji sim Beta

(

Sj + Bj +T (j)sum

i=1

tnji + 1

Wj minus Sj minus Bj +T (j)sum

i=1

twji minus tnji + 1

)

These are updated separately using the Gibbs sampler on each itera-tion

The remaining parameter Wj has a nonstandard conditional pos-terior distribution We use larger updates than in the population modelbecause the initial total population is expected to be larger in size

W prime0 sim Unif

(max(0W0 minus 15N0 + 1N0 + B1)W0 + 15

)

W prime1lejltJ sim Unif

(max(0Wj minus 15 Sj + Bj + 1 Sj + Bj + Bj+1)

Wj + 15)

and

W primeJ sim Unif

(max(0WJ minus 15 SJ + BJ + 1)WJ + 15

)

Recapture Model

For the recapture model we have parameters α βj γd δL εjLζdL δR εjR and ζdR These all have nonstandard conditional pos-terior distributions and we use the MetropolisndashHastings algorithm toupdate the parameters on each iteration We use individual normal dis-tribution proposals centered on the current parameter values and withvariance equal to the prior variance for example

αprime sim N(α185)

[Received June 2005 Revised May 2007]

Corkrey et al Bayesian Recapture Population Model 959

REFERENCES

Brooks S P (1998) ldquoMarkov Chain Monte Carlo Method and Its ApplicationrdquoThe Statistician 47 69ndash100

Brooks S P and Roberts G O (1998) ldquoConvergence Assessment Techniquesfor Markov Chain Monte Carlordquo Statistics and Computing 8 319ndash335

Buckland S T (1982) ldquoA CapturendashRecapture Survival Analysisrdquo Journal ofAnimal Ecology 51 833ndash847

(1990) ldquoEstimation of Survival Rates From Sightings of IndividuallyIdentifiable Whalesrdquo in Individual Recognition of Cetaceans Use of Photo-Identification and Other Techniques to Estimate Population Parameters edsP S Hammond S A Mizroch and G P Donovan Cambridge UK Inter-national Whaling Commission pp 149ndash153

Burnham K and Overton W (1979) ldquoRobust Estimation of Population SizeWhen Capture Probabilities Vary Among Animalsrdquo Ecology 60 927ndash936

Carothers A (1979) ldquoQuantifying Unequal Catchability and Its Effect on Sur-vival Estimates in an Actual Populationrdquo Journal of Animal Ecology 48863ndash869

Chao A (1987) ldquoEstimating the Population Size for CapturendashRecapture DataWith Unequal Catchabilityrdquo Biometrics 43 783ndash791

Chao A Lee S-M and Jeng S-L (1992) ldquoEstimating Population Size forCapturendashRecapture Data When Capture Probabilities Vary by Time and Indi-vidual Animalrdquo Biometrics 48 201ndash216

Chib S and Greenberg E (1995) ldquoUnderstanding the MetropolisndashHastingsAlgorithmrdquo The American Statistician 49 327ndash335

Cormack R (1972) ldquoThe Logic of CapturendashRecapture Experimentsrdquo Biomet-rics 28 337ndash343

Coull B A and Agresti A (1999) ldquoThe Use of Mixed Logit Models to Re-flect Heterogeneity in CapturendashRecapture Studiesrdquo Biometrics 55 294ndash301

Da Silva C Q Rodrigues J Leite J G and Milan L A (2003) ldquoBayesianEstimation of the Size of a Closed Population Using Photo-ID With Part ofthe Population Uncatchablerdquo Communications in Statistics 32 677ndash696

Dorazio R M and Royle J A (2003) ldquoMixture Models for Estimating theSize of a Closed Population When Capture Rates Vary Among IndividualsrdquoBiometrics 59 351ndash364

Durban J Elston D Ellifrit D Dickson E Hammon P and Thompson P(2005) ldquoMultisite Mark-Recapture for Cetaceans Population Estimates WithBayesian Model Averagingrdquo Marine Mammal Science 21 80ndash92

Fienberg S E Johnson M S and Junker B W (1999) ldquoClassical Multi-level and Bayesian Approaches to Population Size Estimation Using MultipleListsrdquo Journal of the Royal Statistical Society Ser A 162 383ndash405

Forcada J and Aguilar A (2000) ldquoUse of Photographic Identification inCapturendashRecapture Studies of Mediterranean Monk Sealsrdquo Marine MammalScience 16 767ndash793

Franklin A B Anderson D R and Burnham K P (2002) ldquoEstimation ofLong-Term Trends and Variation in Avian Survival Probabilities Using Ran-dom Effects Modelsrdquo Journal of Applied Statistics 29 267ndash287

Friday N Smith T D Stevick P T and Allen J (2000) ldquoMeasure-ment of Photographic Quality and Individual Distinctiveness for the Photo-graphic Identification of Humpback Whales Megaptera novaeangliaerdquo Ma-rine Mammal Science 16 355ndash374

Gaspar R (2003) ldquoStatus of the Resident Bottlenose Dolphin Population inthe Sado Estuary Past Present and Futurerdquo unpublished doctoral dissertationUniversity of St Andrews School of Biology

Gelfand A E and Sahu S K (1999) ldquoIdentifiability Improper Priors andGibbs Sampling for Generalized Linear Modelsrdquo Journal of the AmericanStatistical Association 94 247ndash253

Gerrodette T (1987) ldquoA Power Analysis for Detecting Trendsrdquo Ecology 681364ndash1372

Goodman D (2004) ldquoMethods for Joint Inference From Multiple Data Sourcesfor Improved Estimates of Population Size and Survival Ratesrdquo Marine Mam-mal Science 20 401ndash423

Green P J (1995) ldquoReversible-Jump Markov Change Monte Carlo BayesianModel Determinationrdquo Biometrika 82 711ndash732

Grosbois V and Thompson P M (2005) ldquoNorth Atlantic Climate VariationInfluences Survival in Adult Fulmarsrdquo Oikos 109 273ndash290

Haase P (2000) ldquoSocial Organisation Behaviour and Population Parametersof Bottlenose Dolphins in Doubtful Sound Fiordlandrdquo unpublished mastersdissertation University of Otago Dept of Marine Science

Hammond P (1986) ldquoEstimating the Size of Naturally Marked Whale Pop-ulations Using CapturendashRecapture Techniquesrdquo Report of the InternationalWhaling Commission Special Issue 8 253ndash282

(1990) ldquoHeterogeneity in the Gulf of Maine Estimating HumpbackWhale Population Size When Capture Probabilities Are not Equalrdquo in In-dividual Recognition of Cetaceans Use of Photo-Identification and OtherTechniques to Estimate Population Parameters eds P S Hammond S A

Mizroch and G P Donovan Cambridge UK International Whaling Com-mission pp 135ndash153

Huggins R (1989) ldquoOn the Statistical Analysis of Capture Experimentsrdquo Bio-metrika 76 133ndash140

Joyce G G and Dorsey E M (1990) ldquoA Feasibility Study on the Useof Photo-Identification Techniques for Southern Hemisphere Minke WhaleStock Assessmentrdquo in Individual Recognition of Cetaceans Use of Photo-Identification and Other Techniques to Estimate Population Parameters edsP S Hammond S A Mizroch and G P Donovan Cambridge UK Inter-national Whaling Commission pp 419ndash423

King R and Brooks S (2001) ldquoOn the Bayesian Analysis of PopulationSizerdquo Biometrika 88 317ndash336

Lebreton J-D Burnham K P Clobert J and Anderson S R (1992)ldquoModeling Survival and Testing Biological Hypotheses Using Marked An-imals A Unified Approach With Case Studiesrdquo Ecological Monographs 6267ndash118

Lee S-M Huang L-H H and Ou S-T (2004) ldquoBand Recovery Model In-ference With Heterogeoneous Survival Ratesrdquo Statistica Sinica 14 513ndash531

Link W A and Barker R J (2005) ldquoModeling Association Among De-mographic Parameters in Analysis of Open Population CapturendashRecaptureDatardquo Biometrics 61 46ndash54

Mann J and Smuts B (1999) ldquoBehavioral Development in Wild BottlenoseDolphin Newborns (Tursiops sp)rdquo Behaviour 136 529ndash566

McCullagh P (1980) ldquoRegression Models for Ordinal Datardquo Journal of theRoyal Statistical Society Ser B 42 109ndash142

Otis D L Burnham K P White G C and Anderson D R (1978) ldquoStatis-tical Inference From Capture Data on Closed Animal Populationsrdquo WildlifeMonographs 62 135

Pledger S (2000) ldquoUnified Maximum Likelihood Estimates for ClosedCapturendashRecapture Models Using Mixturesrdquo Biometrics 56 434ndash442

Pledger S and Schwarz C J (2002) ldquoModelling Heterogeneity of Survivalin Band-Recovery Data Using Mixturesrdquo Journal of Applied Statistics 29315ndash327

Pledger S Pollock K H and Norris J L (2003) ldquoOpen CapturendashRecaptureModels With Heterogeneity I CormackndashJollyndashSeber Modelrdquo Ecology 59786ndash794

Pollock K H Nichols J D Brownie C and Hines J E (1990) ldquoStatisticalInference for CapturendashRecapture Samplingrdquo Wildlife Monographs 107 1ndash97

Read A Urian K W Wilson B and Waples D (2003) ldquoAbundance andStock Identity of Bottlenose Dolphins in the Bayes Sounds and Estuaries ofNorth Carolinardquo Marine Mammal Science 19 59ndash73

Saacutenchez-Mariacuten F J (2000) ldquoAutomatic Recognition of Biological ShapesWith and Without Representations of Shaperdquo Artificial Intelligence in Medi-cine 18 173ndash186

Saunders-Reed C A Hammond P S Grellier K and Thompson P M(1999) ldquoDevelopment of a Population Model for Bottlenose Dolphinsrdquo Re-search Survey and Monitoring Report 156 Scottish Natural Heritage

Scottish Natural Heritage (1995) Natura 2000 A Guide to the 1992 EC Habi-tats Directive in Scotlandrsquos Marine Environment Perth UK Scottish Nat-ural Heritage

Seber G (2002) The Estimation of Animal Abundance Caldwell NJ Black-burn Press

Stevick P T Palsboslashll P J Smith T D Bravington M V and Ham-mond P S (2001) ldquoErrors in Identification Using Natural Markings RatesSources and Effects on CapturendashRecapture Estimates of Abundancerdquo Cana-dian Journal of Fisheries and Aquatic Science 58 1861ndash1870

Taylor B L Martinez M Gerrodette T Barlow J and Hrovat Y N (2007)ldquoLessons From Monitoring Trends in Abundance of Marine Mammalsrdquo Ma-rine Mammal Science 23 157ndash175

Wells R S and Scott M D (1990) ldquoEstimating Bottlenose Dolphin Popu-lation Parameters From Individual Identification and CapturendashRelease Tech-niquesrdquo in Individual Recognition of Cetaceans Use of Photo-Identificationand Other Techniques to Estimate Population Parameters eds P S Ham-mond S A Mizroch and G P Donovan Cambridge UK InternationalWhaling Commission pp 407ndash415

Whitehead H (2001a) ldquoAnalysis of Animal Movement Using Opportunis-tic Individual Identifications Application to Sperm Whalesrdquo Ecology 821417ndash1432

(2001b) ldquoDirect Estimation of Within-Group Heterogeneity in Photo-Identification of Sperm Whalesrdquo Marine Mammal Science 17 718ndash728

Williams B K Nichols J D and Conroy M J (2001) Analysis and Man-agement of Animal Populations San Diego Academic Press

Wilson B Hammond P S and Thompson P M (1999) ldquoEstimating Size andAssessing Trends in a Coastal Bottlenose Dolphin Populationrdquo EcologicalApplications 9 288ndash300

Wilson B Reid R J Grellier K Thompson P M and Hammond P S(2004) ldquoConsidering the Temporal When Managing the Spatial A Popula-

952 Journal of the American Statistical Association September 2008

We refer to this model as the recapture model It yieldsthe probability P(hdjs |φdj γd πdjLπdjR ζdL ζdR) This isa CormackndashJollyndashSeber model which has been discussed else-where (Seber 2002 Lebreton Burnham Clobert and Anderson1992 Link and Barker 2005)

This completes the model specification To summarize wedefine a population model that describes the recruitment (ωj )and survival (θj ) of animals in the nicked population (Nj ) Therelative sizes of the nicked population (Nj ) and total population(Wj ) are described by the inflation model which makes use ofobserved counts of nicked and unnicked animals Using the cap-ture probability (ρjs ) and the observed number of nicked ani-mals we estimate the nicked population Finally we estimatethe heterogeneous capture (πdjs ) and survival (φdj ) probabili-ties from the capture histories Because this is a Bayesian analy-sis we also specify priors to the following parameters S0 B0ωj W0 N0 α βj γd δs εjs and ζds We describe the priorsin detail in the next section

4 ANALYSIS

In this section we describe the statistical methodology re-quired to analyze the model described in the previous sectionWe adopt a Bayesian approach and describe the prior specifica-tion and simulation techniques required here In this modelingprocess we use a Bayesian approach to allow for uncertaintyin measurement and parameters used The Bayesian approachallows for simultaneous consideration of all data and modelsallowing for the proper propagation of uncertainty throughoutthe model Identifiability is also less of a concern in a Bayesiananalysis in which prior information is supplied (Gelfand andSahu 1999) Moreover the Bayesian approach has an advan-tage in that MCMC methods can be used which greatly simpli-fies the computation compared with the corresponding classi-cal tools The remainder of this section discusses the process ofprior specification and provides details of the implementation

41 Priors

The first stage in the Bayesian analysis is to elicit priors foreach of the model parameters We do so here by grouping theparameters according to the component of the model in whichthey appear The priors used and their associated parameters aresummarized in what follows

The Population Model The model specifies the distribu-tion of parameters Sj and Bj in terms of Sjminus1 and Bjminus1 forj ge 1 Therefore we need a prior for S0 and B0 Becausethey only appear together we set N0 = S0 + B0 and assigna Poisson prior N0 sim Po(λN) Its mean takes a gamma priorλN sim Gam(76 01) with parameters selected to obtain a meanfor the nicked population of 76 as estimated by Wilson et al(1999) and a large variance so that it becomes uninformativecompared with the range of reasonable population values

Because the recruitment rates of nicked dolphins cannot bedirectly estimated from the photo-identification data set thepriors are parameterized with results from a separate analy-sis of randomly selected photographs with at least 2 years ofdata Because it is thought that the rate of nicking might begreater for older animals due to increased social interactionsestimates are estimated from photographs of individuals that

are first identified as known-age calves Identification is doneusing all attributes such as fin shape and the presence of le-sions Photographs of seven of these animals are available forestimating the age at which these dolphins are first seen to havea nick in the dorsal fin We consider these cases a random se-lection from the unnicked population so that the mean num-ber of nicks per year gives an approximate probability of nick-ing per year We set the prior for ωj to be reasonably vagueωj sim Beta(02381) Although rough these assumptions givea crude estimate for the parameter which is then updated bythe data

The Inflation Model The initial population size is assigneda negative-binomial prior W0 minusN0 sim NegBin(N0 υ0) The ini-tial inflation factor υ0 is assigned a beta prior υ0 sim Beta(11)which is equivalent to a noninformative uniform prior

The Recapture Model The recapture model survival pa-rameters have independent normal priors N(0 185) and thecapture parameters have independent normal priors N(0 15)These priors span the range (01) on the logit scale Largervariances are not used because these would obtain a U-shapeddistribution on the logit scale

42 Implementation

Once priors are assigned the full joint distribution can bedetermined as the product of the likelihood with the associatedprior distributions The resulting posterior is complex and high-dimensional and so inference is obtained in the form of poste-rior means and variances obtained through MCMC simulationWe choose to use an implementation in which each parameter isupdated in turn using either a MetropolisndashHastings or a Gibbsupdate depending on the form of the associated posterior con-ditional distributions The updating strategy is detailed in theAppendix All simulation software is written in FORTRAN 77The model was run for 500000 iterations and a 50 burn-in was used Sensitivity studies and standard diagnostic tech-niques were used (Brooks and Roberts 1998) to assess modelvalidity

43 Simulation Study

To test the MCMC code and to demonstrate the utility of theframework developed earlier we construct two artificial pop-ulations one with homogeneous capture and survival proba-bilities and the other with heterogeneous capture and survivalprobabilities These populations are constructed using the fol-lowing procedure An initial population of animals (n = 130)is sampled and a proportion of this initial population is ran-domly marked as nicked At each time period in the simulationanimals are randomly selected to die some of the unnicked be-come nicked and a proportion of the remaining animals arerandomly selected to be observed For each population threesimulations are run with differing priors for the ωj and λN para-meters The priors are such that the three cases represent vaguepriors precise priors and very precise priors but in each casethe means differ from the actual population values One possi-ble outcome is that the precise prior simulations track the priormeans more closely than the vague simulations

The simulations are illustrated in Table 1 which shows theactual total abundances the medians of the posteriors for Wj

Corkrey et al Bayesian Recapture Population Model 953

Table 1 Actual and estimated median total abundances for two artificial populations with 95 HPDI and posterior p values fromthree simulations on each with differing priors

A B C

Population Year Wj Wj p value Wj p value Wj p value

1 1990 124 119(99139) 28 112(97130) 07 107(90124) 021991 120 125(105146) 67 118(95134) 39 114(97135) 251992 114 116(96135) 54 110(93127) 30 106(92123) 171993 109 113(95134) 63 108(92124) 41 105(90121) 261994 103 111(93130) 81 106(88124) 63 103(86118) 491995 97 105(89125) 86 102(87118) 74 100(85114) 621996 95 98(84115) 66 95(81110) 48 93(81107) 381997 91 99(83114) 85 96(80109) 76 95(80108) 651998 89 94(79110) 73 91(77105) 56 90(77103) 521999 85 92(77108) 78 89(75103) 69 87(75103) 632000 76 82(6598) 76 79(6696) 65 79(6594) 592001 70 79(6299) 84 77(5895) 79 76(6095) 752002 68 76(5593) 78 75(5691) 75 73(5892) 70

2 1990 124 135(100178) 71 121(96159) 42 116(90149) 261991 122 123(90158) 51 112(87140) 21 108(84134) 121992 119 114(87148) 35 105(84130) 10 101(78124) 061993 111 106(82138) 34 99(75121) 11 95(74117) 061994 106 102(76130) 33 93(75118) 12 91(68115) 071995 100 96(70122) 38 91(74114) 16 88(67108) 081996 93 93(66119) 48 87(69109) 26 85(65105) 161997 87 91(67115) 60 85(68106) 37 82(62102) 281998 77 79(59101) 56 76(6295) 39 72(5791) 281999 74 84(60108) 80 80(62101) 75 77(5897) 622000 72 78(5799) 67 74(5894) 58 72(5591) 462001 67 71(4992) 62 69(4891) 57 66(4688) 432002 64 68(4191) 59 65(4492) 51 63(4388) 44

NOTE Population 1 has homogeneous capture probability (75) and survival probability (95) while population 2 has heterogeneous capture probability (mean 55 range 05ndash98) andsurvival probability (mean 92 range 68ndash99) The simulations have different priors for ωj and λN Simulation run A has vague priors [ω sim Beta(029412368) λN sim Gam(76 01)]B has more precise priors [ω sim Beta(023710) λN sim Gam(761)] and C has very precise priors [ω sim Beta(929939155) λN sim Gam(7600100)] Their means are the same ineach case (023 and 76) but each differs from the actual population values (06 and 106)

their 95 highest posterior density intervals (HPDI) and a p

value The p values are calculated by counting the proportionof iterations in which the Wj exceeded the true value Examina-tion of the results indicate that the median posterior estimatesgenerally agree with the actual values The agreement is sat-isfactory whether or not there is heterogeneity assumed Thep values are in the range 02ndash88 suggesting that the modelworks correctly The posterior p values for the recapture modelcoefficients are given in Table 2 The p values for α δL andδR are summarized and those for the remaining parameters aresummarized as the proportion of p values in each that lie insidethe range 01ndash99 Almost all p values are within the range 01ndash99 the only exceptions are the capture coefficients for a smallnumber of dolphins

In general the total population estimates for the homoge-neous case are slightly greater and the 95 HPDI is narrowerthen the heterogeneous case but all intervals contain the popu-lation values The estimates also shrink slightly for the mod-els with more precise priors However the p values in eachcase suggest an adequate fit even though some priors are de-liberately set to incorrect values In the case of the recapturecoefficients there is no apparent difference between the homo-geneous and heterogeneous models or with any trend in theprecision of the priors

5 RESULTS

We obtain estimates of heterogeneous capture and survivalcoefficients variation of capture and survival over time over-all survival and capture probabilities the abundance and thetrend in abundance The survival probabilities and capture co-efficients from the left and right sides are shown in Figure 2 andTable 3 We can obtain an overall capture probability for the twosides by calculating exp(δ)(1+exp(δ)) The mean overall cap-

Table 2 Summary of posterior p values for survival and capturecoefficients for two artificial populations from three simulations

on each with differing priors

Population α βj γd δL εjL ζdL δR εjR ζdR

1 16 100 100 66 100 98 60 100 10010 100 100 59 100 98 53 100 10008 100 100 59 100 98 48 100 100

2 34 100 100 17 100 97 50 100 9822 100 100 15 100 97 37 100 9923 100 100 12 100 97 39 100 99

NOTE The populations and simulations are described in Table 1 The p values are calcu-lated as the proportion of iterations greater than the actual population value The p valuesfor α δL and δR are shown Because the remaining parameters are vectors of coefficientsthose p values are summarized as the proportion of p values in each that are found insidethe range 01ndash99

954 Journal of the American Statistical Association September 2008

(a)

(b) (c)

Figure 2 (a) Variation of mean capture coefficients over time forthe left side εL (minusminus) and right side εR (- - bull - - bull) The two hor-izontal lines indicate the global mean capture coefficients δL (mdashmdash)and δR (minus minus minus) On the left vertical axis are the mean left-side het-erogeneity capture coefficients ζL and on the right vertical axis arethe mean right-side heterogeneity capture coefficients ζR Standardserrors for each coefficient are shown as vertical bars (b) Variation ofmean survival probability over time (c) Histogram of mean heteroge-neous survival probabilities

ture probability for the left side is 48 (95 HPDI = 29 68)and for the right side it is 58 (95 HPDI = 39 77) Althoughthis is a small difference the probability of the right capturecoefficient exceeding the left is 81 Examination of Figure 2(a)shows that the variation of capture coefficients over time is largecompared with the overall coefficients and that the left (εL) andright (εR) coefficients tend to track each other The probabilityof the right capture coefficient exceeding the left varies overa wide range 07ndash96 This range results from some years inwhich the probability tends strongly to the left side (1995 and2000 probability of observing the left side 95) or right side(1998 1999 2001 probability of observing the right side 91)Finally the range of the heterogeneous coefficients is compara-ble to that of the time-varying coefficients

The mean heterogeneous survival coefficient is not corre-lated with either of the mean capture heterogeneous coeffi-cients (right r = 009 left r = 022) However the meanleft and right heterogeneous capture coefficients are correlated(r = 714) indicating that animals that are more likely to beseen from one side also are more likely to be seen from theother side However the heterogeneity capture coefficient of theright side (mean 56) exceeded that of the left side (mean 23)with 85 of animals having higher right coefficients than left

Table 3 Left-side and right-side capture probabilities and survivalprobabilities by year

Year Left capture Right capture Survival

1990 1(11)

1991 54(40 68) 60(47 75) 93(84 99)

1992 74(59 86) 75(60 86) 98(94100)

1993 31(20 44) 46(33 61) 95(88100)

1994 36(24 47) 53(35 66) 95(86100)

1995 42(27 55) 71(56 88) 94(86100)

1996 30(17 42) 48(34 63) 93(82100)

1997 23(14 36) 37(23 51) 88(75 98)

1998 39(24 53) 31(20 45) 94(84100)

1999 45(31 63) 40(25 54) 93(82100)

2000 24(14 36) 57(40 72) 96(88100)

2001 87(78 94) 82(73 91) 96(89100)

2002 88(80 96) 89(80 96)

NOTE For each the mean and the 95 HPDI limits are given There is concordancebetween the left-side and right-side results The capture probabilities are more variableand reach minimums in 1997 and 2000 for the left side and in 1998 for the right sidebut both reach maximums in 2002 Survival reaches a maximum in 1992 declines to aminimum in 1997 and then rises again in later years

coefficients Because this is observed after adjustments for theoverall and time-varying components it suggests an effect dueto behavior or sampling protocol

The variation in the capture probability over time is likelyto result from a combination of factors including variation indolphin ranging patterns (see Wilson et al 2004) weather con-ditions and other logistical factors that influence sampling indifferent areas Most notably sampling protocols were changedin 2000 specifically to increase the probability of capture Thischange has proven to have been successful in the latter part ofthe study while estimates of the survival probabilities and to-tal population size remain unaffected Second another study(Wilson et al 2004) has shown that the ranging pattern of thepopulation changed in recent years with some animals spend-ing more time in the southern part of the population range alongthe east coast of Scotland south of the Moray Firth to St An-drews and the Firth of Forth The effect of this would be toreduce the capture probability in the core study area in the in-ner Moray Firth and to potentially increase heterogeneity incapture probabilities The more southerly areas also have beensampled but the data are more sparse and cover only a smallnumber of years Once these histories have become more exten-sive we expect that the capture probabilities will become lessvariable Unlike other models our estimates are also correctedfor individual heterogeneity Variation in capture probabilitiesalso might have arisen from heterogeneous effort in space Al-though this was not explicitly modeled we could have stratifiedthe capture histories by area by including separate capture co-efficients for each area

The overall mean survival estimate through the study periodis 93 (standard deviation 029 95 HPDI = 861 979) Thisis similar to an earlier maximum likelihood estimate of sur-vival for the cohort of nicked individuals from this populationfirst observed in 1990 (Saunders-Reed Hammond Grellier andThompson 1999) But our model estimate is slightly lower thansurvival estimates from other bottlenose populations of 962 plusmn76 in Sarasota Bay Florida (Wells and Scott 1990) 952 plusmn

Corkrey et al Bayesian Recapture Population Model 955

150 in Doubtful Sound New Zealand (Haase 2000) and 95ndash99 in the Sado estuary Portugal (Gaspar 2003) However thismay be because our study included subadults while the otherstudies reported adult survival Variation in survival probabil-ity occurs over time [Fig 2(b) Table 3] It is of interest thatsurvival reached a minimum in 1997 but later returned to lev-els comparable to earlier years The proportion of iterations inwhich the survival for 1997 are above the mean survival forthe remaining years (mean φ = 95 95 HPDI = 85 100)is only 08 The apparent low survival in 1997 may be due tomigration because this is not distinguishable from survival inthe model It is possible that an emigration event occurred in1997 and that those individuals have not yet returned Consid-erable heterogeneity in survival (range 77ndash97) is observedas shown in Figure 2(c) The survival probabilities are not dis-tributed smoothly suggesting multimodality The animals witha low survival coefficient may have died or migrated In thecase of migration this result would indicate that some animalsless consistently reside in the core areas compared with oth-ers

Individuals with higher survival coefficients are frequentlythose seen earliest in the study There is little information avail-able on individuals seen only more recently and consequentlytheir coefficients are close to 0 Some survival coefficients aremore negative and these correspond to animals that have notbeen seen for some time As time goes on and in the absenceof a confirmed death or sighting these coefficients can be ex-pected to become more negative However some animals thatare observed only on a few occasions have negative coefficientsThe estimation of individual survival coefficients means that es-timation of other coefficients in the model are adjusted for vari-able survival The estimation of individual capture probabilitiesallows the results to be combined with other individual covari-ates such as sex or body size if these data become availableSimilarly where individuals differ in their home ranges indi-vidual capture or survival probabilities could be related to localenvironmental conditions within their range to prey availabil-ity or to water quality

Estimates of the total population size from bilateral analy-sis of the left and right photo-identification series and estimatesfrom unilateral analyses of the left and right sides are shown inFigure 3(a) These are similar both to the maximum likelihoodestimate of 129 (95 confidence interval = 110ndash174) for datacollected in 1992 (Wilson et al 1999) and to a Bayesian mul-tisite mark recapture estimate of the nicked population of 85(95 probability interval = 76ndash263) for data collected in 2001(Durban et al 2005) The credible intervals of the abundanceestimates are generally narrower in the bilateral analysis (meanwidth 886) than in left-side analysis (mean width 1373) andright-side analysis (mean width 937) Figure 3(b) shows theprior and mean posterior recruitment probabilities ωj It is in-teresting that higher probabilities occur in 1996 and 2001 Thismight be considered to variations in capture probability butwhereas the capture probability is high in 2001 it remains lowin 1996 [see Fig 2(a)] Instead calculating the probabilities ofthe posterior exceeding the prior recruitment probabilities wefind they are below 5 (range 04ndash44) for all years except 2001in which it is 85 The higher recruitment probability in 2001may be due to an improved protocol in that year that led to the

(a)

(b)

Figure 3 (a) Mean total abundance estimates from a bilateral analy-sis with 95 HPDI Also shown are estimates from separate analysesof left (minus minus minus) and right (middot middot middot) sides along with associated 95 HPDIrepresented by symbols left + right Minimum numbers of ani-mals known to be alive are indicated by diamonds (b) Mean posteriorrecruitment probabilities ωj with 95 HPDI The prior value is rep-resented as a dotted line

identification of previously unrecognized nicked animals Weconsider the rise in 1996 to have insufficient evidence to sug-gest an increase in recruitment in that year

Earlier demographic modeling of this population has pre-dicted that the population is in decline (Saunders-Reed et al1999) To examine this we calculate the statistic [prodJ

j=2 Wj

(Wjminus1)]1(Jminus1) Values of this statistic gt1 indicate an increas-ing population whereas values lt1 indicates a decreasing pop-ulation The mean value is 989 (95 HPDI = 958 102)and the probability of it being lt10 is 77 This suggests thatthe population may be decreasing but that the evidence fora decline is weak By doing this we are able to use the fulltime series of photo identification data to obtain simultaneousposterior estimates of the Wj to compare the probability of apopulation increase or decrease In agreement with the earlierpredictive modeling these data indicate greater probability of adecline than of an increase However because it is likely thatthe population is expanding its range (Wilson et al 2004) thedecline may be confounded with temporary emigration

956 Journal of the American Statistical Association September 2008

6 DISCUSSION

Changes in the size of wildlife populations are often assessedby fitting trend models to independent abundance estimatesbut the power of these techniques to detect trends often canbe low (Gerrodette 1987 Taylor Martinez Gerrodette Bar-low and Hrovat 2007) Here we use a capturendashrecapture modelthat incorporates an underlying population model and an ob-servation model We provide a probability that a population isin decline or increasing Using photoidentification data fromthis bottlenose dolphin population we generate the first em-pirical estimate of trends in abundance for any of the popula-tions of small cetaceans inhabiting European waters Becausethe methodological changes in sampling likely are absorbedby the time-varying capture coefficient the remaining variationdue to individual capture tendencies might reasonably be de-scribed by the individual capture coefficient This then allowsexploration of capture and survival relating to individuals ratherthan to populations Information on individual survival could beimportant for addressing some management issues for exam-ple individuals differing in their spatial distribution

By decomposing the capture probabilities into overall time-varying and heterogeneous terms we find that the time-varyingcontribution and heterogeneous variation are comparable inmagnitude and their variation is greater than that of the overallterm This information can be used in the design of monitoringprograms by suggesting how effort should be distributed Thedifference in overall left and right capture probabilities mightbe interpretable as an artifact of the sampling techniques or ofbehavioral effects However after allowing for this the prob-ability of observing one side also varied over time with someyears strongly favoring one side or the other This might be ex-plained by variation in protocol between years such as changesin personnel The relative sizes of these effects may be of usein evaluating consistency of protocol over time Finally afterallowing for the overall difference and time variation the het-erogeneous capture coefficients exhibit a bias to the one sidepossibly due to behavior or protocol These findings demon-strate that bilateral decomposition of capture probabilities canprovide useful information

We decompose the survival probability similarly and we findit to be unusually low in 1997 suggesting a possible multimodalheterogeneous survival probability that may be the result of mi-gration or survival variation This type of decomposition allowsexploration of such effects in more detail In particular the mul-timodality of heterogeneous survival suggests that the popula-tion comprises individuals with different strategies some vis-iting the core study area more or less often Ongoing work isattempting to quantify individual heterogeneity such as in ex-posure to sewage contaminated waters and boat-based tourismTherefore this technique could be used to determine whetherexposure to different anthropogenic stressors is related to sur-vival

Our model also makes use of priors such as those on ωj thatare not informed by the recapture data set but make use of in-formation from other sources An alternative approach mightbe to specify these as informative priors based on expert opin-ion or tuned to obtain satisfactory results Our application il-lustrates how disparate data sources may be combined into a

single analysis Although the data source for the prior recruit-ment probability is small and problematic it is the only oneavailable Even when some priors are defined in the simulationstudy to incorrect values the estimates of parameters of interestremain valid The posterior probability also adds to the state ofknowledge of this parameter

Previous use of capturendashrecapture models with cetacean pho-toidentification data highlights the potential violation of modelassumptions due to heterogeneity in capture andor survivalprobabilities This model explicitly accounts for heterogene-ity and also provides simultaneous estimates of individual sur-vival and capture probabilities This can be used both to ex-plore the biological basis of heterogeneity and to improve sam-pling programs The individual survival coefficients also can beused to explore aspects of biology where correlates are avail-able for example the relationship of survival to age sex orother variables may be of interest Abundance under the as-sumption of heterogeneous survival has been estimated by LeeHuang and Ou (2004) whereas individual and time-varyingsurvival estimates have been obtained using band-recoverymodels (Grosbois and Thompson 2005) Other studies have ex-amined survival by group for example Franklin Andersonand Burnham (2002) obtained separate estimates of group-levelsurvival for males and females using stratification in a bandingstudy Although we lack such variables as age and sex theo-retically it would be possible to examine individual survivalby sex either as a post hoc comparison or through their inte-gration into the model itself For example survival and cap-ture could be modeled as logit(φ) = α + βj + γd + τ andlogit(π) = δ + εj + ζd + χ where τ corresponds to stage-specific (eg young immature reproductive female senes-cent) survival and χ corresponds to capture probabilities forvarious categories (eg young reproductive adult)

Other cetacean studies have examined heterogeneity White-head (2001b) obtained an estimate of the coefficient of variation(CV) for identification from a randomization test of successiveidentifications in whale track data (The CV is also a measure ofcapture heterogeneity) They found that the CV was significantin one of the two years and that probably depended on the sizesof the animals being observed This is not comparable to thedolphin observations in which heterogeneity is more likely toresult from differences in animal behavior In the present studywe use a likelihood in which heterogeneity of survival and cap-ture has been incorporated by a logit function This approachwas described previously (Pledger Pollock and Norris 2003Pledger and Schwarz 2002) within a non-Bayesian contextPledger et al (2003) classified animals into an unknown set oflatent classes each of which could have a separate survival andcapture probability A simplification allows for time-varyingand animal-level heterogeneity This approach is similar to oursbut we also integrate a dynamic population model and an ob-servation model into the estimation process Goodman (2004)also implemented a population model in a Bayesian context thatcombines capturendashrecapture data with carcass-recovery databut assumes the absence of heterogeneity in capture probabil-ities Durban et al (2005) estimated abundance for the samepopulation as us but used a multisite Bayesian log-linear modeland model averaging (King and Brooks 2001) across the pos-sible models with varying combinations of interactions Our

Corkrey et al Bayesian Recapture Population Model 957

model could be extended to include multiple sites and yearsby using a capturendashrecapture history that includes site informa-tion and a yearndashsite term in the capture probability expressionBy doing this we suggest that geographic areas with more datacan be used to strengthen estimates from poorly sampled areasThis also may account for some individual variation currentlyidentified as individual capture heterogeneity because some in-dividuals may be more closely associated with some areas thanothers and may attenuate a possible confounding of migrationwith survival

We use a separate data set and a negative-binomial model toinflate the estimate of the proportion of nicked animals in thesamples Another study (Da Silva Rodrigues Leite and Mi-lan 2003) obtained an abundance estimate for bowhead whalesusing an inflation factor derived from the numbers of good andbad photographs of marked and unmarked whales Our analysisavoids the lack of independence between good photographs anddistinctively marked animals in the photographs that might in-validate the estimate of variance in the inflation factor Otherstudies avoid this issue by including only separately catego-rized good-quality photographs (eg Wilson et al 1999 ReadUrian Wilson and Waples 2003) However independent es-timates of inflation when available such as in our study arepreferred

We do not know of other cetacean studies that combinephoto-identification series of the left and right sides but ananalogous approach is that of log-linear capturendashrecapturemodeling (King and Brooks 2001) in which animals are identi-fied from separate lists Some photo-identification studies mightuse information from both sides in the process of identifyinganimals (Joyce and Dorsey 1990) A more robust approach isto perform the identification on each side separately We haveshown how these separate data can then be combined

A major advantage to our modeling approach is its extensibil-ity Within the Bayesian framework models can be comparedby such methods as reversible-jump Markov chain Monte Carlotechniques (Green 1995) Model extensions include extendingthe population model considering an individual-level approachThis would allow incorporation of the heterogeneity into thispart of the model and also modeling of the unobserved popu-lation More detailed modeling of heterogeneity could be ex-plored by including hyperpriors for the variances of the recap-ture model terms For convenience we analyze the data on anannual basis Additional information would be available if theanalyses were conducted on a finer time scale such as monthsor by individual trip This would allow the model to be extendedto include seasonal effects For example dolphins are observedmore frequently in summer months than in winter months andthis tendency could be included in an extended model by chang-ing the capture probability to be logit(π) = δt + εj + ζd whereδt t isin [1 12] is a monthly capture coefficient This maybe of use in modeling fine-scale movements The response vari-able itself also can be generalized Although we are careful toonly include data from high-quality photographs and nicked an-imals there is still the possibility of some identification errorsparticularly for more subtly marked individuals A possible ex-tension of the model could allow identification to be assigneddifferent levels of certainty for example by incorporating anintermediate state between nonrecapture and recapture that we

term ldquopossible recapturerdquo This could be done by replacing thelogit function logit(ν) = log(ν(1 minus ν)) where ν correspondsto the probability of recapture by a proportional odds function(McCullagh 1980) This technique has previously found appli-cation in a study of photoidentification used to model photo-graphic quality scores and individual distinctiveness of hump-back whale tail flukes (Friday Smith Stevick and Allen 2000)In this approach the probability of ldquopossible recapturerdquo is givenby ν1 = π1 the probability of certain recapture is given byν2 = π1 + π2 and includes the possible recapture category andthe probability of no capture is given by ν0 = 1 minus π1 + π2There are two proportional odds functions corresponding topossible recapture and certain recapture which are logit(ν1) =log(π1(π2 +π3)) and logit(ν2) = log((π1 +π2)π3) We thencould extend the recapture model using logit(ν1) = α+βj +γk1

and logit(ν2) = α + βj + γk2 This proportional odds approachmay be particularly useful in situations where automatic match-ing algorithms (Saacutenchez-Mariacuten 2000) provide an estimate ofthe likelihood of a match which could then be incorporatedinto abundance estimate models

APPENDIX DERIVATION OF POSTERIORS

Here we show how we derive the posterior conditional distributionsfor the model parameters and describe how each is updated within theMCMC algorithm The full joint distribution has the form shown in(A1) In (A1) the first two lines correspond to the population modelthe third line corresponds to the inflation model the fourth line cor-responds to the observation model the fifth line corresponds to therecapture model and the remaining lines correspond to priors

π = P(B1|ω1N0W0)P (S1|αN0)

(Sec 31)

timesSprod

j=2

P(Bj |Sjminus1Bjminus1Wjminus1ωj )

times P(Sj |Sjminus1Bjminus1 αβjminus1)

(Sec 31)

timesSprod

j=1

P(Wj |Sj Bj υj )

Tjprod

i=1

P(tnji |υj twji )

(Sec 32)

timesSprod

j=1

P(nLj|δL εjLSj Bj )P (nRj

|δR εjRSj Bj )

(Sec 33)

timesSprod

j=1

P(Hdj |αβj γd δL εjL ζdL δR εjR ζdR)

(Sec 34)

times P(N0|λN)P (W0|N0 υ0)P (α δL δRλN υ0)

(Sec 41)

timesSprod

j=1

P(βj εjL εjRωj υj )

nprod

d=1

P(γd)P (ζdL)P (ζdR)

(Sec 41) (A1)

958 Journal of the American Statistical Association September 2008

For each parameter we obtain the corresponding posterior condi-tional distribution by extracting all terms containing that parameterfrom the full joint distribution When the conditional distributions havea standard form we use the Gibbs sampler (Brooks 1998) to update theparameters otherwise we use MetropolisndashHastings updates (Chib andGreenberg 1995)

Within a MetropolisndashHastings update we propose a new valuefor a parameter x and denote the proposed value by xprime To decidewhether to accept the proposed value we calculate an acceptance ra-tio α = min(1A) where A is the acceptance ratio The acceptanceratio is given by A = π(xprime)q(x)π(x)q(xprime) where π(x) is the poste-rior probability of x and q(x) is the probability of the proposed valueWe accept the new value xprime with probability α otherwise we leavethe value of x unchanged (See Chib and Greenberg 1995 for an exam-ple)

Normal proposal distributions are used for continuous parameterscentered around the current parameter value Discrete uniform pro-posals are used for discrete variables each centered around the currentparameter values The discrete proposals also require constraints to en-sure that the proposed values are valid For example a proposed valuefor N0 must be less than the current value of W0 We provide a detaileddescription of each update next

Population Model

The population model has parameters ωj λ Bj Sj and N0 andare updated using Gibbs or MetropolisndashHastings samplers The condi-tional posterior distributions for ωj and λN can be derived as

ωj |Bj Sjminus1Bjminus1

sim Beta(Bj + 0238Wjminus1 minus Sjminus1 minus Bjminus1 minus Bj + 1)

and

λN |N0 sim Gam(76 + N0101)

These are individually updated using the Gibbs sampler during eachiteration of the Markov chain

The parameters Bj Sj and N0 are updated by the MetropolisndashHastings algorithm using acceptance ratios calculated as described ear-lier We use uniform proposals centered on the current parameter valueand limited to the range of possible values

For the Sj rsquos we use three alternative proposals

Sprimej=1 sim Unif

(max

(0 S1 minus 5 S2 minus B1max(n1Ln1R) minus N1

2 minus B1)

min(S1 + 5N0W1 minus B1W1 minus B1 minus B2

W1 minus B1 minus 1))

Sprime2lejltJ sim Unif

(max

(0 Sj minus 5 Sj+1 minus Bj max(njLnjR) minus Bj

2 minus Bj

)

min(Sj + 5 Sjminus1 + Bjminus1Wj minus Bj Wj minus Bj minus Bj+1

Wj minus Bj minus 1))

and

Sprimej=J sim Unif

(max

(0 SJ minus 5max(nJLnJR) minus BJ 2 minus NJ

)

min(SJ minus 5 Sjminus1 + Bjminus1Wj minus Bj Wj minus Bj minus 1))

For Bj we use three alternative proposals

B prime1 sim Unif

(max

(0B1 minus 15 S2 minus S1max(n1Ln1R) minus S1

2 minus S1)

min(B1 + 15W0 minus N0W1 minus S1W1 minus S1 minus B2

W1 minus S1 minus 1))

B prime2lejltJ sim Unif

(max

(0Bj minus 15 Sj+1 minus Sj max(njLnjR) minus Sj

2 minus Sj

)

min(Bj + 15Wjminus1 minus Sjminus1 minus Bjminus1Wj minus Sj

Wj minus Sj minus Bj+1Wj minus Sj minus 1))

and

B primeJ sim Unif

(max

(0BJ minus 15max(nJLnJR) minus SJ 2 minus SJ

)

min(BJ + 15WJminus1 minus SJminus1 minus BJminus1

WJ minus SJ WJ minus SJ minus 1))

For N0 we use

N prime0 sim Unif

(max(5N0 minus 15 S1)min(N0 + 15W0 minus B1W0 minus 1)

)

Inflation Model

The inflation model parameters Wj and υj are updated usingMetropolisndashHastings and Gibbs updates The conditional posterior dis-tributions for υj can be derived as

υ0|N0W0 sim Beta(N0 + 1W0 minus N0 + 1)

and

υjgt0|Sj Bj Wj tnji twji sim Beta

(

Sj + Bj +T (j)sum

i=1

tnji + 1

Wj minus Sj minus Bj +T (j)sum

i=1

twji minus tnji + 1

)

These are updated separately using the Gibbs sampler on each itera-tion

The remaining parameter Wj has a nonstandard conditional pos-terior distribution We use larger updates than in the population modelbecause the initial total population is expected to be larger in size

W prime0 sim Unif

(max(0W0 minus 15N0 + 1N0 + B1)W0 + 15

)

W prime1lejltJ sim Unif

(max(0Wj minus 15 Sj + Bj + 1 Sj + Bj + Bj+1)

Wj + 15)

and

W primeJ sim Unif

(max(0WJ minus 15 SJ + BJ + 1)WJ + 15

)

Recapture Model

For the recapture model we have parameters α βj γd δL εjLζdL δR εjR and ζdR These all have nonstandard conditional pos-terior distributions and we use the MetropolisndashHastings algorithm toupdate the parameters on each iteration We use individual normal dis-tribution proposals centered on the current parameter values and withvariance equal to the prior variance for example

αprime sim N(α185)

[Received June 2005 Revised May 2007]

Corkrey et al Bayesian Recapture Population Model 959

REFERENCES

Brooks S P (1998) ldquoMarkov Chain Monte Carlo Method and Its ApplicationrdquoThe Statistician 47 69ndash100

Brooks S P and Roberts G O (1998) ldquoConvergence Assessment Techniquesfor Markov Chain Monte Carlordquo Statistics and Computing 8 319ndash335

Buckland S T (1982) ldquoA CapturendashRecapture Survival Analysisrdquo Journal ofAnimal Ecology 51 833ndash847

(1990) ldquoEstimation of Survival Rates From Sightings of IndividuallyIdentifiable Whalesrdquo in Individual Recognition of Cetaceans Use of Photo-Identification and Other Techniques to Estimate Population Parameters edsP S Hammond S A Mizroch and G P Donovan Cambridge UK Inter-national Whaling Commission pp 149ndash153

Burnham K and Overton W (1979) ldquoRobust Estimation of Population SizeWhen Capture Probabilities Vary Among Animalsrdquo Ecology 60 927ndash936

Carothers A (1979) ldquoQuantifying Unequal Catchability and Its Effect on Sur-vival Estimates in an Actual Populationrdquo Journal of Animal Ecology 48863ndash869

Chao A (1987) ldquoEstimating the Population Size for CapturendashRecapture DataWith Unequal Catchabilityrdquo Biometrics 43 783ndash791

Chao A Lee S-M and Jeng S-L (1992) ldquoEstimating Population Size forCapturendashRecapture Data When Capture Probabilities Vary by Time and Indi-vidual Animalrdquo Biometrics 48 201ndash216

Chib S and Greenberg E (1995) ldquoUnderstanding the MetropolisndashHastingsAlgorithmrdquo The American Statistician 49 327ndash335

Cormack R (1972) ldquoThe Logic of CapturendashRecapture Experimentsrdquo Biomet-rics 28 337ndash343

Coull B A and Agresti A (1999) ldquoThe Use of Mixed Logit Models to Re-flect Heterogeneity in CapturendashRecapture Studiesrdquo Biometrics 55 294ndash301

Da Silva C Q Rodrigues J Leite J G and Milan L A (2003) ldquoBayesianEstimation of the Size of a Closed Population Using Photo-ID With Part ofthe Population Uncatchablerdquo Communications in Statistics 32 677ndash696

Dorazio R M and Royle J A (2003) ldquoMixture Models for Estimating theSize of a Closed Population When Capture Rates Vary Among IndividualsrdquoBiometrics 59 351ndash364

Durban J Elston D Ellifrit D Dickson E Hammon P and Thompson P(2005) ldquoMultisite Mark-Recapture for Cetaceans Population Estimates WithBayesian Model Averagingrdquo Marine Mammal Science 21 80ndash92

Fienberg S E Johnson M S and Junker B W (1999) ldquoClassical Multi-level and Bayesian Approaches to Population Size Estimation Using MultipleListsrdquo Journal of the Royal Statistical Society Ser A 162 383ndash405

Forcada J and Aguilar A (2000) ldquoUse of Photographic Identification inCapturendashRecapture Studies of Mediterranean Monk Sealsrdquo Marine MammalScience 16 767ndash793

Franklin A B Anderson D R and Burnham K P (2002) ldquoEstimation ofLong-Term Trends and Variation in Avian Survival Probabilities Using Ran-dom Effects Modelsrdquo Journal of Applied Statistics 29 267ndash287

Friday N Smith T D Stevick P T and Allen J (2000) ldquoMeasure-ment of Photographic Quality and Individual Distinctiveness for the Photo-graphic Identification of Humpback Whales Megaptera novaeangliaerdquo Ma-rine Mammal Science 16 355ndash374

Gaspar R (2003) ldquoStatus of the Resident Bottlenose Dolphin Population inthe Sado Estuary Past Present and Futurerdquo unpublished doctoral dissertationUniversity of St Andrews School of Biology

Gelfand A E and Sahu S K (1999) ldquoIdentifiability Improper Priors andGibbs Sampling for Generalized Linear Modelsrdquo Journal of the AmericanStatistical Association 94 247ndash253

Gerrodette T (1987) ldquoA Power Analysis for Detecting Trendsrdquo Ecology 681364ndash1372

Goodman D (2004) ldquoMethods for Joint Inference From Multiple Data Sourcesfor Improved Estimates of Population Size and Survival Ratesrdquo Marine Mam-mal Science 20 401ndash423

Green P J (1995) ldquoReversible-Jump Markov Change Monte Carlo BayesianModel Determinationrdquo Biometrika 82 711ndash732

Grosbois V and Thompson P M (2005) ldquoNorth Atlantic Climate VariationInfluences Survival in Adult Fulmarsrdquo Oikos 109 273ndash290

Haase P (2000) ldquoSocial Organisation Behaviour and Population Parametersof Bottlenose Dolphins in Doubtful Sound Fiordlandrdquo unpublished mastersdissertation University of Otago Dept of Marine Science

Hammond P (1986) ldquoEstimating the Size of Naturally Marked Whale Pop-ulations Using CapturendashRecapture Techniquesrdquo Report of the InternationalWhaling Commission Special Issue 8 253ndash282

(1990) ldquoHeterogeneity in the Gulf of Maine Estimating HumpbackWhale Population Size When Capture Probabilities Are not Equalrdquo in In-dividual Recognition of Cetaceans Use of Photo-Identification and OtherTechniques to Estimate Population Parameters eds P S Hammond S A

Mizroch and G P Donovan Cambridge UK International Whaling Com-mission pp 135ndash153

Huggins R (1989) ldquoOn the Statistical Analysis of Capture Experimentsrdquo Bio-metrika 76 133ndash140

Joyce G G and Dorsey E M (1990) ldquoA Feasibility Study on the Useof Photo-Identification Techniques for Southern Hemisphere Minke WhaleStock Assessmentrdquo in Individual Recognition of Cetaceans Use of Photo-Identification and Other Techniques to Estimate Population Parameters edsP S Hammond S A Mizroch and G P Donovan Cambridge UK Inter-national Whaling Commission pp 419ndash423

King R and Brooks S (2001) ldquoOn the Bayesian Analysis of PopulationSizerdquo Biometrika 88 317ndash336

Lebreton J-D Burnham K P Clobert J and Anderson S R (1992)ldquoModeling Survival and Testing Biological Hypotheses Using Marked An-imals A Unified Approach With Case Studiesrdquo Ecological Monographs 6267ndash118

Lee S-M Huang L-H H and Ou S-T (2004) ldquoBand Recovery Model In-ference With Heterogeoneous Survival Ratesrdquo Statistica Sinica 14 513ndash531

Link W A and Barker R J (2005) ldquoModeling Association Among De-mographic Parameters in Analysis of Open Population CapturendashRecaptureDatardquo Biometrics 61 46ndash54

Mann J and Smuts B (1999) ldquoBehavioral Development in Wild BottlenoseDolphin Newborns (Tursiops sp)rdquo Behaviour 136 529ndash566

McCullagh P (1980) ldquoRegression Models for Ordinal Datardquo Journal of theRoyal Statistical Society Ser B 42 109ndash142

Otis D L Burnham K P White G C and Anderson D R (1978) ldquoStatis-tical Inference From Capture Data on Closed Animal Populationsrdquo WildlifeMonographs 62 135

Pledger S (2000) ldquoUnified Maximum Likelihood Estimates for ClosedCapturendashRecapture Models Using Mixturesrdquo Biometrics 56 434ndash442

Pledger S and Schwarz C J (2002) ldquoModelling Heterogeneity of Survivalin Band-Recovery Data Using Mixturesrdquo Journal of Applied Statistics 29315ndash327

Pledger S Pollock K H and Norris J L (2003) ldquoOpen CapturendashRecaptureModels With Heterogeneity I CormackndashJollyndashSeber Modelrdquo Ecology 59786ndash794

Pollock K H Nichols J D Brownie C and Hines J E (1990) ldquoStatisticalInference for CapturendashRecapture Samplingrdquo Wildlife Monographs 107 1ndash97

Read A Urian K W Wilson B and Waples D (2003) ldquoAbundance andStock Identity of Bottlenose Dolphins in the Bayes Sounds and Estuaries ofNorth Carolinardquo Marine Mammal Science 19 59ndash73

Saacutenchez-Mariacuten F J (2000) ldquoAutomatic Recognition of Biological ShapesWith and Without Representations of Shaperdquo Artificial Intelligence in Medi-cine 18 173ndash186

Saunders-Reed C A Hammond P S Grellier K and Thompson P M(1999) ldquoDevelopment of a Population Model for Bottlenose Dolphinsrdquo Re-search Survey and Monitoring Report 156 Scottish Natural Heritage

Scottish Natural Heritage (1995) Natura 2000 A Guide to the 1992 EC Habi-tats Directive in Scotlandrsquos Marine Environment Perth UK Scottish Nat-ural Heritage

Seber G (2002) The Estimation of Animal Abundance Caldwell NJ Black-burn Press

Stevick P T Palsboslashll P J Smith T D Bravington M V and Ham-mond P S (2001) ldquoErrors in Identification Using Natural Markings RatesSources and Effects on CapturendashRecapture Estimates of Abundancerdquo Cana-dian Journal of Fisheries and Aquatic Science 58 1861ndash1870

Taylor B L Martinez M Gerrodette T Barlow J and Hrovat Y N (2007)ldquoLessons From Monitoring Trends in Abundance of Marine Mammalsrdquo Ma-rine Mammal Science 23 157ndash175

Wells R S and Scott M D (1990) ldquoEstimating Bottlenose Dolphin Popu-lation Parameters From Individual Identification and CapturendashRelease Tech-niquesrdquo in Individual Recognition of Cetaceans Use of Photo-Identificationand Other Techniques to Estimate Population Parameters eds P S Ham-mond S A Mizroch and G P Donovan Cambridge UK InternationalWhaling Commission pp 407ndash415

Whitehead H (2001a) ldquoAnalysis of Animal Movement Using Opportunis-tic Individual Identifications Application to Sperm Whalesrdquo Ecology 821417ndash1432

(2001b) ldquoDirect Estimation of Within-Group Heterogeneity in Photo-Identification of Sperm Whalesrdquo Marine Mammal Science 17 718ndash728

Williams B K Nichols J D and Conroy M J (2001) Analysis and Man-agement of Animal Populations San Diego Academic Press

Wilson B Hammond P S and Thompson P M (1999) ldquoEstimating Size andAssessing Trends in a Coastal Bottlenose Dolphin Populationrdquo EcologicalApplications 9 288ndash300

Wilson B Reid R J Grellier K Thompson P M and Hammond P S(2004) ldquoConsidering the Temporal When Managing the Spatial A Popula-

Corkrey et al Bayesian Recapture Population Model 953

Table 1 Actual and estimated median total abundances for two artificial populations with 95 HPDI and posterior p values fromthree simulations on each with differing priors

A B C

Population Year Wj Wj p value Wj p value Wj p value

1 1990 124 119(99139) 28 112(97130) 07 107(90124) 021991 120 125(105146) 67 118(95134) 39 114(97135) 251992 114 116(96135) 54 110(93127) 30 106(92123) 171993 109 113(95134) 63 108(92124) 41 105(90121) 261994 103 111(93130) 81 106(88124) 63 103(86118) 491995 97 105(89125) 86 102(87118) 74 100(85114) 621996 95 98(84115) 66 95(81110) 48 93(81107) 381997 91 99(83114) 85 96(80109) 76 95(80108) 651998 89 94(79110) 73 91(77105) 56 90(77103) 521999 85 92(77108) 78 89(75103) 69 87(75103) 632000 76 82(6598) 76 79(6696) 65 79(6594) 592001 70 79(6299) 84 77(5895) 79 76(6095) 752002 68 76(5593) 78 75(5691) 75 73(5892) 70

2 1990 124 135(100178) 71 121(96159) 42 116(90149) 261991 122 123(90158) 51 112(87140) 21 108(84134) 121992 119 114(87148) 35 105(84130) 10 101(78124) 061993 111 106(82138) 34 99(75121) 11 95(74117) 061994 106 102(76130) 33 93(75118) 12 91(68115) 071995 100 96(70122) 38 91(74114) 16 88(67108) 081996 93 93(66119) 48 87(69109) 26 85(65105) 161997 87 91(67115) 60 85(68106) 37 82(62102) 281998 77 79(59101) 56 76(6295) 39 72(5791) 281999 74 84(60108) 80 80(62101) 75 77(5897) 622000 72 78(5799) 67 74(5894) 58 72(5591) 462001 67 71(4992) 62 69(4891) 57 66(4688) 432002 64 68(4191) 59 65(4492) 51 63(4388) 44

NOTE Population 1 has homogeneous capture probability (75) and survival probability (95) while population 2 has heterogeneous capture probability (mean 55 range 05ndash98) andsurvival probability (mean 92 range 68ndash99) The simulations have different priors for ωj and λN Simulation run A has vague priors [ω sim Beta(029412368) λN sim Gam(76 01)]B has more precise priors [ω sim Beta(023710) λN sim Gam(761)] and C has very precise priors [ω sim Beta(929939155) λN sim Gam(7600100)] Their means are the same ineach case (023 and 76) but each differs from the actual population values (06 and 106)

their 95 highest posterior density intervals (HPDI) and a p

value The p values are calculated by counting the proportionof iterations in which the Wj exceeded the true value Examina-tion of the results indicate that the median posterior estimatesgenerally agree with the actual values The agreement is sat-isfactory whether or not there is heterogeneity assumed Thep values are in the range 02ndash88 suggesting that the modelworks correctly The posterior p values for the recapture modelcoefficients are given in Table 2 The p values for α δL andδR are summarized and those for the remaining parameters aresummarized as the proportion of p values in each that lie insidethe range 01ndash99 Almost all p values are within the range 01ndash99 the only exceptions are the capture coefficients for a smallnumber of dolphins

In general the total population estimates for the homoge-neous case are slightly greater and the 95 HPDI is narrowerthen the heterogeneous case but all intervals contain the popu-lation values The estimates also shrink slightly for the mod-els with more precise priors However the p values in eachcase suggest an adequate fit even though some priors are de-liberately set to incorrect values In the case of the recapturecoefficients there is no apparent difference between the homo-geneous and heterogeneous models or with any trend in theprecision of the priors

5 RESULTS

We obtain estimates of heterogeneous capture and survivalcoefficients variation of capture and survival over time over-all survival and capture probabilities the abundance and thetrend in abundance The survival probabilities and capture co-efficients from the left and right sides are shown in Figure 2 andTable 3 We can obtain an overall capture probability for the twosides by calculating exp(δ)(1+exp(δ)) The mean overall cap-

Table 2 Summary of posterior p values for survival and capturecoefficients for two artificial populations from three simulations

on each with differing priors

Population α βj γd δL εjL ζdL δR εjR ζdR

1 16 100 100 66 100 98 60 100 10010 100 100 59 100 98 53 100 10008 100 100 59 100 98 48 100 100

2 34 100 100 17 100 97 50 100 9822 100 100 15 100 97 37 100 9923 100 100 12 100 97 39 100 99

NOTE The populations and simulations are described in Table 1 The p values are calcu-lated as the proportion of iterations greater than the actual population value The p valuesfor α δL and δR are shown Because the remaining parameters are vectors of coefficientsthose p values are summarized as the proportion of p values in each that are found insidethe range 01ndash99

954 Journal of the American Statistical Association September 2008

(a)

(b) (c)

Figure 2 (a) Variation of mean capture coefficients over time forthe left side εL (minusminus) and right side εR (- - bull - - bull) The two hor-izontal lines indicate the global mean capture coefficients δL (mdashmdash)and δR (minus minus minus) On the left vertical axis are the mean left-side het-erogeneity capture coefficients ζL and on the right vertical axis arethe mean right-side heterogeneity capture coefficients ζR Standardserrors for each coefficient are shown as vertical bars (b) Variation ofmean survival probability over time (c) Histogram of mean heteroge-neous survival probabilities

ture probability for the left side is 48 (95 HPDI = 29 68)and for the right side it is 58 (95 HPDI = 39 77) Althoughthis is a small difference the probability of the right capturecoefficient exceeding the left is 81 Examination of Figure 2(a)shows that the variation of capture coefficients over time is largecompared with the overall coefficients and that the left (εL) andright (εR) coefficients tend to track each other The probabilityof the right capture coefficient exceeding the left varies overa wide range 07ndash96 This range results from some years inwhich the probability tends strongly to the left side (1995 and2000 probability of observing the left side 95) or right side(1998 1999 2001 probability of observing the right side 91)Finally the range of the heterogeneous coefficients is compara-ble to that of the time-varying coefficients

The mean heterogeneous survival coefficient is not corre-lated with either of the mean capture heterogeneous coeffi-cients (right r = 009 left r = 022) However the meanleft and right heterogeneous capture coefficients are correlated(r = 714) indicating that animals that are more likely to beseen from one side also are more likely to be seen from theother side However the heterogeneity capture coefficient of theright side (mean 56) exceeded that of the left side (mean 23)with 85 of animals having higher right coefficients than left

Table 3 Left-side and right-side capture probabilities and survivalprobabilities by year

Year Left capture Right capture Survival

1990 1(11)

1991 54(40 68) 60(47 75) 93(84 99)

1992 74(59 86) 75(60 86) 98(94100)

1993 31(20 44) 46(33 61) 95(88100)

1994 36(24 47) 53(35 66) 95(86100)

1995 42(27 55) 71(56 88) 94(86100)

1996 30(17 42) 48(34 63) 93(82100)

1997 23(14 36) 37(23 51) 88(75 98)

1998 39(24 53) 31(20 45) 94(84100)

1999 45(31 63) 40(25 54) 93(82100)

2000 24(14 36) 57(40 72) 96(88100)

2001 87(78 94) 82(73 91) 96(89100)

2002 88(80 96) 89(80 96)

NOTE For each the mean and the 95 HPDI limits are given There is concordancebetween the left-side and right-side results The capture probabilities are more variableand reach minimums in 1997 and 2000 for the left side and in 1998 for the right sidebut both reach maximums in 2002 Survival reaches a maximum in 1992 declines to aminimum in 1997 and then rises again in later years

coefficients Because this is observed after adjustments for theoverall and time-varying components it suggests an effect dueto behavior or sampling protocol

The variation in the capture probability over time is likelyto result from a combination of factors including variation indolphin ranging patterns (see Wilson et al 2004) weather con-ditions and other logistical factors that influence sampling indifferent areas Most notably sampling protocols were changedin 2000 specifically to increase the probability of capture Thischange has proven to have been successful in the latter part ofthe study while estimates of the survival probabilities and to-tal population size remain unaffected Second another study(Wilson et al 2004) has shown that the ranging pattern of thepopulation changed in recent years with some animals spend-ing more time in the southern part of the population range alongthe east coast of Scotland south of the Moray Firth to St An-drews and the Firth of Forth The effect of this would be toreduce the capture probability in the core study area in the in-ner Moray Firth and to potentially increase heterogeneity incapture probabilities The more southerly areas also have beensampled but the data are more sparse and cover only a smallnumber of years Once these histories have become more exten-sive we expect that the capture probabilities will become lessvariable Unlike other models our estimates are also correctedfor individual heterogeneity Variation in capture probabilitiesalso might have arisen from heterogeneous effort in space Al-though this was not explicitly modeled we could have stratifiedthe capture histories by area by including separate capture co-efficients for each area

The overall mean survival estimate through the study periodis 93 (standard deviation 029 95 HPDI = 861 979) Thisis similar to an earlier maximum likelihood estimate of sur-vival for the cohort of nicked individuals from this populationfirst observed in 1990 (Saunders-Reed Hammond Grellier andThompson 1999) But our model estimate is slightly lower thansurvival estimates from other bottlenose populations of 962 plusmn76 in Sarasota Bay Florida (Wells and Scott 1990) 952 plusmn

Corkrey et al Bayesian Recapture Population Model 955

150 in Doubtful Sound New Zealand (Haase 2000) and 95ndash99 in the Sado estuary Portugal (Gaspar 2003) However thismay be because our study included subadults while the otherstudies reported adult survival Variation in survival probabil-ity occurs over time [Fig 2(b) Table 3] It is of interest thatsurvival reached a minimum in 1997 but later returned to lev-els comparable to earlier years The proportion of iterations inwhich the survival for 1997 are above the mean survival forthe remaining years (mean φ = 95 95 HPDI = 85 100)is only 08 The apparent low survival in 1997 may be due tomigration because this is not distinguishable from survival inthe model It is possible that an emigration event occurred in1997 and that those individuals have not yet returned Consid-erable heterogeneity in survival (range 77ndash97) is observedas shown in Figure 2(c) The survival probabilities are not dis-tributed smoothly suggesting multimodality The animals witha low survival coefficient may have died or migrated In thecase of migration this result would indicate that some animalsless consistently reside in the core areas compared with oth-ers

Individuals with higher survival coefficients are frequentlythose seen earliest in the study There is little information avail-able on individuals seen only more recently and consequentlytheir coefficients are close to 0 Some survival coefficients aremore negative and these correspond to animals that have notbeen seen for some time As time goes on and in the absenceof a confirmed death or sighting these coefficients can be ex-pected to become more negative However some animals thatare observed only on a few occasions have negative coefficientsThe estimation of individual survival coefficients means that es-timation of other coefficients in the model are adjusted for vari-able survival The estimation of individual capture probabilitiesallows the results to be combined with other individual covari-ates such as sex or body size if these data become availableSimilarly where individuals differ in their home ranges indi-vidual capture or survival probabilities could be related to localenvironmental conditions within their range to prey availabil-ity or to water quality

Estimates of the total population size from bilateral analy-sis of the left and right photo-identification series and estimatesfrom unilateral analyses of the left and right sides are shown inFigure 3(a) These are similar both to the maximum likelihoodestimate of 129 (95 confidence interval = 110ndash174) for datacollected in 1992 (Wilson et al 1999) and to a Bayesian mul-tisite mark recapture estimate of the nicked population of 85(95 probability interval = 76ndash263) for data collected in 2001(Durban et al 2005) The credible intervals of the abundanceestimates are generally narrower in the bilateral analysis (meanwidth 886) than in left-side analysis (mean width 1373) andright-side analysis (mean width 937) Figure 3(b) shows theprior and mean posterior recruitment probabilities ωj It is in-teresting that higher probabilities occur in 1996 and 2001 Thismight be considered to variations in capture probability butwhereas the capture probability is high in 2001 it remains lowin 1996 [see Fig 2(a)] Instead calculating the probabilities ofthe posterior exceeding the prior recruitment probabilities wefind they are below 5 (range 04ndash44) for all years except 2001in which it is 85 The higher recruitment probability in 2001may be due to an improved protocol in that year that led to the

(a)

(b)

Figure 3 (a) Mean total abundance estimates from a bilateral analy-sis with 95 HPDI Also shown are estimates from separate analysesof left (minus minus minus) and right (middot middot middot) sides along with associated 95 HPDIrepresented by symbols left + right Minimum numbers of ani-mals known to be alive are indicated by diamonds (b) Mean posteriorrecruitment probabilities ωj with 95 HPDI The prior value is rep-resented as a dotted line

identification of previously unrecognized nicked animals Weconsider the rise in 1996 to have insufficient evidence to sug-gest an increase in recruitment in that year

Earlier demographic modeling of this population has pre-dicted that the population is in decline (Saunders-Reed et al1999) To examine this we calculate the statistic [prodJ

j=2 Wj

(Wjminus1)]1(Jminus1) Values of this statistic gt1 indicate an increas-ing population whereas values lt1 indicates a decreasing pop-ulation The mean value is 989 (95 HPDI = 958 102)and the probability of it being lt10 is 77 This suggests thatthe population may be decreasing but that the evidence fora decline is weak By doing this we are able to use the fulltime series of photo identification data to obtain simultaneousposterior estimates of the Wj to compare the probability of apopulation increase or decrease In agreement with the earlierpredictive modeling these data indicate greater probability of adecline than of an increase However because it is likely thatthe population is expanding its range (Wilson et al 2004) thedecline may be confounded with temporary emigration

956 Journal of the American Statistical Association September 2008

6 DISCUSSION

Changes in the size of wildlife populations are often assessedby fitting trend models to independent abundance estimatesbut the power of these techniques to detect trends often canbe low (Gerrodette 1987 Taylor Martinez Gerrodette Bar-low and Hrovat 2007) Here we use a capturendashrecapture modelthat incorporates an underlying population model and an ob-servation model We provide a probability that a population isin decline or increasing Using photoidentification data fromthis bottlenose dolphin population we generate the first em-pirical estimate of trends in abundance for any of the popula-tions of small cetaceans inhabiting European waters Becausethe methodological changes in sampling likely are absorbedby the time-varying capture coefficient the remaining variationdue to individual capture tendencies might reasonably be de-scribed by the individual capture coefficient This then allowsexploration of capture and survival relating to individuals ratherthan to populations Information on individual survival could beimportant for addressing some management issues for exam-ple individuals differing in their spatial distribution

By decomposing the capture probabilities into overall time-varying and heterogeneous terms we find that the time-varyingcontribution and heterogeneous variation are comparable inmagnitude and their variation is greater than that of the overallterm This information can be used in the design of monitoringprograms by suggesting how effort should be distributed Thedifference in overall left and right capture probabilities mightbe interpretable as an artifact of the sampling techniques or ofbehavioral effects However after allowing for this the prob-ability of observing one side also varied over time with someyears strongly favoring one side or the other This might be ex-plained by variation in protocol between years such as changesin personnel The relative sizes of these effects may be of usein evaluating consistency of protocol over time Finally afterallowing for the overall difference and time variation the het-erogeneous capture coefficients exhibit a bias to the one sidepossibly due to behavior or protocol These findings demon-strate that bilateral decomposition of capture probabilities canprovide useful information

We decompose the survival probability similarly and we findit to be unusually low in 1997 suggesting a possible multimodalheterogeneous survival probability that may be the result of mi-gration or survival variation This type of decomposition allowsexploration of such effects in more detail In particular the mul-timodality of heterogeneous survival suggests that the popula-tion comprises individuals with different strategies some vis-iting the core study area more or less often Ongoing work isattempting to quantify individual heterogeneity such as in ex-posure to sewage contaminated waters and boat-based tourismTherefore this technique could be used to determine whetherexposure to different anthropogenic stressors is related to sur-vival

Our model also makes use of priors such as those on ωj thatare not informed by the recapture data set but make use of in-formation from other sources An alternative approach mightbe to specify these as informative priors based on expert opin-ion or tuned to obtain satisfactory results Our application il-lustrates how disparate data sources may be combined into a

single analysis Although the data source for the prior recruit-ment probability is small and problematic it is the only oneavailable Even when some priors are defined in the simulationstudy to incorrect values the estimates of parameters of interestremain valid The posterior probability also adds to the state ofknowledge of this parameter

Previous use of capturendashrecapture models with cetacean pho-toidentification data highlights the potential violation of modelassumptions due to heterogeneity in capture andor survivalprobabilities This model explicitly accounts for heterogene-ity and also provides simultaneous estimates of individual sur-vival and capture probabilities This can be used both to ex-plore the biological basis of heterogeneity and to improve sam-pling programs The individual survival coefficients also can beused to explore aspects of biology where correlates are avail-able for example the relationship of survival to age sex orother variables may be of interest Abundance under the as-sumption of heterogeneous survival has been estimated by LeeHuang and Ou (2004) whereas individual and time-varyingsurvival estimates have been obtained using band-recoverymodels (Grosbois and Thompson 2005) Other studies have ex-amined survival by group for example Franklin Andersonand Burnham (2002) obtained separate estimates of group-levelsurvival for males and females using stratification in a bandingstudy Although we lack such variables as age and sex theo-retically it would be possible to examine individual survivalby sex either as a post hoc comparison or through their inte-gration into the model itself For example survival and cap-ture could be modeled as logit(φ) = α + βj + γd + τ andlogit(π) = δ + εj + ζd + χ where τ corresponds to stage-specific (eg young immature reproductive female senes-cent) survival and χ corresponds to capture probabilities forvarious categories (eg young reproductive adult)

Other cetacean studies have examined heterogeneity White-head (2001b) obtained an estimate of the coefficient of variation(CV) for identification from a randomization test of successiveidentifications in whale track data (The CV is also a measure ofcapture heterogeneity) They found that the CV was significantin one of the two years and that probably depended on the sizesof the animals being observed This is not comparable to thedolphin observations in which heterogeneity is more likely toresult from differences in animal behavior In the present studywe use a likelihood in which heterogeneity of survival and cap-ture has been incorporated by a logit function This approachwas described previously (Pledger Pollock and Norris 2003Pledger and Schwarz 2002) within a non-Bayesian contextPledger et al (2003) classified animals into an unknown set oflatent classes each of which could have a separate survival andcapture probability A simplification allows for time-varyingand animal-level heterogeneity This approach is similar to oursbut we also integrate a dynamic population model and an ob-servation model into the estimation process Goodman (2004)also implemented a population model in a Bayesian context thatcombines capturendashrecapture data with carcass-recovery databut assumes the absence of heterogeneity in capture probabil-ities Durban et al (2005) estimated abundance for the samepopulation as us but used a multisite Bayesian log-linear modeland model averaging (King and Brooks 2001) across the pos-sible models with varying combinations of interactions Our

Corkrey et al Bayesian Recapture Population Model 957

model could be extended to include multiple sites and yearsby using a capturendashrecapture history that includes site informa-tion and a yearndashsite term in the capture probability expressionBy doing this we suggest that geographic areas with more datacan be used to strengthen estimates from poorly sampled areasThis also may account for some individual variation currentlyidentified as individual capture heterogeneity because some in-dividuals may be more closely associated with some areas thanothers and may attenuate a possible confounding of migrationwith survival

We use a separate data set and a negative-binomial model toinflate the estimate of the proportion of nicked animals in thesamples Another study (Da Silva Rodrigues Leite and Mi-lan 2003) obtained an abundance estimate for bowhead whalesusing an inflation factor derived from the numbers of good andbad photographs of marked and unmarked whales Our analysisavoids the lack of independence between good photographs anddistinctively marked animals in the photographs that might in-validate the estimate of variance in the inflation factor Otherstudies avoid this issue by including only separately catego-rized good-quality photographs (eg Wilson et al 1999 ReadUrian Wilson and Waples 2003) However independent es-timates of inflation when available such as in our study arepreferred

We do not know of other cetacean studies that combinephoto-identification series of the left and right sides but ananalogous approach is that of log-linear capturendashrecapturemodeling (King and Brooks 2001) in which animals are identi-fied from separate lists Some photo-identification studies mightuse information from both sides in the process of identifyinganimals (Joyce and Dorsey 1990) A more robust approach isto perform the identification on each side separately We haveshown how these separate data can then be combined

A major advantage to our modeling approach is its extensibil-ity Within the Bayesian framework models can be comparedby such methods as reversible-jump Markov chain Monte Carlotechniques (Green 1995) Model extensions include extendingthe population model considering an individual-level approachThis would allow incorporation of the heterogeneity into thispart of the model and also modeling of the unobserved popu-lation More detailed modeling of heterogeneity could be ex-plored by including hyperpriors for the variances of the recap-ture model terms For convenience we analyze the data on anannual basis Additional information would be available if theanalyses were conducted on a finer time scale such as monthsor by individual trip This would allow the model to be extendedto include seasonal effects For example dolphins are observedmore frequently in summer months than in winter months andthis tendency could be included in an extended model by chang-ing the capture probability to be logit(π) = δt + εj + ζd whereδt t isin [1 12] is a monthly capture coefficient This maybe of use in modeling fine-scale movements The response vari-able itself also can be generalized Although we are careful toonly include data from high-quality photographs and nicked an-imals there is still the possibility of some identification errorsparticularly for more subtly marked individuals A possible ex-tension of the model could allow identification to be assigneddifferent levels of certainty for example by incorporating anintermediate state between nonrecapture and recapture that we

term ldquopossible recapturerdquo This could be done by replacing thelogit function logit(ν) = log(ν(1 minus ν)) where ν correspondsto the probability of recapture by a proportional odds function(McCullagh 1980) This technique has previously found appli-cation in a study of photoidentification used to model photo-graphic quality scores and individual distinctiveness of hump-back whale tail flukes (Friday Smith Stevick and Allen 2000)In this approach the probability of ldquopossible recapturerdquo is givenby ν1 = π1 the probability of certain recapture is given byν2 = π1 + π2 and includes the possible recapture category andthe probability of no capture is given by ν0 = 1 minus π1 + π2There are two proportional odds functions corresponding topossible recapture and certain recapture which are logit(ν1) =log(π1(π2 +π3)) and logit(ν2) = log((π1 +π2)π3) We thencould extend the recapture model using logit(ν1) = α+βj +γk1

and logit(ν2) = α + βj + γk2 This proportional odds approachmay be particularly useful in situations where automatic match-ing algorithms (Saacutenchez-Mariacuten 2000) provide an estimate ofthe likelihood of a match which could then be incorporatedinto abundance estimate models

APPENDIX DERIVATION OF POSTERIORS

Here we show how we derive the posterior conditional distributionsfor the model parameters and describe how each is updated within theMCMC algorithm The full joint distribution has the form shown in(A1) In (A1) the first two lines correspond to the population modelthe third line corresponds to the inflation model the fourth line cor-responds to the observation model the fifth line corresponds to therecapture model and the remaining lines correspond to priors

π = P(B1|ω1N0W0)P (S1|αN0)

(Sec 31)

timesSprod

j=2

P(Bj |Sjminus1Bjminus1Wjminus1ωj )

times P(Sj |Sjminus1Bjminus1 αβjminus1)

(Sec 31)

timesSprod

j=1

P(Wj |Sj Bj υj )

Tjprod

i=1

P(tnji |υj twji )

(Sec 32)

timesSprod

j=1

P(nLj|δL εjLSj Bj )P (nRj

|δR εjRSj Bj )

(Sec 33)

timesSprod

j=1

P(Hdj |αβj γd δL εjL ζdL δR εjR ζdR)

(Sec 34)

times P(N0|λN)P (W0|N0 υ0)P (α δL δRλN υ0)

(Sec 41)

timesSprod

j=1

P(βj εjL εjRωj υj )

nprod

d=1

P(γd)P (ζdL)P (ζdR)

(Sec 41) (A1)

958 Journal of the American Statistical Association September 2008

For each parameter we obtain the corresponding posterior condi-tional distribution by extracting all terms containing that parameterfrom the full joint distribution When the conditional distributions havea standard form we use the Gibbs sampler (Brooks 1998) to update theparameters otherwise we use MetropolisndashHastings updates (Chib andGreenberg 1995)

Within a MetropolisndashHastings update we propose a new valuefor a parameter x and denote the proposed value by xprime To decidewhether to accept the proposed value we calculate an acceptance ra-tio α = min(1A) where A is the acceptance ratio The acceptanceratio is given by A = π(xprime)q(x)π(x)q(xprime) where π(x) is the poste-rior probability of x and q(x) is the probability of the proposed valueWe accept the new value xprime with probability α otherwise we leavethe value of x unchanged (See Chib and Greenberg 1995 for an exam-ple)

Normal proposal distributions are used for continuous parameterscentered around the current parameter value Discrete uniform pro-posals are used for discrete variables each centered around the currentparameter values The discrete proposals also require constraints to en-sure that the proposed values are valid For example a proposed valuefor N0 must be less than the current value of W0 We provide a detaileddescription of each update next

Population Model

The population model has parameters ωj λ Bj Sj and N0 andare updated using Gibbs or MetropolisndashHastings samplers The condi-tional posterior distributions for ωj and λN can be derived as

ωj |Bj Sjminus1Bjminus1

sim Beta(Bj + 0238Wjminus1 minus Sjminus1 minus Bjminus1 minus Bj + 1)

and

λN |N0 sim Gam(76 + N0101)

These are individually updated using the Gibbs sampler during eachiteration of the Markov chain

The parameters Bj Sj and N0 are updated by the MetropolisndashHastings algorithm using acceptance ratios calculated as described ear-lier We use uniform proposals centered on the current parameter valueand limited to the range of possible values

For the Sj rsquos we use three alternative proposals

Sprimej=1 sim Unif

(max

(0 S1 minus 5 S2 minus B1max(n1Ln1R) minus N1

2 minus B1)

min(S1 + 5N0W1 minus B1W1 minus B1 minus B2

W1 minus B1 minus 1))

Sprime2lejltJ sim Unif

(max

(0 Sj minus 5 Sj+1 minus Bj max(njLnjR) minus Bj

2 minus Bj

)

min(Sj + 5 Sjminus1 + Bjminus1Wj minus Bj Wj minus Bj minus Bj+1

Wj minus Bj minus 1))

and

Sprimej=J sim Unif

(max

(0 SJ minus 5max(nJLnJR) minus BJ 2 minus NJ

)

min(SJ minus 5 Sjminus1 + Bjminus1Wj minus Bj Wj minus Bj minus 1))

For Bj we use three alternative proposals

B prime1 sim Unif

(max

(0B1 minus 15 S2 minus S1max(n1Ln1R) minus S1

2 minus S1)

min(B1 + 15W0 minus N0W1 minus S1W1 minus S1 minus B2

W1 minus S1 minus 1))

B prime2lejltJ sim Unif

(max

(0Bj minus 15 Sj+1 minus Sj max(njLnjR) minus Sj

2 minus Sj

)

min(Bj + 15Wjminus1 minus Sjminus1 minus Bjminus1Wj minus Sj

Wj minus Sj minus Bj+1Wj minus Sj minus 1))

and

B primeJ sim Unif

(max

(0BJ minus 15max(nJLnJR) minus SJ 2 minus SJ

)

min(BJ + 15WJminus1 minus SJminus1 minus BJminus1

WJ minus SJ WJ minus SJ minus 1))

For N0 we use

N prime0 sim Unif

(max(5N0 minus 15 S1)min(N0 + 15W0 minus B1W0 minus 1)

)

Inflation Model

The inflation model parameters Wj and υj are updated usingMetropolisndashHastings and Gibbs updates The conditional posterior dis-tributions for υj can be derived as

υ0|N0W0 sim Beta(N0 + 1W0 minus N0 + 1)

and

υjgt0|Sj Bj Wj tnji twji sim Beta

(

Sj + Bj +T (j)sum

i=1

tnji + 1

Wj minus Sj minus Bj +T (j)sum

i=1

twji minus tnji + 1

)

These are updated separately using the Gibbs sampler on each itera-tion

The remaining parameter Wj has a nonstandard conditional pos-terior distribution We use larger updates than in the population modelbecause the initial total population is expected to be larger in size

W prime0 sim Unif

(max(0W0 minus 15N0 + 1N0 + B1)W0 + 15

)

W prime1lejltJ sim Unif

(max(0Wj minus 15 Sj + Bj + 1 Sj + Bj + Bj+1)

Wj + 15)

and

W primeJ sim Unif

(max(0WJ minus 15 SJ + BJ + 1)WJ + 15

)

Recapture Model

For the recapture model we have parameters α βj γd δL εjLζdL δR εjR and ζdR These all have nonstandard conditional pos-terior distributions and we use the MetropolisndashHastings algorithm toupdate the parameters on each iteration We use individual normal dis-tribution proposals centered on the current parameter values and withvariance equal to the prior variance for example

αprime sim N(α185)

[Received June 2005 Revised May 2007]

Corkrey et al Bayesian Recapture Population Model 959

REFERENCES

Brooks S P (1998) ldquoMarkov Chain Monte Carlo Method and Its ApplicationrdquoThe Statistician 47 69ndash100

Brooks S P and Roberts G O (1998) ldquoConvergence Assessment Techniquesfor Markov Chain Monte Carlordquo Statistics and Computing 8 319ndash335

Buckland S T (1982) ldquoA CapturendashRecapture Survival Analysisrdquo Journal ofAnimal Ecology 51 833ndash847

(1990) ldquoEstimation of Survival Rates From Sightings of IndividuallyIdentifiable Whalesrdquo in Individual Recognition of Cetaceans Use of Photo-Identification and Other Techniques to Estimate Population Parameters edsP S Hammond S A Mizroch and G P Donovan Cambridge UK Inter-national Whaling Commission pp 149ndash153

Burnham K and Overton W (1979) ldquoRobust Estimation of Population SizeWhen Capture Probabilities Vary Among Animalsrdquo Ecology 60 927ndash936

Carothers A (1979) ldquoQuantifying Unequal Catchability and Its Effect on Sur-vival Estimates in an Actual Populationrdquo Journal of Animal Ecology 48863ndash869

Chao A (1987) ldquoEstimating the Population Size for CapturendashRecapture DataWith Unequal Catchabilityrdquo Biometrics 43 783ndash791

Chao A Lee S-M and Jeng S-L (1992) ldquoEstimating Population Size forCapturendashRecapture Data When Capture Probabilities Vary by Time and Indi-vidual Animalrdquo Biometrics 48 201ndash216

Chib S and Greenberg E (1995) ldquoUnderstanding the MetropolisndashHastingsAlgorithmrdquo The American Statistician 49 327ndash335

Cormack R (1972) ldquoThe Logic of CapturendashRecapture Experimentsrdquo Biomet-rics 28 337ndash343

Coull B A and Agresti A (1999) ldquoThe Use of Mixed Logit Models to Re-flect Heterogeneity in CapturendashRecapture Studiesrdquo Biometrics 55 294ndash301

Da Silva C Q Rodrigues J Leite J G and Milan L A (2003) ldquoBayesianEstimation of the Size of a Closed Population Using Photo-ID With Part ofthe Population Uncatchablerdquo Communications in Statistics 32 677ndash696

Dorazio R M and Royle J A (2003) ldquoMixture Models for Estimating theSize of a Closed Population When Capture Rates Vary Among IndividualsrdquoBiometrics 59 351ndash364

Durban J Elston D Ellifrit D Dickson E Hammon P and Thompson P(2005) ldquoMultisite Mark-Recapture for Cetaceans Population Estimates WithBayesian Model Averagingrdquo Marine Mammal Science 21 80ndash92

Fienberg S E Johnson M S and Junker B W (1999) ldquoClassical Multi-level and Bayesian Approaches to Population Size Estimation Using MultipleListsrdquo Journal of the Royal Statistical Society Ser A 162 383ndash405

Forcada J and Aguilar A (2000) ldquoUse of Photographic Identification inCapturendashRecapture Studies of Mediterranean Monk Sealsrdquo Marine MammalScience 16 767ndash793

Franklin A B Anderson D R and Burnham K P (2002) ldquoEstimation ofLong-Term Trends and Variation in Avian Survival Probabilities Using Ran-dom Effects Modelsrdquo Journal of Applied Statistics 29 267ndash287

Friday N Smith T D Stevick P T and Allen J (2000) ldquoMeasure-ment of Photographic Quality and Individual Distinctiveness for the Photo-graphic Identification of Humpback Whales Megaptera novaeangliaerdquo Ma-rine Mammal Science 16 355ndash374

Gaspar R (2003) ldquoStatus of the Resident Bottlenose Dolphin Population inthe Sado Estuary Past Present and Futurerdquo unpublished doctoral dissertationUniversity of St Andrews School of Biology

Gelfand A E and Sahu S K (1999) ldquoIdentifiability Improper Priors andGibbs Sampling for Generalized Linear Modelsrdquo Journal of the AmericanStatistical Association 94 247ndash253

Gerrodette T (1987) ldquoA Power Analysis for Detecting Trendsrdquo Ecology 681364ndash1372

Goodman D (2004) ldquoMethods for Joint Inference From Multiple Data Sourcesfor Improved Estimates of Population Size and Survival Ratesrdquo Marine Mam-mal Science 20 401ndash423

Green P J (1995) ldquoReversible-Jump Markov Change Monte Carlo BayesianModel Determinationrdquo Biometrika 82 711ndash732

Grosbois V and Thompson P M (2005) ldquoNorth Atlantic Climate VariationInfluences Survival in Adult Fulmarsrdquo Oikos 109 273ndash290

Haase P (2000) ldquoSocial Organisation Behaviour and Population Parametersof Bottlenose Dolphins in Doubtful Sound Fiordlandrdquo unpublished mastersdissertation University of Otago Dept of Marine Science

Hammond P (1986) ldquoEstimating the Size of Naturally Marked Whale Pop-ulations Using CapturendashRecapture Techniquesrdquo Report of the InternationalWhaling Commission Special Issue 8 253ndash282

(1990) ldquoHeterogeneity in the Gulf of Maine Estimating HumpbackWhale Population Size When Capture Probabilities Are not Equalrdquo in In-dividual Recognition of Cetaceans Use of Photo-Identification and OtherTechniques to Estimate Population Parameters eds P S Hammond S A

Mizroch and G P Donovan Cambridge UK International Whaling Com-mission pp 135ndash153

Huggins R (1989) ldquoOn the Statistical Analysis of Capture Experimentsrdquo Bio-metrika 76 133ndash140

Joyce G G and Dorsey E M (1990) ldquoA Feasibility Study on the Useof Photo-Identification Techniques for Southern Hemisphere Minke WhaleStock Assessmentrdquo in Individual Recognition of Cetaceans Use of Photo-Identification and Other Techniques to Estimate Population Parameters edsP S Hammond S A Mizroch and G P Donovan Cambridge UK Inter-national Whaling Commission pp 419ndash423

King R and Brooks S (2001) ldquoOn the Bayesian Analysis of PopulationSizerdquo Biometrika 88 317ndash336

Lebreton J-D Burnham K P Clobert J and Anderson S R (1992)ldquoModeling Survival and Testing Biological Hypotheses Using Marked An-imals A Unified Approach With Case Studiesrdquo Ecological Monographs 6267ndash118

Lee S-M Huang L-H H and Ou S-T (2004) ldquoBand Recovery Model In-ference With Heterogeoneous Survival Ratesrdquo Statistica Sinica 14 513ndash531

Link W A and Barker R J (2005) ldquoModeling Association Among De-mographic Parameters in Analysis of Open Population CapturendashRecaptureDatardquo Biometrics 61 46ndash54

Mann J and Smuts B (1999) ldquoBehavioral Development in Wild BottlenoseDolphin Newborns (Tursiops sp)rdquo Behaviour 136 529ndash566

McCullagh P (1980) ldquoRegression Models for Ordinal Datardquo Journal of theRoyal Statistical Society Ser B 42 109ndash142

Otis D L Burnham K P White G C and Anderson D R (1978) ldquoStatis-tical Inference From Capture Data on Closed Animal Populationsrdquo WildlifeMonographs 62 135

Pledger S (2000) ldquoUnified Maximum Likelihood Estimates for ClosedCapturendashRecapture Models Using Mixturesrdquo Biometrics 56 434ndash442

Pledger S and Schwarz C J (2002) ldquoModelling Heterogeneity of Survivalin Band-Recovery Data Using Mixturesrdquo Journal of Applied Statistics 29315ndash327

Pledger S Pollock K H and Norris J L (2003) ldquoOpen CapturendashRecaptureModels With Heterogeneity I CormackndashJollyndashSeber Modelrdquo Ecology 59786ndash794

Pollock K H Nichols J D Brownie C and Hines J E (1990) ldquoStatisticalInference for CapturendashRecapture Samplingrdquo Wildlife Monographs 107 1ndash97

Read A Urian K W Wilson B and Waples D (2003) ldquoAbundance andStock Identity of Bottlenose Dolphins in the Bayes Sounds and Estuaries ofNorth Carolinardquo Marine Mammal Science 19 59ndash73

Saacutenchez-Mariacuten F J (2000) ldquoAutomatic Recognition of Biological ShapesWith and Without Representations of Shaperdquo Artificial Intelligence in Medi-cine 18 173ndash186

Saunders-Reed C A Hammond P S Grellier K and Thompson P M(1999) ldquoDevelopment of a Population Model for Bottlenose Dolphinsrdquo Re-search Survey and Monitoring Report 156 Scottish Natural Heritage

Scottish Natural Heritage (1995) Natura 2000 A Guide to the 1992 EC Habi-tats Directive in Scotlandrsquos Marine Environment Perth UK Scottish Nat-ural Heritage

Seber G (2002) The Estimation of Animal Abundance Caldwell NJ Black-burn Press

Stevick P T Palsboslashll P J Smith T D Bravington M V and Ham-mond P S (2001) ldquoErrors in Identification Using Natural Markings RatesSources and Effects on CapturendashRecapture Estimates of Abundancerdquo Cana-dian Journal of Fisheries and Aquatic Science 58 1861ndash1870

Taylor B L Martinez M Gerrodette T Barlow J and Hrovat Y N (2007)ldquoLessons From Monitoring Trends in Abundance of Marine Mammalsrdquo Ma-rine Mammal Science 23 157ndash175

Wells R S and Scott M D (1990) ldquoEstimating Bottlenose Dolphin Popu-lation Parameters From Individual Identification and CapturendashRelease Tech-niquesrdquo in Individual Recognition of Cetaceans Use of Photo-Identificationand Other Techniques to Estimate Population Parameters eds P S Ham-mond S A Mizroch and G P Donovan Cambridge UK InternationalWhaling Commission pp 407ndash415

Whitehead H (2001a) ldquoAnalysis of Animal Movement Using Opportunis-tic Individual Identifications Application to Sperm Whalesrdquo Ecology 821417ndash1432

(2001b) ldquoDirect Estimation of Within-Group Heterogeneity in Photo-Identification of Sperm Whalesrdquo Marine Mammal Science 17 718ndash728

Williams B K Nichols J D and Conroy M J (2001) Analysis and Man-agement of Animal Populations San Diego Academic Press

Wilson B Hammond P S and Thompson P M (1999) ldquoEstimating Size andAssessing Trends in a Coastal Bottlenose Dolphin Populationrdquo EcologicalApplications 9 288ndash300

Wilson B Reid R J Grellier K Thompson P M and Hammond P S(2004) ldquoConsidering the Temporal When Managing the Spatial A Popula-

954 Journal of the American Statistical Association September 2008

(a)

(b) (c)

Figure 2 (a) Variation of mean capture coefficients over time forthe left side εL (minusminus) and right side εR (- - bull - - bull) The two hor-izontal lines indicate the global mean capture coefficients δL (mdashmdash)and δR (minus minus minus) On the left vertical axis are the mean left-side het-erogeneity capture coefficients ζL and on the right vertical axis arethe mean right-side heterogeneity capture coefficients ζR Standardserrors for each coefficient are shown as vertical bars (b) Variation ofmean survival probability over time (c) Histogram of mean heteroge-neous survival probabilities

ture probability for the left side is 48 (95 HPDI = 29 68)and for the right side it is 58 (95 HPDI = 39 77) Althoughthis is a small difference the probability of the right capturecoefficient exceeding the left is 81 Examination of Figure 2(a)shows that the variation of capture coefficients over time is largecompared with the overall coefficients and that the left (εL) andright (εR) coefficients tend to track each other The probabilityof the right capture coefficient exceeding the left varies overa wide range 07ndash96 This range results from some years inwhich the probability tends strongly to the left side (1995 and2000 probability of observing the left side 95) or right side(1998 1999 2001 probability of observing the right side 91)Finally the range of the heterogeneous coefficients is compara-ble to that of the time-varying coefficients

The mean heterogeneous survival coefficient is not corre-lated with either of the mean capture heterogeneous coeffi-cients (right r = 009 left r = 022) However the meanleft and right heterogeneous capture coefficients are correlated(r = 714) indicating that animals that are more likely to beseen from one side also are more likely to be seen from theother side However the heterogeneity capture coefficient of theright side (mean 56) exceeded that of the left side (mean 23)with 85 of animals having higher right coefficients than left

Table 3 Left-side and right-side capture probabilities and survivalprobabilities by year

Year Left capture Right capture Survival

1990 1(11)

1991 54(40 68) 60(47 75) 93(84 99)

1992 74(59 86) 75(60 86) 98(94100)

1993 31(20 44) 46(33 61) 95(88100)

1994 36(24 47) 53(35 66) 95(86100)

1995 42(27 55) 71(56 88) 94(86100)

1996 30(17 42) 48(34 63) 93(82100)

1997 23(14 36) 37(23 51) 88(75 98)

1998 39(24 53) 31(20 45) 94(84100)

1999 45(31 63) 40(25 54) 93(82100)

2000 24(14 36) 57(40 72) 96(88100)

2001 87(78 94) 82(73 91) 96(89100)

2002 88(80 96) 89(80 96)

NOTE For each the mean and the 95 HPDI limits are given There is concordancebetween the left-side and right-side results The capture probabilities are more variableand reach minimums in 1997 and 2000 for the left side and in 1998 for the right sidebut both reach maximums in 2002 Survival reaches a maximum in 1992 declines to aminimum in 1997 and then rises again in later years

coefficients Because this is observed after adjustments for theoverall and time-varying components it suggests an effect dueto behavior or sampling protocol

The variation in the capture probability over time is likelyto result from a combination of factors including variation indolphin ranging patterns (see Wilson et al 2004) weather con-ditions and other logistical factors that influence sampling indifferent areas Most notably sampling protocols were changedin 2000 specifically to increase the probability of capture Thischange has proven to have been successful in the latter part ofthe study while estimates of the survival probabilities and to-tal population size remain unaffected Second another study(Wilson et al 2004) has shown that the ranging pattern of thepopulation changed in recent years with some animals spend-ing more time in the southern part of the population range alongthe east coast of Scotland south of the Moray Firth to St An-drews and the Firth of Forth The effect of this would be toreduce the capture probability in the core study area in the in-ner Moray Firth and to potentially increase heterogeneity incapture probabilities The more southerly areas also have beensampled but the data are more sparse and cover only a smallnumber of years Once these histories have become more exten-sive we expect that the capture probabilities will become lessvariable Unlike other models our estimates are also correctedfor individual heterogeneity Variation in capture probabilitiesalso might have arisen from heterogeneous effort in space Al-though this was not explicitly modeled we could have stratifiedthe capture histories by area by including separate capture co-efficients for each area

The overall mean survival estimate through the study periodis 93 (standard deviation 029 95 HPDI = 861 979) Thisis similar to an earlier maximum likelihood estimate of sur-vival for the cohort of nicked individuals from this populationfirst observed in 1990 (Saunders-Reed Hammond Grellier andThompson 1999) But our model estimate is slightly lower thansurvival estimates from other bottlenose populations of 962 plusmn76 in Sarasota Bay Florida (Wells and Scott 1990) 952 plusmn

Corkrey et al Bayesian Recapture Population Model 955

150 in Doubtful Sound New Zealand (Haase 2000) and 95ndash99 in the Sado estuary Portugal (Gaspar 2003) However thismay be because our study included subadults while the otherstudies reported adult survival Variation in survival probabil-ity occurs over time [Fig 2(b) Table 3] It is of interest thatsurvival reached a minimum in 1997 but later returned to lev-els comparable to earlier years The proportion of iterations inwhich the survival for 1997 are above the mean survival forthe remaining years (mean φ = 95 95 HPDI = 85 100)is only 08 The apparent low survival in 1997 may be due tomigration because this is not distinguishable from survival inthe model It is possible that an emigration event occurred in1997 and that those individuals have not yet returned Consid-erable heterogeneity in survival (range 77ndash97) is observedas shown in Figure 2(c) The survival probabilities are not dis-tributed smoothly suggesting multimodality The animals witha low survival coefficient may have died or migrated In thecase of migration this result would indicate that some animalsless consistently reside in the core areas compared with oth-ers

Individuals with higher survival coefficients are frequentlythose seen earliest in the study There is little information avail-able on individuals seen only more recently and consequentlytheir coefficients are close to 0 Some survival coefficients aremore negative and these correspond to animals that have notbeen seen for some time As time goes on and in the absenceof a confirmed death or sighting these coefficients can be ex-pected to become more negative However some animals thatare observed only on a few occasions have negative coefficientsThe estimation of individual survival coefficients means that es-timation of other coefficients in the model are adjusted for vari-able survival The estimation of individual capture probabilitiesallows the results to be combined with other individual covari-ates such as sex or body size if these data become availableSimilarly where individuals differ in their home ranges indi-vidual capture or survival probabilities could be related to localenvironmental conditions within their range to prey availabil-ity or to water quality

Estimates of the total population size from bilateral analy-sis of the left and right photo-identification series and estimatesfrom unilateral analyses of the left and right sides are shown inFigure 3(a) These are similar both to the maximum likelihoodestimate of 129 (95 confidence interval = 110ndash174) for datacollected in 1992 (Wilson et al 1999) and to a Bayesian mul-tisite mark recapture estimate of the nicked population of 85(95 probability interval = 76ndash263) for data collected in 2001(Durban et al 2005) The credible intervals of the abundanceestimates are generally narrower in the bilateral analysis (meanwidth 886) than in left-side analysis (mean width 1373) andright-side analysis (mean width 937) Figure 3(b) shows theprior and mean posterior recruitment probabilities ωj It is in-teresting that higher probabilities occur in 1996 and 2001 Thismight be considered to variations in capture probability butwhereas the capture probability is high in 2001 it remains lowin 1996 [see Fig 2(a)] Instead calculating the probabilities ofthe posterior exceeding the prior recruitment probabilities wefind they are below 5 (range 04ndash44) for all years except 2001in which it is 85 The higher recruitment probability in 2001may be due to an improved protocol in that year that led to the

(a)

(b)

Figure 3 (a) Mean total abundance estimates from a bilateral analy-sis with 95 HPDI Also shown are estimates from separate analysesof left (minus minus minus) and right (middot middot middot) sides along with associated 95 HPDIrepresented by symbols left + right Minimum numbers of ani-mals known to be alive are indicated by diamonds (b) Mean posteriorrecruitment probabilities ωj with 95 HPDI The prior value is rep-resented as a dotted line

identification of previously unrecognized nicked animals Weconsider the rise in 1996 to have insufficient evidence to sug-gest an increase in recruitment in that year

Earlier demographic modeling of this population has pre-dicted that the population is in decline (Saunders-Reed et al1999) To examine this we calculate the statistic [prodJ

j=2 Wj

(Wjminus1)]1(Jminus1) Values of this statistic gt1 indicate an increas-ing population whereas values lt1 indicates a decreasing pop-ulation The mean value is 989 (95 HPDI = 958 102)and the probability of it being lt10 is 77 This suggests thatthe population may be decreasing but that the evidence fora decline is weak By doing this we are able to use the fulltime series of photo identification data to obtain simultaneousposterior estimates of the Wj to compare the probability of apopulation increase or decrease In agreement with the earlierpredictive modeling these data indicate greater probability of adecline than of an increase However because it is likely thatthe population is expanding its range (Wilson et al 2004) thedecline may be confounded with temporary emigration

956 Journal of the American Statistical Association September 2008

6 DISCUSSION

Changes in the size of wildlife populations are often assessedby fitting trend models to independent abundance estimatesbut the power of these techniques to detect trends often canbe low (Gerrodette 1987 Taylor Martinez Gerrodette Bar-low and Hrovat 2007) Here we use a capturendashrecapture modelthat incorporates an underlying population model and an ob-servation model We provide a probability that a population isin decline or increasing Using photoidentification data fromthis bottlenose dolphin population we generate the first em-pirical estimate of trends in abundance for any of the popula-tions of small cetaceans inhabiting European waters Becausethe methodological changes in sampling likely are absorbedby the time-varying capture coefficient the remaining variationdue to individual capture tendencies might reasonably be de-scribed by the individual capture coefficient This then allowsexploration of capture and survival relating to individuals ratherthan to populations Information on individual survival could beimportant for addressing some management issues for exam-ple individuals differing in their spatial distribution

By decomposing the capture probabilities into overall time-varying and heterogeneous terms we find that the time-varyingcontribution and heterogeneous variation are comparable inmagnitude and their variation is greater than that of the overallterm This information can be used in the design of monitoringprograms by suggesting how effort should be distributed Thedifference in overall left and right capture probabilities mightbe interpretable as an artifact of the sampling techniques or ofbehavioral effects However after allowing for this the prob-ability of observing one side also varied over time with someyears strongly favoring one side or the other This might be ex-plained by variation in protocol between years such as changesin personnel The relative sizes of these effects may be of usein evaluating consistency of protocol over time Finally afterallowing for the overall difference and time variation the het-erogeneous capture coefficients exhibit a bias to the one sidepossibly due to behavior or protocol These findings demon-strate that bilateral decomposition of capture probabilities canprovide useful information

We decompose the survival probability similarly and we findit to be unusually low in 1997 suggesting a possible multimodalheterogeneous survival probability that may be the result of mi-gration or survival variation This type of decomposition allowsexploration of such effects in more detail In particular the mul-timodality of heterogeneous survival suggests that the popula-tion comprises individuals with different strategies some vis-iting the core study area more or less often Ongoing work isattempting to quantify individual heterogeneity such as in ex-posure to sewage contaminated waters and boat-based tourismTherefore this technique could be used to determine whetherexposure to different anthropogenic stressors is related to sur-vival

Our model also makes use of priors such as those on ωj thatare not informed by the recapture data set but make use of in-formation from other sources An alternative approach mightbe to specify these as informative priors based on expert opin-ion or tuned to obtain satisfactory results Our application il-lustrates how disparate data sources may be combined into a

single analysis Although the data source for the prior recruit-ment probability is small and problematic it is the only oneavailable Even when some priors are defined in the simulationstudy to incorrect values the estimates of parameters of interestremain valid The posterior probability also adds to the state ofknowledge of this parameter

Previous use of capturendashrecapture models with cetacean pho-toidentification data highlights the potential violation of modelassumptions due to heterogeneity in capture andor survivalprobabilities This model explicitly accounts for heterogene-ity and also provides simultaneous estimates of individual sur-vival and capture probabilities This can be used both to ex-plore the biological basis of heterogeneity and to improve sam-pling programs The individual survival coefficients also can beused to explore aspects of biology where correlates are avail-able for example the relationship of survival to age sex orother variables may be of interest Abundance under the as-sumption of heterogeneous survival has been estimated by LeeHuang and Ou (2004) whereas individual and time-varyingsurvival estimates have been obtained using band-recoverymodels (Grosbois and Thompson 2005) Other studies have ex-amined survival by group for example Franklin Andersonand Burnham (2002) obtained separate estimates of group-levelsurvival for males and females using stratification in a bandingstudy Although we lack such variables as age and sex theo-retically it would be possible to examine individual survivalby sex either as a post hoc comparison or through their inte-gration into the model itself For example survival and cap-ture could be modeled as logit(φ) = α + βj + γd + τ andlogit(π) = δ + εj + ζd + χ where τ corresponds to stage-specific (eg young immature reproductive female senes-cent) survival and χ corresponds to capture probabilities forvarious categories (eg young reproductive adult)

Other cetacean studies have examined heterogeneity White-head (2001b) obtained an estimate of the coefficient of variation(CV) for identification from a randomization test of successiveidentifications in whale track data (The CV is also a measure ofcapture heterogeneity) They found that the CV was significantin one of the two years and that probably depended on the sizesof the animals being observed This is not comparable to thedolphin observations in which heterogeneity is more likely toresult from differences in animal behavior In the present studywe use a likelihood in which heterogeneity of survival and cap-ture has been incorporated by a logit function This approachwas described previously (Pledger Pollock and Norris 2003Pledger and Schwarz 2002) within a non-Bayesian contextPledger et al (2003) classified animals into an unknown set oflatent classes each of which could have a separate survival andcapture probability A simplification allows for time-varyingand animal-level heterogeneity This approach is similar to oursbut we also integrate a dynamic population model and an ob-servation model into the estimation process Goodman (2004)also implemented a population model in a Bayesian context thatcombines capturendashrecapture data with carcass-recovery databut assumes the absence of heterogeneity in capture probabil-ities Durban et al (2005) estimated abundance for the samepopulation as us but used a multisite Bayesian log-linear modeland model averaging (King and Brooks 2001) across the pos-sible models with varying combinations of interactions Our

Corkrey et al Bayesian Recapture Population Model 957

model could be extended to include multiple sites and yearsby using a capturendashrecapture history that includes site informa-tion and a yearndashsite term in the capture probability expressionBy doing this we suggest that geographic areas with more datacan be used to strengthen estimates from poorly sampled areasThis also may account for some individual variation currentlyidentified as individual capture heterogeneity because some in-dividuals may be more closely associated with some areas thanothers and may attenuate a possible confounding of migrationwith survival

We use a separate data set and a negative-binomial model toinflate the estimate of the proportion of nicked animals in thesamples Another study (Da Silva Rodrigues Leite and Mi-lan 2003) obtained an abundance estimate for bowhead whalesusing an inflation factor derived from the numbers of good andbad photographs of marked and unmarked whales Our analysisavoids the lack of independence between good photographs anddistinctively marked animals in the photographs that might in-validate the estimate of variance in the inflation factor Otherstudies avoid this issue by including only separately catego-rized good-quality photographs (eg Wilson et al 1999 ReadUrian Wilson and Waples 2003) However independent es-timates of inflation when available such as in our study arepreferred

We do not know of other cetacean studies that combinephoto-identification series of the left and right sides but ananalogous approach is that of log-linear capturendashrecapturemodeling (King and Brooks 2001) in which animals are identi-fied from separate lists Some photo-identification studies mightuse information from both sides in the process of identifyinganimals (Joyce and Dorsey 1990) A more robust approach isto perform the identification on each side separately We haveshown how these separate data can then be combined

A major advantage to our modeling approach is its extensibil-ity Within the Bayesian framework models can be comparedby such methods as reversible-jump Markov chain Monte Carlotechniques (Green 1995) Model extensions include extendingthe population model considering an individual-level approachThis would allow incorporation of the heterogeneity into thispart of the model and also modeling of the unobserved popu-lation More detailed modeling of heterogeneity could be ex-plored by including hyperpriors for the variances of the recap-ture model terms For convenience we analyze the data on anannual basis Additional information would be available if theanalyses were conducted on a finer time scale such as monthsor by individual trip This would allow the model to be extendedto include seasonal effects For example dolphins are observedmore frequently in summer months than in winter months andthis tendency could be included in an extended model by chang-ing the capture probability to be logit(π) = δt + εj + ζd whereδt t isin [1 12] is a monthly capture coefficient This maybe of use in modeling fine-scale movements The response vari-able itself also can be generalized Although we are careful toonly include data from high-quality photographs and nicked an-imals there is still the possibility of some identification errorsparticularly for more subtly marked individuals A possible ex-tension of the model could allow identification to be assigneddifferent levels of certainty for example by incorporating anintermediate state between nonrecapture and recapture that we

term ldquopossible recapturerdquo This could be done by replacing thelogit function logit(ν) = log(ν(1 minus ν)) where ν correspondsto the probability of recapture by a proportional odds function(McCullagh 1980) This technique has previously found appli-cation in a study of photoidentification used to model photo-graphic quality scores and individual distinctiveness of hump-back whale tail flukes (Friday Smith Stevick and Allen 2000)In this approach the probability of ldquopossible recapturerdquo is givenby ν1 = π1 the probability of certain recapture is given byν2 = π1 + π2 and includes the possible recapture category andthe probability of no capture is given by ν0 = 1 minus π1 + π2There are two proportional odds functions corresponding topossible recapture and certain recapture which are logit(ν1) =log(π1(π2 +π3)) and logit(ν2) = log((π1 +π2)π3) We thencould extend the recapture model using logit(ν1) = α+βj +γk1

and logit(ν2) = α + βj + γk2 This proportional odds approachmay be particularly useful in situations where automatic match-ing algorithms (Saacutenchez-Mariacuten 2000) provide an estimate ofthe likelihood of a match which could then be incorporatedinto abundance estimate models

APPENDIX DERIVATION OF POSTERIORS

Here we show how we derive the posterior conditional distributionsfor the model parameters and describe how each is updated within theMCMC algorithm The full joint distribution has the form shown in(A1) In (A1) the first two lines correspond to the population modelthe third line corresponds to the inflation model the fourth line cor-responds to the observation model the fifth line corresponds to therecapture model and the remaining lines correspond to priors

π = P(B1|ω1N0W0)P (S1|αN0)

(Sec 31)

timesSprod

j=2

P(Bj |Sjminus1Bjminus1Wjminus1ωj )

times P(Sj |Sjminus1Bjminus1 αβjminus1)

(Sec 31)

timesSprod

j=1

P(Wj |Sj Bj υj )

Tjprod

i=1

P(tnji |υj twji )

(Sec 32)

timesSprod

j=1

P(nLj|δL εjLSj Bj )P (nRj

|δR εjRSj Bj )

(Sec 33)

timesSprod

j=1

P(Hdj |αβj γd δL εjL ζdL δR εjR ζdR)

(Sec 34)

times P(N0|λN)P (W0|N0 υ0)P (α δL δRλN υ0)

(Sec 41)

timesSprod

j=1

P(βj εjL εjRωj υj )

nprod

d=1

P(γd)P (ζdL)P (ζdR)

(Sec 41) (A1)

958 Journal of the American Statistical Association September 2008

For each parameter we obtain the corresponding posterior condi-tional distribution by extracting all terms containing that parameterfrom the full joint distribution When the conditional distributions havea standard form we use the Gibbs sampler (Brooks 1998) to update theparameters otherwise we use MetropolisndashHastings updates (Chib andGreenberg 1995)

Within a MetropolisndashHastings update we propose a new valuefor a parameter x and denote the proposed value by xprime To decidewhether to accept the proposed value we calculate an acceptance ra-tio α = min(1A) where A is the acceptance ratio The acceptanceratio is given by A = π(xprime)q(x)π(x)q(xprime) where π(x) is the poste-rior probability of x and q(x) is the probability of the proposed valueWe accept the new value xprime with probability α otherwise we leavethe value of x unchanged (See Chib and Greenberg 1995 for an exam-ple)

Normal proposal distributions are used for continuous parameterscentered around the current parameter value Discrete uniform pro-posals are used for discrete variables each centered around the currentparameter values The discrete proposals also require constraints to en-sure that the proposed values are valid For example a proposed valuefor N0 must be less than the current value of W0 We provide a detaileddescription of each update next

Population Model

The population model has parameters ωj λ Bj Sj and N0 andare updated using Gibbs or MetropolisndashHastings samplers The condi-tional posterior distributions for ωj and λN can be derived as

ωj |Bj Sjminus1Bjminus1

sim Beta(Bj + 0238Wjminus1 minus Sjminus1 minus Bjminus1 minus Bj + 1)

and

λN |N0 sim Gam(76 + N0101)

These are individually updated using the Gibbs sampler during eachiteration of the Markov chain

The parameters Bj Sj and N0 are updated by the MetropolisndashHastings algorithm using acceptance ratios calculated as described ear-lier We use uniform proposals centered on the current parameter valueand limited to the range of possible values

For the Sj rsquos we use three alternative proposals

Sprimej=1 sim Unif

(max

(0 S1 minus 5 S2 minus B1max(n1Ln1R) minus N1

2 minus B1)

min(S1 + 5N0W1 minus B1W1 minus B1 minus B2

W1 minus B1 minus 1))

Sprime2lejltJ sim Unif

(max

(0 Sj minus 5 Sj+1 minus Bj max(njLnjR) minus Bj

2 minus Bj

)

min(Sj + 5 Sjminus1 + Bjminus1Wj minus Bj Wj minus Bj minus Bj+1

Wj minus Bj minus 1))

and

Sprimej=J sim Unif

(max

(0 SJ minus 5max(nJLnJR) minus BJ 2 minus NJ

)

min(SJ minus 5 Sjminus1 + Bjminus1Wj minus Bj Wj minus Bj minus 1))

For Bj we use three alternative proposals

B prime1 sim Unif

(max

(0B1 minus 15 S2 minus S1max(n1Ln1R) minus S1

2 minus S1)

min(B1 + 15W0 minus N0W1 minus S1W1 minus S1 minus B2

W1 minus S1 minus 1))

B prime2lejltJ sim Unif

(max

(0Bj minus 15 Sj+1 minus Sj max(njLnjR) minus Sj

2 minus Sj

)

min(Bj + 15Wjminus1 minus Sjminus1 minus Bjminus1Wj minus Sj

Wj minus Sj minus Bj+1Wj minus Sj minus 1))

and

B primeJ sim Unif

(max

(0BJ minus 15max(nJLnJR) minus SJ 2 minus SJ

)

min(BJ + 15WJminus1 minus SJminus1 minus BJminus1

WJ minus SJ WJ minus SJ minus 1))

For N0 we use

N prime0 sim Unif

(max(5N0 minus 15 S1)min(N0 + 15W0 minus B1W0 minus 1)

)

Inflation Model

The inflation model parameters Wj and υj are updated usingMetropolisndashHastings and Gibbs updates The conditional posterior dis-tributions for υj can be derived as

υ0|N0W0 sim Beta(N0 + 1W0 minus N0 + 1)

and

υjgt0|Sj Bj Wj tnji twji sim Beta

(

Sj + Bj +T (j)sum

i=1

tnji + 1

Wj minus Sj minus Bj +T (j)sum

i=1

twji minus tnji + 1

)

These are updated separately using the Gibbs sampler on each itera-tion

The remaining parameter Wj has a nonstandard conditional pos-terior distribution We use larger updates than in the population modelbecause the initial total population is expected to be larger in size

W prime0 sim Unif

(max(0W0 minus 15N0 + 1N0 + B1)W0 + 15

)

W prime1lejltJ sim Unif

(max(0Wj minus 15 Sj + Bj + 1 Sj + Bj + Bj+1)

Wj + 15)

and

W primeJ sim Unif

(max(0WJ minus 15 SJ + BJ + 1)WJ + 15

)

Recapture Model

For the recapture model we have parameters α βj γd δL εjLζdL δR εjR and ζdR These all have nonstandard conditional pos-terior distributions and we use the MetropolisndashHastings algorithm toupdate the parameters on each iteration We use individual normal dis-tribution proposals centered on the current parameter values and withvariance equal to the prior variance for example

αprime sim N(α185)

[Received June 2005 Revised May 2007]

Corkrey et al Bayesian Recapture Population Model 959

REFERENCES

Brooks S P (1998) ldquoMarkov Chain Monte Carlo Method and Its ApplicationrdquoThe Statistician 47 69ndash100

Brooks S P and Roberts G O (1998) ldquoConvergence Assessment Techniquesfor Markov Chain Monte Carlordquo Statistics and Computing 8 319ndash335

Buckland S T (1982) ldquoA CapturendashRecapture Survival Analysisrdquo Journal ofAnimal Ecology 51 833ndash847

(1990) ldquoEstimation of Survival Rates From Sightings of IndividuallyIdentifiable Whalesrdquo in Individual Recognition of Cetaceans Use of Photo-Identification and Other Techniques to Estimate Population Parameters edsP S Hammond S A Mizroch and G P Donovan Cambridge UK Inter-national Whaling Commission pp 149ndash153

Burnham K and Overton W (1979) ldquoRobust Estimation of Population SizeWhen Capture Probabilities Vary Among Animalsrdquo Ecology 60 927ndash936

Carothers A (1979) ldquoQuantifying Unequal Catchability and Its Effect on Sur-vival Estimates in an Actual Populationrdquo Journal of Animal Ecology 48863ndash869

Chao A (1987) ldquoEstimating the Population Size for CapturendashRecapture DataWith Unequal Catchabilityrdquo Biometrics 43 783ndash791

Chao A Lee S-M and Jeng S-L (1992) ldquoEstimating Population Size forCapturendashRecapture Data When Capture Probabilities Vary by Time and Indi-vidual Animalrdquo Biometrics 48 201ndash216

Chib S and Greenberg E (1995) ldquoUnderstanding the MetropolisndashHastingsAlgorithmrdquo The American Statistician 49 327ndash335

Cormack R (1972) ldquoThe Logic of CapturendashRecapture Experimentsrdquo Biomet-rics 28 337ndash343

Coull B A and Agresti A (1999) ldquoThe Use of Mixed Logit Models to Re-flect Heterogeneity in CapturendashRecapture Studiesrdquo Biometrics 55 294ndash301

Da Silva C Q Rodrigues J Leite J G and Milan L A (2003) ldquoBayesianEstimation of the Size of a Closed Population Using Photo-ID With Part ofthe Population Uncatchablerdquo Communications in Statistics 32 677ndash696

Dorazio R M and Royle J A (2003) ldquoMixture Models for Estimating theSize of a Closed Population When Capture Rates Vary Among IndividualsrdquoBiometrics 59 351ndash364

Durban J Elston D Ellifrit D Dickson E Hammon P and Thompson P(2005) ldquoMultisite Mark-Recapture for Cetaceans Population Estimates WithBayesian Model Averagingrdquo Marine Mammal Science 21 80ndash92

Fienberg S E Johnson M S and Junker B W (1999) ldquoClassical Multi-level and Bayesian Approaches to Population Size Estimation Using MultipleListsrdquo Journal of the Royal Statistical Society Ser A 162 383ndash405

Forcada J and Aguilar A (2000) ldquoUse of Photographic Identification inCapturendashRecapture Studies of Mediterranean Monk Sealsrdquo Marine MammalScience 16 767ndash793

Franklin A B Anderson D R and Burnham K P (2002) ldquoEstimation ofLong-Term Trends and Variation in Avian Survival Probabilities Using Ran-dom Effects Modelsrdquo Journal of Applied Statistics 29 267ndash287

Friday N Smith T D Stevick P T and Allen J (2000) ldquoMeasure-ment of Photographic Quality and Individual Distinctiveness for the Photo-graphic Identification of Humpback Whales Megaptera novaeangliaerdquo Ma-rine Mammal Science 16 355ndash374

Gaspar R (2003) ldquoStatus of the Resident Bottlenose Dolphin Population inthe Sado Estuary Past Present and Futurerdquo unpublished doctoral dissertationUniversity of St Andrews School of Biology

Gelfand A E and Sahu S K (1999) ldquoIdentifiability Improper Priors andGibbs Sampling for Generalized Linear Modelsrdquo Journal of the AmericanStatistical Association 94 247ndash253

Gerrodette T (1987) ldquoA Power Analysis for Detecting Trendsrdquo Ecology 681364ndash1372

Goodman D (2004) ldquoMethods for Joint Inference From Multiple Data Sourcesfor Improved Estimates of Population Size and Survival Ratesrdquo Marine Mam-mal Science 20 401ndash423

Green P J (1995) ldquoReversible-Jump Markov Change Monte Carlo BayesianModel Determinationrdquo Biometrika 82 711ndash732

Grosbois V and Thompson P M (2005) ldquoNorth Atlantic Climate VariationInfluences Survival in Adult Fulmarsrdquo Oikos 109 273ndash290

Haase P (2000) ldquoSocial Organisation Behaviour and Population Parametersof Bottlenose Dolphins in Doubtful Sound Fiordlandrdquo unpublished mastersdissertation University of Otago Dept of Marine Science

Hammond P (1986) ldquoEstimating the Size of Naturally Marked Whale Pop-ulations Using CapturendashRecapture Techniquesrdquo Report of the InternationalWhaling Commission Special Issue 8 253ndash282

(1990) ldquoHeterogeneity in the Gulf of Maine Estimating HumpbackWhale Population Size When Capture Probabilities Are not Equalrdquo in In-dividual Recognition of Cetaceans Use of Photo-Identification and OtherTechniques to Estimate Population Parameters eds P S Hammond S A

Mizroch and G P Donovan Cambridge UK International Whaling Com-mission pp 135ndash153

Huggins R (1989) ldquoOn the Statistical Analysis of Capture Experimentsrdquo Bio-metrika 76 133ndash140

Joyce G G and Dorsey E M (1990) ldquoA Feasibility Study on the Useof Photo-Identification Techniques for Southern Hemisphere Minke WhaleStock Assessmentrdquo in Individual Recognition of Cetaceans Use of Photo-Identification and Other Techniques to Estimate Population Parameters edsP S Hammond S A Mizroch and G P Donovan Cambridge UK Inter-national Whaling Commission pp 419ndash423

King R and Brooks S (2001) ldquoOn the Bayesian Analysis of PopulationSizerdquo Biometrika 88 317ndash336

Lebreton J-D Burnham K P Clobert J and Anderson S R (1992)ldquoModeling Survival and Testing Biological Hypotheses Using Marked An-imals A Unified Approach With Case Studiesrdquo Ecological Monographs 6267ndash118

Lee S-M Huang L-H H and Ou S-T (2004) ldquoBand Recovery Model In-ference With Heterogeoneous Survival Ratesrdquo Statistica Sinica 14 513ndash531

Link W A and Barker R J (2005) ldquoModeling Association Among De-mographic Parameters in Analysis of Open Population CapturendashRecaptureDatardquo Biometrics 61 46ndash54

Mann J and Smuts B (1999) ldquoBehavioral Development in Wild BottlenoseDolphin Newborns (Tursiops sp)rdquo Behaviour 136 529ndash566

McCullagh P (1980) ldquoRegression Models for Ordinal Datardquo Journal of theRoyal Statistical Society Ser B 42 109ndash142

Otis D L Burnham K P White G C and Anderson D R (1978) ldquoStatis-tical Inference From Capture Data on Closed Animal Populationsrdquo WildlifeMonographs 62 135

Pledger S (2000) ldquoUnified Maximum Likelihood Estimates for ClosedCapturendashRecapture Models Using Mixturesrdquo Biometrics 56 434ndash442

Pledger S and Schwarz C J (2002) ldquoModelling Heterogeneity of Survivalin Band-Recovery Data Using Mixturesrdquo Journal of Applied Statistics 29315ndash327

Pledger S Pollock K H and Norris J L (2003) ldquoOpen CapturendashRecaptureModels With Heterogeneity I CormackndashJollyndashSeber Modelrdquo Ecology 59786ndash794

Pollock K H Nichols J D Brownie C and Hines J E (1990) ldquoStatisticalInference for CapturendashRecapture Samplingrdquo Wildlife Monographs 107 1ndash97

Read A Urian K W Wilson B and Waples D (2003) ldquoAbundance andStock Identity of Bottlenose Dolphins in the Bayes Sounds and Estuaries ofNorth Carolinardquo Marine Mammal Science 19 59ndash73

Saacutenchez-Mariacuten F J (2000) ldquoAutomatic Recognition of Biological ShapesWith and Without Representations of Shaperdquo Artificial Intelligence in Medi-cine 18 173ndash186

Saunders-Reed C A Hammond P S Grellier K and Thompson P M(1999) ldquoDevelopment of a Population Model for Bottlenose Dolphinsrdquo Re-search Survey and Monitoring Report 156 Scottish Natural Heritage

Scottish Natural Heritage (1995) Natura 2000 A Guide to the 1992 EC Habi-tats Directive in Scotlandrsquos Marine Environment Perth UK Scottish Nat-ural Heritage

Seber G (2002) The Estimation of Animal Abundance Caldwell NJ Black-burn Press

Stevick P T Palsboslashll P J Smith T D Bravington M V and Ham-mond P S (2001) ldquoErrors in Identification Using Natural Markings RatesSources and Effects on CapturendashRecapture Estimates of Abundancerdquo Cana-dian Journal of Fisheries and Aquatic Science 58 1861ndash1870

Taylor B L Martinez M Gerrodette T Barlow J and Hrovat Y N (2007)ldquoLessons From Monitoring Trends in Abundance of Marine Mammalsrdquo Ma-rine Mammal Science 23 157ndash175

Wells R S and Scott M D (1990) ldquoEstimating Bottlenose Dolphin Popu-lation Parameters From Individual Identification and CapturendashRelease Tech-niquesrdquo in Individual Recognition of Cetaceans Use of Photo-Identificationand Other Techniques to Estimate Population Parameters eds P S Ham-mond S A Mizroch and G P Donovan Cambridge UK InternationalWhaling Commission pp 407ndash415

Whitehead H (2001a) ldquoAnalysis of Animal Movement Using Opportunis-tic Individual Identifications Application to Sperm Whalesrdquo Ecology 821417ndash1432

(2001b) ldquoDirect Estimation of Within-Group Heterogeneity in Photo-Identification of Sperm Whalesrdquo Marine Mammal Science 17 718ndash728

Williams B K Nichols J D and Conroy M J (2001) Analysis and Man-agement of Animal Populations San Diego Academic Press

Wilson B Hammond P S and Thompson P M (1999) ldquoEstimating Size andAssessing Trends in a Coastal Bottlenose Dolphin Populationrdquo EcologicalApplications 9 288ndash300

Wilson B Reid R J Grellier K Thompson P M and Hammond P S(2004) ldquoConsidering the Temporal When Managing the Spatial A Popula-

Corkrey et al Bayesian Recapture Population Model 955

150 in Doubtful Sound New Zealand (Haase 2000) and 95ndash99 in the Sado estuary Portugal (Gaspar 2003) However thismay be because our study included subadults while the otherstudies reported adult survival Variation in survival probabil-ity occurs over time [Fig 2(b) Table 3] It is of interest thatsurvival reached a minimum in 1997 but later returned to lev-els comparable to earlier years The proportion of iterations inwhich the survival for 1997 are above the mean survival forthe remaining years (mean φ = 95 95 HPDI = 85 100)is only 08 The apparent low survival in 1997 may be due tomigration because this is not distinguishable from survival inthe model It is possible that an emigration event occurred in1997 and that those individuals have not yet returned Consid-erable heterogeneity in survival (range 77ndash97) is observedas shown in Figure 2(c) The survival probabilities are not dis-tributed smoothly suggesting multimodality The animals witha low survival coefficient may have died or migrated In thecase of migration this result would indicate that some animalsless consistently reside in the core areas compared with oth-ers

Individuals with higher survival coefficients are frequentlythose seen earliest in the study There is little information avail-able on individuals seen only more recently and consequentlytheir coefficients are close to 0 Some survival coefficients aremore negative and these correspond to animals that have notbeen seen for some time As time goes on and in the absenceof a confirmed death or sighting these coefficients can be ex-pected to become more negative However some animals thatare observed only on a few occasions have negative coefficientsThe estimation of individual survival coefficients means that es-timation of other coefficients in the model are adjusted for vari-able survival The estimation of individual capture probabilitiesallows the results to be combined with other individual covari-ates such as sex or body size if these data become availableSimilarly where individuals differ in their home ranges indi-vidual capture or survival probabilities could be related to localenvironmental conditions within their range to prey availabil-ity or to water quality

Estimates of the total population size from bilateral analy-sis of the left and right photo-identification series and estimatesfrom unilateral analyses of the left and right sides are shown inFigure 3(a) These are similar both to the maximum likelihoodestimate of 129 (95 confidence interval = 110ndash174) for datacollected in 1992 (Wilson et al 1999) and to a Bayesian mul-tisite mark recapture estimate of the nicked population of 85(95 probability interval = 76ndash263) for data collected in 2001(Durban et al 2005) The credible intervals of the abundanceestimates are generally narrower in the bilateral analysis (meanwidth 886) than in left-side analysis (mean width 1373) andright-side analysis (mean width 937) Figure 3(b) shows theprior and mean posterior recruitment probabilities ωj It is in-teresting that higher probabilities occur in 1996 and 2001 Thismight be considered to variations in capture probability butwhereas the capture probability is high in 2001 it remains lowin 1996 [see Fig 2(a)] Instead calculating the probabilities ofthe posterior exceeding the prior recruitment probabilities wefind they are below 5 (range 04ndash44) for all years except 2001in which it is 85 The higher recruitment probability in 2001may be due to an improved protocol in that year that led to the

(a)

(b)

Figure 3 (a) Mean total abundance estimates from a bilateral analy-sis with 95 HPDI Also shown are estimates from separate analysesof left (minus minus minus) and right (middot middot middot) sides along with associated 95 HPDIrepresented by symbols left + right Minimum numbers of ani-mals known to be alive are indicated by diamonds (b) Mean posteriorrecruitment probabilities ωj with 95 HPDI The prior value is rep-resented as a dotted line

identification of previously unrecognized nicked animals Weconsider the rise in 1996 to have insufficient evidence to sug-gest an increase in recruitment in that year

Earlier demographic modeling of this population has pre-dicted that the population is in decline (Saunders-Reed et al1999) To examine this we calculate the statistic [prodJ

j=2 Wj

(Wjminus1)]1(Jminus1) Values of this statistic gt1 indicate an increas-ing population whereas values lt1 indicates a decreasing pop-ulation The mean value is 989 (95 HPDI = 958 102)and the probability of it being lt10 is 77 This suggests thatthe population may be decreasing but that the evidence fora decline is weak By doing this we are able to use the fulltime series of photo identification data to obtain simultaneousposterior estimates of the Wj to compare the probability of apopulation increase or decrease In agreement with the earlierpredictive modeling these data indicate greater probability of adecline than of an increase However because it is likely thatthe population is expanding its range (Wilson et al 2004) thedecline may be confounded with temporary emigration

956 Journal of the American Statistical Association September 2008

6 DISCUSSION

Changes in the size of wildlife populations are often assessedby fitting trend models to independent abundance estimatesbut the power of these techniques to detect trends often canbe low (Gerrodette 1987 Taylor Martinez Gerrodette Bar-low and Hrovat 2007) Here we use a capturendashrecapture modelthat incorporates an underlying population model and an ob-servation model We provide a probability that a population isin decline or increasing Using photoidentification data fromthis bottlenose dolphin population we generate the first em-pirical estimate of trends in abundance for any of the popula-tions of small cetaceans inhabiting European waters Becausethe methodological changes in sampling likely are absorbedby the time-varying capture coefficient the remaining variationdue to individual capture tendencies might reasonably be de-scribed by the individual capture coefficient This then allowsexploration of capture and survival relating to individuals ratherthan to populations Information on individual survival could beimportant for addressing some management issues for exam-ple individuals differing in their spatial distribution

By decomposing the capture probabilities into overall time-varying and heterogeneous terms we find that the time-varyingcontribution and heterogeneous variation are comparable inmagnitude and their variation is greater than that of the overallterm This information can be used in the design of monitoringprograms by suggesting how effort should be distributed Thedifference in overall left and right capture probabilities mightbe interpretable as an artifact of the sampling techniques or ofbehavioral effects However after allowing for this the prob-ability of observing one side also varied over time with someyears strongly favoring one side or the other This might be ex-plained by variation in protocol between years such as changesin personnel The relative sizes of these effects may be of usein evaluating consistency of protocol over time Finally afterallowing for the overall difference and time variation the het-erogeneous capture coefficients exhibit a bias to the one sidepossibly due to behavior or protocol These findings demon-strate that bilateral decomposition of capture probabilities canprovide useful information

We decompose the survival probability similarly and we findit to be unusually low in 1997 suggesting a possible multimodalheterogeneous survival probability that may be the result of mi-gration or survival variation This type of decomposition allowsexploration of such effects in more detail In particular the mul-timodality of heterogeneous survival suggests that the popula-tion comprises individuals with different strategies some vis-iting the core study area more or less often Ongoing work isattempting to quantify individual heterogeneity such as in ex-posure to sewage contaminated waters and boat-based tourismTherefore this technique could be used to determine whetherexposure to different anthropogenic stressors is related to sur-vival

Our model also makes use of priors such as those on ωj thatare not informed by the recapture data set but make use of in-formation from other sources An alternative approach mightbe to specify these as informative priors based on expert opin-ion or tuned to obtain satisfactory results Our application il-lustrates how disparate data sources may be combined into a

single analysis Although the data source for the prior recruit-ment probability is small and problematic it is the only oneavailable Even when some priors are defined in the simulationstudy to incorrect values the estimates of parameters of interestremain valid The posterior probability also adds to the state ofknowledge of this parameter

Previous use of capturendashrecapture models with cetacean pho-toidentification data highlights the potential violation of modelassumptions due to heterogeneity in capture andor survivalprobabilities This model explicitly accounts for heterogene-ity and also provides simultaneous estimates of individual sur-vival and capture probabilities This can be used both to ex-plore the biological basis of heterogeneity and to improve sam-pling programs The individual survival coefficients also can beused to explore aspects of biology where correlates are avail-able for example the relationship of survival to age sex orother variables may be of interest Abundance under the as-sumption of heterogeneous survival has been estimated by LeeHuang and Ou (2004) whereas individual and time-varyingsurvival estimates have been obtained using band-recoverymodels (Grosbois and Thompson 2005) Other studies have ex-amined survival by group for example Franklin Andersonand Burnham (2002) obtained separate estimates of group-levelsurvival for males and females using stratification in a bandingstudy Although we lack such variables as age and sex theo-retically it would be possible to examine individual survivalby sex either as a post hoc comparison or through their inte-gration into the model itself For example survival and cap-ture could be modeled as logit(φ) = α + βj + γd + τ andlogit(π) = δ + εj + ζd + χ where τ corresponds to stage-specific (eg young immature reproductive female senes-cent) survival and χ corresponds to capture probabilities forvarious categories (eg young reproductive adult)

Other cetacean studies have examined heterogeneity White-head (2001b) obtained an estimate of the coefficient of variation(CV) for identification from a randomization test of successiveidentifications in whale track data (The CV is also a measure ofcapture heterogeneity) They found that the CV was significantin one of the two years and that probably depended on the sizesof the animals being observed This is not comparable to thedolphin observations in which heterogeneity is more likely toresult from differences in animal behavior In the present studywe use a likelihood in which heterogeneity of survival and cap-ture has been incorporated by a logit function This approachwas described previously (Pledger Pollock and Norris 2003Pledger and Schwarz 2002) within a non-Bayesian contextPledger et al (2003) classified animals into an unknown set oflatent classes each of which could have a separate survival andcapture probability A simplification allows for time-varyingand animal-level heterogeneity This approach is similar to oursbut we also integrate a dynamic population model and an ob-servation model into the estimation process Goodman (2004)also implemented a population model in a Bayesian context thatcombines capturendashrecapture data with carcass-recovery databut assumes the absence of heterogeneity in capture probabil-ities Durban et al (2005) estimated abundance for the samepopulation as us but used a multisite Bayesian log-linear modeland model averaging (King and Brooks 2001) across the pos-sible models with varying combinations of interactions Our

Corkrey et al Bayesian Recapture Population Model 957

model could be extended to include multiple sites and yearsby using a capturendashrecapture history that includes site informa-tion and a yearndashsite term in the capture probability expressionBy doing this we suggest that geographic areas with more datacan be used to strengthen estimates from poorly sampled areasThis also may account for some individual variation currentlyidentified as individual capture heterogeneity because some in-dividuals may be more closely associated with some areas thanothers and may attenuate a possible confounding of migrationwith survival

We use a separate data set and a negative-binomial model toinflate the estimate of the proportion of nicked animals in thesamples Another study (Da Silva Rodrigues Leite and Mi-lan 2003) obtained an abundance estimate for bowhead whalesusing an inflation factor derived from the numbers of good andbad photographs of marked and unmarked whales Our analysisavoids the lack of independence between good photographs anddistinctively marked animals in the photographs that might in-validate the estimate of variance in the inflation factor Otherstudies avoid this issue by including only separately catego-rized good-quality photographs (eg Wilson et al 1999 ReadUrian Wilson and Waples 2003) However independent es-timates of inflation when available such as in our study arepreferred

We do not know of other cetacean studies that combinephoto-identification series of the left and right sides but ananalogous approach is that of log-linear capturendashrecapturemodeling (King and Brooks 2001) in which animals are identi-fied from separate lists Some photo-identification studies mightuse information from both sides in the process of identifyinganimals (Joyce and Dorsey 1990) A more robust approach isto perform the identification on each side separately We haveshown how these separate data can then be combined

A major advantage to our modeling approach is its extensibil-ity Within the Bayesian framework models can be comparedby such methods as reversible-jump Markov chain Monte Carlotechniques (Green 1995) Model extensions include extendingthe population model considering an individual-level approachThis would allow incorporation of the heterogeneity into thispart of the model and also modeling of the unobserved popu-lation More detailed modeling of heterogeneity could be ex-plored by including hyperpriors for the variances of the recap-ture model terms For convenience we analyze the data on anannual basis Additional information would be available if theanalyses were conducted on a finer time scale such as monthsor by individual trip This would allow the model to be extendedto include seasonal effects For example dolphins are observedmore frequently in summer months than in winter months andthis tendency could be included in an extended model by chang-ing the capture probability to be logit(π) = δt + εj + ζd whereδt t isin [1 12] is a monthly capture coefficient This maybe of use in modeling fine-scale movements The response vari-able itself also can be generalized Although we are careful toonly include data from high-quality photographs and nicked an-imals there is still the possibility of some identification errorsparticularly for more subtly marked individuals A possible ex-tension of the model could allow identification to be assigneddifferent levels of certainty for example by incorporating anintermediate state between nonrecapture and recapture that we

term ldquopossible recapturerdquo This could be done by replacing thelogit function logit(ν) = log(ν(1 minus ν)) where ν correspondsto the probability of recapture by a proportional odds function(McCullagh 1980) This technique has previously found appli-cation in a study of photoidentification used to model photo-graphic quality scores and individual distinctiveness of hump-back whale tail flukes (Friday Smith Stevick and Allen 2000)In this approach the probability of ldquopossible recapturerdquo is givenby ν1 = π1 the probability of certain recapture is given byν2 = π1 + π2 and includes the possible recapture category andthe probability of no capture is given by ν0 = 1 minus π1 + π2There are two proportional odds functions corresponding topossible recapture and certain recapture which are logit(ν1) =log(π1(π2 +π3)) and logit(ν2) = log((π1 +π2)π3) We thencould extend the recapture model using logit(ν1) = α+βj +γk1

and logit(ν2) = α + βj + γk2 This proportional odds approachmay be particularly useful in situations where automatic match-ing algorithms (Saacutenchez-Mariacuten 2000) provide an estimate ofthe likelihood of a match which could then be incorporatedinto abundance estimate models

APPENDIX DERIVATION OF POSTERIORS

Here we show how we derive the posterior conditional distributionsfor the model parameters and describe how each is updated within theMCMC algorithm The full joint distribution has the form shown in(A1) In (A1) the first two lines correspond to the population modelthe third line corresponds to the inflation model the fourth line cor-responds to the observation model the fifth line corresponds to therecapture model and the remaining lines correspond to priors

π = P(B1|ω1N0W0)P (S1|αN0)

(Sec 31)

timesSprod

j=2

P(Bj |Sjminus1Bjminus1Wjminus1ωj )

times P(Sj |Sjminus1Bjminus1 αβjminus1)

(Sec 31)

timesSprod

j=1

P(Wj |Sj Bj υj )

Tjprod

i=1

P(tnji |υj twji )

(Sec 32)

timesSprod

j=1

P(nLj|δL εjLSj Bj )P (nRj

|δR εjRSj Bj )

(Sec 33)

timesSprod

j=1

P(Hdj |αβj γd δL εjL ζdL δR εjR ζdR)

(Sec 34)

times P(N0|λN)P (W0|N0 υ0)P (α δL δRλN υ0)

(Sec 41)

timesSprod

j=1

P(βj εjL εjRωj υj )

nprod

d=1

P(γd)P (ζdL)P (ζdR)

(Sec 41) (A1)

958 Journal of the American Statistical Association September 2008

For each parameter we obtain the corresponding posterior condi-tional distribution by extracting all terms containing that parameterfrom the full joint distribution When the conditional distributions havea standard form we use the Gibbs sampler (Brooks 1998) to update theparameters otherwise we use MetropolisndashHastings updates (Chib andGreenberg 1995)

Within a MetropolisndashHastings update we propose a new valuefor a parameter x and denote the proposed value by xprime To decidewhether to accept the proposed value we calculate an acceptance ra-tio α = min(1A) where A is the acceptance ratio The acceptanceratio is given by A = π(xprime)q(x)π(x)q(xprime) where π(x) is the poste-rior probability of x and q(x) is the probability of the proposed valueWe accept the new value xprime with probability α otherwise we leavethe value of x unchanged (See Chib and Greenberg 1995 for an exam-ple)

Normal proposal distributions are used for continuous parameterscentered around the current parameter value Discrete uniform pro-posals are used for discrete variables each centered around the currentparameter values The discrete proposals also require constraints to en-sure that the proposed values are valid For example a proposed valuefor N0 must be less than the current value of W0 We provide a detaileddescription of each update next

Population Model

The population model has parameters ωj λ Bj Sj and N0 andare updated using Gibbs or MetropolisndashHastings samplers The condi-tional posterior distributions for ωj and λN can be derived as

ωj |Bj Sjminus1Bjminus1

sim Beta(Bj + 0238Wjminus1 minus Sjminus1 minus Bjminus1 minus Bj + 1)

and

λN |N0 sim Gam(76 + N0101)

These are individually updated using the Gibbs sampler during eachiteration of the Markov chain

The parameters Bj Sj and N0 are updated by the MetropolisndashHastings algorithm using acceptance ratios calculated as described ear-lier We use uniform proposals centered on the current parameter valueand limited to the range of possible values

For the Sj rsquos we use three alternative proposals

Sprimej=1 sim Unif

(max

(0 S1 minus 5 S2 minus B1max(n1Ln1R) minus N1

2 minus B1)

min(S1 + 5N0W1 minus B1W1 minus B1 minus B2

W1 minus B1 minus 1))

Sprime2lejltJ sim Unif

(max

(0 Sj minus 5 Sj+1 minus Bj max(njLnjR) minus Bj

2 minus Bj

)

min(Sj + 5 Sjminus1 + Bjminus1Wj minus Bj Wj minus Bj minus Bj+1

Wj minus Bj minus 1))

and

Sprimej=J sim Unif

(max

(0 SJ minus 5max(nJLnJR) minus BJ 2 minus NJ

)

min(SJ minus 5 Sjminus1 + Bjminus1Wj minus Bj Wj minus Bj minus 1))

For Bj we use three alternative proposals

B prime1 sim Unif

(max

(0B1 minus 15 S2 minus S1max(n1Ln1R) minus S1

2 minus S1)

min(B1 + 15W0 minus N0W1 minus S1W1 minus S1 minus B2

W1 minus S1 minus 1))

B prime2lejltJ sim Unif

(max

(0Bj minus 15 Sj+1 minus Sj max(njLnjR) minus Sj

2 minus Sj

)

min(Bj + 15Wjminus1 minus Sjminus1 minus Bjminus1Wj minus Sj

Wj minus Sj minus Bj+1Wj minus Sj minus 1))

and

B primeJ sim Unif

(max

(0BJ minus 15max(nJLnJR) minus SJ 2 minus SJ

)

min(BJ + 15WJminus1 minus SJminus1 minus BJminus1

WJ minus SJ WJ minus SJ minus 1))

For N0 we use

N prime0 sim Unif

(max(5N0 minus 15 S1)min(N0 + 15W0 minus B1W0 minus 1)

)

Inflation Model

The inflation model parameters Wj and υj are updated usingMetropolisndashHastings and Gibbs updates The conditional posterior dis-tributions for υj can be derived as

υ0|N0W0 sim Beta(N0 + 1W0 minus N0 + 1)

and

υjgt0|Sj Bj Wj tnji twji sim Beta

(

Sj + Bj +T (j)sum

i=1

tnji + 1

Wj minus Sj minus Bj +T (j)sum

i=1

twji minus tnji + 1

)

These are updated separately using the Gibbs sampler on each itera-tion

The remaining parameter Wj has a nonstandard conditional pos-terior distribution We use larger updates than in the population modelbecause the initial total population is expected to be larger in size

W prime0 sim Unif

(max(0W0 minus 15N0 + 1N0 + B1)W0 + 15

)

W prime1lejltJ sim Unif

(max(0Wj minus 15 Sj + Bj + 1 Sj + Bj + Bj+1)

Wj + 15)

and

W primeJ sim Unif

(max(0WJ minus 15 SJ + BJ + 1)WJ + 15

)

Recapture Model

For the recapture model we have parameters α βj γd δL εjLζdL δR εjR and ζdR These all have nonstandard conditional pos-terior distributions and we use the MetropolisndashHastings algorithm toupdate the parameters on each iteration We use individual normal dis-tribution proposals centered on the current parameter values and withvariance equal to the prior variance for example

αprime sim N(α185)

[Received June 2005 Revised May 2007]

Corkrey et al Bayesian Recapture Population Model 959

REFERENCES

Brooks S P (1998) ldquoMarkov Chain Monte Carlo Method and Its ApplicationrdquoThe Statistician 47 69ndash100

Brooks S P and Roberts G O (1998) ldquoConvergence Assessment Techniquesfor Markov Chain Monte Carlordquo Statistics and Computing 8 319ndash335

Buckland S T (1982) ldquoA CapturendashRecapture Survival Analysisrdquo Journal ofAnimal Ecology 51 833ndash847

(1990) ldquoEstimation of Survival Rates From Sightings of IndividuallyIdentifiable Whalesrdquo in Individual Recognition of Cetaceans Use of Photo-Identification and Other Techniques to Estimate Population Parameters edsP S Hammond S A Mizroch and G P Donovan Cambridge UK Inter-national Whaling Commission pp 149ndash153

Burnham K and Overton W (1979) ldquoRobust Estimation of Population SizeWhen Capture Probabilities Vary Among Animalsrdquo Ecology 60 927ndash936

Carothers A (1979) ldquoQuantifying Unequal Catchability and Its Effect on Sur-vival Estimates in an Actual Populationrdquo Journal of Animal Ecology 48863ndash869

Chao A (1987) ldquoEstimating the Population Size for CapturendashRecapture DataWith Unequal Catchabilityrdquo Biometrics 43 783ndash791

Chao A Lee S-M and Jeng S-L (1992) ldquoEstimating Population Size forCapturendashRecapture Data When Capture Probabilities Vary by Time and Indi-vidual Animalrdquo Biometrics 48 201ndash216

Chib S and Greenberg E (1995) ldquoUnderstanding the MetropolisndashHastingsAlgorithmrdquo The American Statistician 49 327ndash335

Cormack R (1972) ldquoThe Logic of CapturendashRecapture Experimentsrdquo Biomet-rics 28 337ndash343

Coull B A and Agresti A (1999) ldquoThe Use of Mixed Logit Models to Re-flect Heterogeneity in CapturendashRecapture Studiesrdquo Biometrics 55 294ndash301

Da Silva C Q Rodrigues J Leite J G and Milan L A (2003) ldquoBayesianEstimation of the Size of a Closed Population Using Photo-ID With Part ofthe Population Uncatchablerdquo Communications in Statistics 32 677ndash696

Dorazio R M and Royle J A (2003) ldquoMixture Models for Estimating theSize of a Closed Population When Capture Rates Vary Among IndividualsrdquoBiometrics 59 351ndash364

Durban J Elston D Ellifrit D Dickson E Hammon P and Thompson P(2005) ldquoMultisite Mark-Recapture for Cetaceans Population Estimates WithBayesian Model Averagingrdquo Marine Mammal Science 21 80ndash92

Fienberg S E Johnson M S and Junker B W (1999) ldquoClassical Multi-level and Bayesian Approaches to Population Size Estimation Using MultipleListsrdquo Journal of the Royal Statistical Society Ser A 162 383ndash405

Forcada J and Aguilar A (2000) ldquoUse of Photographic Identification inCapturendashRecapture Studies of Mediterranean Monk Sealsrdquo Marine MammalScience 16 767ndash793

Franklin A B Anderson D R and Burnham K P (2002) ldquoEstimation ofLong-Term Trends and Variation in Avian Survival Probabilities Using Ran-dom Effects Modelsrdquo Journal of Applied Statistics 29 267ndash287

Friday N Smith T D Stevick P T and Allen J (2000) ldquoMeasure-ment of Photographic Quality and Individual Distinctiveness for the Photo-graphic Identification of Humpback Whales Megaptera novaeangliaerdquo Ma-rine Mammal Science 16 355ndash374

Gaspar R (2003) ldquoStatus of the Resident Bottlenose Dolphin Population inthe Sado Estuary Past Present and Futurerdquo unpublished doctoral dissertationUniversity of St Andrews School of Biology

Gelfand A E and Sahu S K (1999) ldquoIdentifiability Improper Priors andGibbs Sampling for Generalized Linear Modelsrdquo Journal of the AmericanStatistical Association 94 247ndash253

Gerrodette T (1987) ldquoA Power Analysis for Detecting Trendsrdquo Ecology 681364ndash1372

Goodman D (2004) ldquoMethods for Joint Inference From Multiple Data Sourcesfor Improved Estimates of Population Size and Survival Ratesrdquo Marine Mam-mal Science 20 401ndash423

Green P J (1995) ldquoReversible-Jump Markov Change Monte Carlo BayesianModel Determinationrdquo Biometrika 82 711ndash732

Grosbois V and Thompson P M (2005) ldquoNorth Atlantic Climate VariationInfluences Survival in Adult Fulmarsrdquo Oikos 109 273ndash290

Haase P (2000) ldquoSocial Organisation Behaviour and Population Parametersof Bottlenose Dolphins in Doubtful Sound Fiordlandrdquo unpublished mastersdissertation University of Otago Dept of Marine Science

Hammond P (1986) ldquoEstimating the Size of Naturally Marked Whale Pop-ulations Using CapturendashRecapture Techniquesrdquo Report of the InternationalWhaling Commission Special Issue 8 253ndash282

(1990) ldquoHeterogeneity in the Gulf of Maine Estimating HumpbackWhale Population Size When Capture Probabilities Are not Equalrdquo in In-dividual Recognition of Cetaceans Use of Photo-Identification and OtherTechniques to Estimate Population Parameters eds P S Hammond S A

Mizroch and G P Donovan Cambridge UK International Whaling Com-mission pp 135ndash153

Huggins R (1989) ldquoOn the Statistical Analysis of Capture Experimentsrdquo Bio-metrika 76 133ndash140

Joyce G G and Dorsey E M (1990) ldquoA Feasibility Study on the Useof Photo-Identification Techniques for Southern Hemisphere Minke WhaleStock Assessmentrdquo in Individual Recognition of Cetaceans Use of Photo-Identification and Other Techniques to Estimate Population Parameters edsP S Hammond S A Mizroch and G P Donovan Cambridge UK Inter-national Whaling Commission pp 419ndash423

King R and Brooks S (2001) ldquoOn the Bayesian Analysis of PopulationSizerdquo Biometrika 88 317ndash336

Lebreton J-D Burnham K P Clobert J and Anderson S R (1992)ldquoModeling Survival and Testing Biological Hypotheses Using Marked An-imals A Unified Approach With Case Studiesrdquo Ecological Monographs 6267ndash118

Lee S-M Huang L-H H and Ou S-T (2004) ldquoBand Recovery Model In-ference With Heterogeoneous Survival Ratesrdquo Statistica Sinica 14 513ndash531

Link W A and Barker R J (2005) ldquoModeling Association Among De-mographic Parameters in Analysis of Open Population CapturendashRecaptureDatardquo Biometrics 61 46ndash54

Mann J and Smuts B (1999) ldquoBehavioral Development in Wild BottlenoseDolphin Newborns (Tursiops sp)rdquo Behaviour 136 529ndash566

McCullagh P (1980) ldquoRegression Models for Ordinal Datardquo Journal of theRoyal Statistical Society Ser B 42 109ndash142

Otis D L Burnham K P White G C and Anderson D R (1978) ldquoStatis-tical Inference From Capture Data on Closed Animal Populationsrdquo WildlifeMonographs 62 135

Pledger S (2000) ldquoUnified Maximum Likelihood Estimates for ClosedCapturendashRecapture Models Using Mixturesrdquo Biometrics 56 434ndash442

Pledger S and Schwarz C J (2002) ldquoModelling Heterogeneity of Survivalin Band-Recovery Data Using Mixturesrdquo Journal of Applied Statistics 29315ndash327

Pledger S Pollock K H and Norris J L (2003) ldquoOpen CapturendashRecaptureModels With Heterogeneity I CormackndashJollyndashSeber Modelrdquo Ecology 59786ndash794

Pollock K H Nichols J D Brownie C and Hines J E (1990) ldquoStatisticalInference for CapturendashRecapture Samplingrdquo Wildlife Monographs 107 1ndash97

Read A Urian K W Wilson B and Waples D (2003) ldquoAbundance andStock Identity of Bottlenose Dolphins in the Bayes Sounds and Estuaries ofNorth Carolinardquo Marine Mammal Science 19 59ndash73

Saacutenchez-Mariacuten F J (2000) ldquoAutomatic Recognition of Biological ShapesWith and Without Representations of Shaperdquo Artificial Intelligence in Medi-cine 18 173ndash186

Saunders-Reed C A Hammond P S Grellier K and Thompson P M(1999) ldquoDevelopment of a Population Model for Bottlenose Dolphinsrdquo Re-search Survey and Monitoring Report 156 Scottish Natural Heritage

Scottish Natural Heritage (1995) Natura 2000 A Guide to the 1992 EC Habi-tats Directive in Scotlandrsquos Marine Environment Perth UK Scottish Nat-ural Heritage

Seber G (2002) The Estimation of Animal Abundance Caldwell NJ Black-burn Press

Stevick P T Palsboslashll P J Smith T D Bravington M V and Ham-mond P S (2001) ldquoErrors in Identification Using Natural Markings RatesSources and Effects on CapturendashRecapture Estimates of Abundancerdquo Cana-dian Journal of Fisheries and Aquatic Science 58 1861ndash1870

Taylor B L Martinez M Gerrodette T Barlow J and Hrovat Y N (2007)ldquoLessons From Monitoring Trends in Abundance of Marine Mammalsrdquo Ma-rine Mammal Science 23 157ndash175

Wells R S and Scott M D (1990) ldquoEstimating Bottlenose Dolphin Popu-lation Parameters From Individual Identification and CapturendashRelease Tech-niquesrdquo in Individual Recognition of Cetaceans Use of Photo-Identificationand Other Techniques to Estimate Population Parameters eds P S Ham-mond S A Mizroch and G P Donovan Cambridge UK InternationalWhaling Commission pp 407ndash415

Whitehead H (2001a) ldquoAnalysis of Animal Movement Using Opportunis-tic Individual Identifications Application to Sperm Whalesrdquo Ecology 821417ndash1432

(2001b) ldquoDirect Estimation of Within-Group Heterogeneity in Photo-Identification of Sperm Whalesrdquo Marine Mammal Science 17 718ndash728

Williams B K Nichols J D and Conroy M J (2001) Analysis and Man-agement of Animal Populations San Diego Academic Press

Wilson B Hammond P S and Thompson P M (1999) ldquoEstimating Size andAssessing Trends in a Coastal Bottlenose Dolphin Populationrdquo EcologicalApplications 9 288ndash300

Wilson B Reid R J Grellier K Thompson P M and Hammond P S(2004) ldquoConsidering the Temporal When Managing the Spatial A Popula-

956 Journal of the American Statistical Association September 2008

6 DISCUSSION

Changes in the size of wildlife populations are often assessedby fitting trend models to independent abundance estimatesbut the power of these techniques to detect trends often canbe low (Gerrodette 1987 Taylor Martinez Gerrodette Bar-low and Hrovat 2007) Here we use a capturendashrecapture modelthat incorporates an underlying population model and an ob-servation model We provide a probability that a population isin decline or increasing Using photoidentification data fromthis bottlenose dolphin population we generate the first em-pirical estimate of trends in abundance for any of the popula-tions of small cetaceans inhabiting European waters Becausethe methodological changes in sampling likely are absorbedby the time-varying capture coefficient the remaining variationdue to individual capture tendencies might reasonably be de-scribed by the individual capture coefficient This then allowsexploration of capture and survival relating to individuals ratherthan to populations Information on individual survival could beimportant for addressing some management issues for exam-ple individuals differing in their spatial distribution

By decomposing the capture probabilities into overall time-varying and heterogeneous terms we find that the time-varyingcontribution and heterogeneous variation are comparable inmagnitude and their variation is greater than that of the overallterm This information can be used in the design of monitoringprograms by suggesting how effort should be distributed Thedifference in overall left and right capture probabilities mightbe interpretable as an artifact of the sampling techniques or ofbehavioral effects However after allowing for this the prob-ability of observing one side also varied over time with someyears strongly favoring one side or the other This might be ex-plained by variation in protocol between years such as changesin personnel The relative sizes of these effects may be of usein evaluating consistency of protocol over time Finally afterallowing for the overall difference and time variation the het-erogeneous capture coefficients exhibit a bias to the one sidepossibly due to behavior or protocol These findings demon-strate that bilateral decomposition of capture probabilities canprovide useful information

We decompose the survival probability similarly and we findit to be unusually low in 1997 suggesting a possible multimodalheterogeneous survival probability that may be the result of mi-gration or survival variation This type of decomposition allowsexploration of such effects in more detail In particular the mul-timodality of heterogeneous survival suggests that the popula-tion comprises individuals with different strategies some vis-iting the core study area more or less often Ongoing work isattempting to quantify individual heterogeneity such as in ex-posure to sewage contaminated waters and boat-based tourismTherefore this technique could be used to determine whetherexposure to different anthropogenic stressors is related to sur-vival

Our model also makes use of priors such as those on ωj thatare not informed by the recapture data set but make use of in-formation from other sources An alternative approach mightbe to specify these as informative priors based on expert opin-ion or tuned to obtain satisfactory results Our application il-lustrates how disparate data sources may be combined into a

single analysis Although the data source for the prior recruit-ment probability is small and problematic it is the only oneavailable Even when some priors are defined in the simulationstudy to incorrect values the estimates of parameters of interestremain valid The posterior probability also adds to the state ofknowledge of this parameter

Previous use of capturendashrecapture models with cetacean pho-toidentification data highlights the potential violation of modelassumptions due to heterogeneity in capture andor survivalprobabilities This model explicitly accounts for heterogene-ity and also provides simultaneous estimates of individual sur-vival and capture probabilities This can be used both to ex-plore the biological basis of heterogeneity and to improve sam-pling programs The individual survival coefficients also can beused to explore aspects of biology where correlates are avail-able for example the relationship of survival to age sex orother variables may be of interest Abundance under the as-sumption of heterogeneous survival has been estimated by LeeHuang and Ou (2004) whereas individual and time-varyingsurvival estimates have been obtained using band-recoverymodels (Grosbois and Thompson 2005) Other studies have ex-amined survival by group for example Franklin Andersonand Burnham (2002) obtained separate estimates of group-levelsurvival for males and females using stratification in a bandingstudy Although we lack such variables as age and sex theo-retically it would be possible to examine individual survivalby sex either as a post hoc comparison or through their inte-gration into the model itself For example survival and cap-ture could be modeled as logit(φ) = α + βj + γd + τ andlogit(π) = δ + εj + ζd + χ where τ corresponds to stage-specific (eg young immature reproductive female senes-cent) survival and χ corresponds to capture probabilities forvarious categories (eg young reproductive adult)

Other cetacean studies have examined heterogeneity White-head (2001b) obtained an estimate of the coefficient of variation(CV) for identification from a randomization test of successiveidentifications in whale track data (The CV is also a measure ofcapture heterogeneity) They found that the CV was significantin one of the two years and that probably depended on the sizesof the animals being observed This is not comparable to thedolphin observations in which heterogeneity is more likely toresult from differences in animal behavior In the present studywe use a likelihood in which heterogeneity of survival and cap-ture has been incorporated by a logit function This approachwas described previously (Pledger Pollock and Norris 2003Pledger and Schwarz 2002) within a non-Bayesian contextPledger et al (2003) classified animals into an unknown set oflatent classes each of which could have a separate survival andcapture probability A simplification allows for time-varyingand animal-level heterogeneity This approach is similar to oursbut we also integrate a dynamic population model and an ob-servation model into the estimation process Goodman (2004)also implemented a population model in a Bayesian context thatcombines capturendashrecapture data with carcass-recovery databut assumes the absence of heterogeneity in capture probabil-ities Durban et al (2005) estimated abundance for the samepopulation as us but used a multisite Bayesian log-linear modeland model averaging (King and Brooks 2001) across the pos-sible models with varying combinations of interactions Our

Corkrey et al Bayesian Recapture Population Model 957

model could be extended to include multiple sites and yearsby using a capturendashrecapture history that includes site informa-tion and a yearndashsite term in the capture probability expressionBy doing this we suggest that geographic areas with more datacan be used to strengthen estimates from poorly sampled areasThis also may account for some individual variation currentlyidentified as individual capture heterogeneity because some in-dividuals may be more closely associated with some areas thanothers and may attenuate a possible confounding of migrationwith survival

We use a separate data set and a negative-binomial model toinflate the estimate of the proportion of nicked animals in thesamples Another study (Da Silva Rodrigues Leite and Mi-lan 2003) obtained an abundance estimate for bowhead whalesusing an inflation factor derived from the numbers of good andbad photographs of marked and unmarked whales Our analysisavoids the lack of independence between good photographs anddistinctively marked animals in the photographs that might in-validate the estimate of variance in the inflation factor Otherstudies avoid this issue by including only separately catego-rized good-quality photographs (eg Wilson et al 1999 ReadUrian Wilson and Waples 2003) However independent es-timates of inflation when available such as in our study arepreferred

We do not know of other cetacean studies that combinephoto-identification series of the left and right sides but ananalogous approach is that of log-linear capturendashrecapturemodeling (King and Brooks 2001) in which animals are identi-fied from separate lists Some photo-identification studies mightuse information from both sides in the process of identifyinganimals (Joyce and Dorsey 1990) A more robust approach isto perform the identification on each side separately We haveshown how these separate data can then be combined

A major advantage to our modeling approach is its extensibil-ity Within the Bayesian framework models can be comparedby such methods as reversible-jump Markov chain Monte Carlotechniques (Green 1995) Model extensions include extendingthe population model considering an individual-level approachThis would allow incorporation of the heterogeneity into thispart of the model and also modeling of the unobserved popu-lation More detailed modeling of heterogeneity could be ex-plored by including hyperpriors for the variances of the recap-ture model terms For convenience we analyze the data on anannual basis Additional information would be available if theanalyses were conducted on a finer time scale such as monthsor by individual trip This would allow the model to be extendedto include seasonal effects For example dolphins are observedmore frequently in summer months than in winter months andthis tendency could be included in an extended model by chang-ing the capture probability to be logit(π) = δt + εj + ζd whereδt t isin [1 12] is a monthly capture coefficient This maybe of use in modeling fine-scale movements The response vari-able itself also can be generalized Although we are careful toonly include data from high-quality photographs and nicked an-imals there is still the possibility of some identification errorsparticularly for more subtly marked individuals A possible ex-tension of the model could allow identification to be assigneddifferent levels of certainty for example by incorporating anintermediate state between nonrecapture and recapture that we

term ldquopossible recapturerdquo This could be done by replacing thelogit function logit(ν) = log(ν(1 minus ν)) where ν correspondsto the probability of recapture by a proportional odds function(McCullagh 1980) This technique has previously found appli-cation in a study of photoidentification used to model photo-graphic quality scores and individual distinctiveness of hump-back whale tail flukes (Friday Smith Stevick and Allen 2000)In this approach the probability of ldquopossible recapturerdquo is givenby ν1 = π1 the probability of certain recapture is given byν2 = π1 + π2 and includes the possible recapture category andthe probability of no capture is given by ν0 = 1 minus π1 + π2There are two proportional odds functions corresponding topossible recapture and certain recapture which are logit(ν1) =log(π1(π2 +π3)) and logit(ν2) = log((π1 +π2)π3) We thencould extend the recapture model using logit(ν1) = α+βj +γk1

and logit(ν2) = α + βj + γk2 This proportional odds approachmay be particularly useful in situations where automatic match-ing algorithms (Saacutenchez-Mariacuten 2000) provide an estimate ofthe likelihood of a match which could then be incorporatedinto abundance estimate models

APPENDIX DERIVATION OF POSTERIORS

Here we show how we derive the posterior conditional distributionsfor the model parameters and describe how each is updated within theMCMC algorithm The full joint distribution has the form shown in(A1) In (A1) the first two lines correspond to the population modelthe third line corresponds to the inflation model the fourth line cor-responds to the observation model the fifth line corresponds to therecapture model and the remaining lines correspond to priors

π = P(B1|ω1N0W0)P (S1|αN0)

(Sec 31)

timesSprod

j=2

P(Bj |Sjminus1Bjminus1Wjminus1ωj )

times P(Sj |Sjminus1Bjminus1 αβjminus1)

(Sec 31)

timesSprod

j=1

P(Wj |Sj Bj υj )

Tjprod

i=1

P(tnji |υj twji )

(Sec 32)

timesSprod

j=1

P(nLj|δL εjLSj Bj )P (nRj

|δR εjRSj Bj )

(Sec 33)

timesSprod

j=1

P(Hdj |αβj γd δL εjL ζdL δR εjR ζdR)

(Sec 34)

times P(N0|λN)P (W0|N0 υ0)P (α δL δRλN υ0)

(Sec 41)

timesSprod

j=1

P(βj εjL εjRωj υj )

nprod

d=1

P(γd)P (ζdL)P (ζdR)

(Sec 41) (A1)

958 Journal of the American Statistical Association September 2008

For each parameter we obtain the corresponding posterior condi-tional distribution by extracting all terms containing that parameterfrom the full joint distribution When the conditional distributions havea standard form we use the Gibbs sampler (Brooks 1998) to update theparameters otherwise we use MetropolisndashHastings updates (Chib andGreenberg 1995)

Within a MetropolisndashHastings update we propose a new valuefor a parameter x and denote the proposed value by xprime To decidewhether to accept the proposed value we calculate an acceptance ra-tio α = min(1A) where A is the acceptance ratio The acceptanceratio is given by A = π(xprime)q(x)π(x)q(xprime) where π(x) is the poste-rior probability of x and q(x) is the probability of the proposed valueWe accept the new value xprime with probability α otherwise we leavethe value of x unchanged (See Chib and Greenberg 1995 for an exam-ple)

Normal proposal distributions are used for continuous parameterscentered around the current parameter value Discrete uniform pro-posals are used for discrete variables each centered around the currentparameter values The discrete proposals also require constraints to en-sure that the proposed values are valid For example a proposed valuefor N0 must be less than the current value of W0 We provide a detaileddescription of each update next

Population Model

The population model has parameters ωj λ Bj Sj and N0 andare updated using Gibbs or MetropolisndashHastings samplers The condi-tional posterior distributions for ωj and λN can be derived as

ωj |Bj Sjminus1Bjminus1

sim Beta(Bj + 0238Wjminus1 minus Sjminus1 minus Bjminus1 minus Bj + 1)

and

λN |N0 sim Gam(76 + N0101)

These are individually updated using the Gibbs sampler during eachiteration of the Markov chain

The parameters Bj Sj and N0 are updated by the MetropolisndashHastings algorithm using acceptance ratios calculated as described ear-lier We use uniform proposals centered on the current parameter valueand limited to the range of possible values

For the Sj rsquos we use three alternative proposals

Sprimej=1 sim Unif

(max

(0 S1 minus 5 S2 minus B1max(n1Ln1R) minus N1

2 minus B1)

min(S1 + 5N0W1 minus B1W1 minus B1 minus B2

W1 minus B1 minus 1))

Sprime2lejltJ sim Unif

(max

(0 Sj minus 5 Sj+1 minus Bj max(njLnjR) minus Bj

2 minus Bj

)

min(Sj + 5 Sjminus1 + Bjminus1Wj minus Bj Wj minus Bj minus Bj+1

Wj minus Bj minus 1))

and

Sprimej=J sim Unif

(max

(0 SJ minus 5max(nJLnJR) minus BJ 2 minus NJ

)

min(SJ minus 5 Sjminus1 + Bjminus1Wj minus Bj Wj minus Bj minus 1))

For Bj we use three alternative proposals

B prime1 sim Unif

(max

(0B1 minus 15 S2 minus S1max(n1Ln1R) minus S1

2 minus S1)

min(B1 + 15W0 minus N0W1 minus S1W1 minus S1 minus B2

W1 minus S1 minus 1))

B prime2lejltJ sim Unif

(max

(0Bj minus 15 Sj+1 minus Sj max(njLnjR) minus Sj

2 minus Sj

)

min(Bj + 15Wjminus1 minus Sjminus1 minus Bjminus1Wj minus Sj

Wj minus Sj minus Bj+1Wj minus Sj minus 1))

and

B primeJ sim Unif

(max

(0BJ minus 15max(nJLnJR) minus SJ 2 minus SJ

)

min(BJ + 15WJminus1 minus SJminus1 minus BJminus1

WJ minus SJ WJ minus SJ minus 1))

For N0 we use

N prime0 sim Unif

(max(5N0 minus 15 S1)min(N0 + 15W0 minus B1W0 minus 1)

)

Inflation Model

The inflation model parameters Wj and υj are updated usingMetropolisndashHastings and Gibbs updates The conditional posterior dis-tributions for υj can be derived as

υ0|N0W0 sim Beta(N0 + 1W0 minus N0 + 1)

and

υjgt0|Sj Bj Wj tnji twji sim Beta

(

Sj + Bj +T (j)sum

i=1

tnji + 1

Wj minus Sj minus Bj +T (j)sum

i=1

twji minus tnji + 1

)

These are updated separately using the Gibbs sampler on each itera-tion

The remaining parameter Wj has a nonstandard conditional pos-terior distribution We use larger updates than in the population modelbecause the initial total population is expected to be larger in size

W prime0 sim Unif

(max(0W0 minus 15N0 + 1N0 + B1)W0 + 15

)

W prime1lejltJ sim Unif

(max(0Wj minus 15 Sj + Bj + 1 Sj + Bj + Bj+1)

Wj + 15)

and

W primeJ sim Unif

(max(0WJ minus 15 SJ + BJ + 1)WJ + 15

)

Recapture Model

For the recapture model we have parameters α βj γd δL εjLζdL δR εjR and ζdR These all have nonstandard conditional pos-terior distributions and we use the MetropolisndashHastings algorithm toupdate the parameters on each iteration We use individual normal dis-tribution proposals centered on the current parameter values and withvariance equal to the prior variance for example

αprime sim N(α185)

[Received June 2005 Revised May 2007]

Corkrey et al Bayesian Recapture Population Model 959

REFERENCES

Brooks S P (1998) ldquoMarkov Chain Monte Carlo Method and Its ApplicationrdquoThe Statistician 47 69ndash100

Brooks S P and Roberts G O (1998) ldquoConvergence Assessment Techniquesfor Markov Chain Monte Carlordquo Statistics and Computing 8 319ndash335

Buckland S T (1982) ldquoA CapturendashRecapture Survival Analysisrdquo Journal ofAnimal Ecology 51 833ndash847

(1990) ldquoEstimation of Survival Rates From Sightings of IndividuallyIdentifiable Whalesrdquo in Individual Recognition of Cetaceans Use of Photo-Identification and Other Techniques to Estimate Population Parameters edsP S Hammond S A Mizroch and G P Donovan Cambridge UK Inter-national Whaling Commission pp 149ndash153

Burnham K and Overton W (1979) ldquoRobust Estimation of Population SizeWhen Capture Probabilities Vary Among Animalsrdquo Ecology 60 927ndash936

Carothers A (1979) ldquoQuantifying Unequal Catchability and Its Effect on Sur-vival Estimates in an Actual Populationrdquo Journal of Animal Ecology 48863ndash869

Chao A (1987) ldquoEstimating the Population Size for CapturendashRecapture DataWith Unequal Catchabilityrdquo Biometrics 43 783ndash791

Chao A Lee S-M and Jeng S-L (1992) ldquoEstimating Population Size forCapturendashRecapture Data When Capture Probabilities Vary by Time and Indi-vidual Animalrdquo Biometrics 48 201ndash216

Chib S and Greenberg E (1995) ldquoUnderstanding the MetropolisndashHastingsAlgorithmrdquo The American Statistician 49 327ndash335

Cormack R (1972) ldquoThe Logic of CapturendashRecapture Experimentsrdquo Biomet-rics 28 337ndash343

Coull B A and Agresti A (1999) ldquoThe Use of Mixed Logit Models to Re-flect Heterogeneity in CapturendashRecapture Studiesrdquo Biometrics 55 294ndash301

Da Silva C Q Rodrigues J Leite J G and Milan L A (2003) ldquoBayesianEstimation of the Size of a Closed Population Using Photo-ID With Part ofthe Population Uncatchablerdquo Communications in Statistics 32 677ndash696

Dorazio R M and Royle J A (2003) ldquoMixture Models for Estimating theSize of a Closed Population When Capture Rates Vary Among IndividualsrdquoBiometrics 59 351ndash364

Durban J Elston D Ellifrit D Dickson E Hammon P and Thompson P(2005) ldquoMultisite Mark-Recapture for Cetaceans Population Estimates WithBayesian Model Averagingrdquo Marine Mammal Science 21 80ndash92

Fienberg S E Johnson M S and Junker B W (1999) ldquoClassical Multi-level and Bayesian Approaches to Population Size Estimation Using MultipleListsrdquo Journal of the Royal Statistical Society Ser A 162 383ndash405

Forcada J and Aguilar A (2000) ldquoUse of Photographic Identification inCapturendashRecapture Studies of Mediterranean Monk Sealsrdquo Marine MammalScience 16 767ndash793

Franklin A B Anderson D R and Burnham K P (2002) ldquoEstimation ofLong-Term Trends and Variation in Avian Survival Probabilities Using Ran-dom Effects Modelsrdquo Journal of Applied Statistics 29 267ndash287

Friday N Smith T D Stevick P T and Allen J (2000) ldquoMeasure-ment of Photographic Quality and Individual Distinctiveness for the Photo-graphic Identification of Humpback Whales Megaptera novaeangliaerdquo Ma-rine Mammal Science 16 355ndash374

Gaspar R (2003) ldquoStatus of the Resident Bottlenose Dolphin Population inthe Sado Estuary Past Present and Futurerdquo unpublished doctoral dissertationUniversity of St Andrews School of Biology

Gelfand A E and Sahu S K (1999) ldquoIdentifiability Improper Priors andGibbs Sampling for Generalized Linear Modelsrdquo Journal of the AmericanStatistical Association 94 247ndash253

Gerrodette T (1987) ldquoA Power Analysis for Detecting Trendsrdquo Ecology 681364ndash1372

Goodman D (2004) ldquoMethods for Joint Inference From Multiple Data Sourcesfor Improved Estimates of Population Size and Survival Ratesrdquo Marine Mam-mal Science 20 401ndash423

Green P J (1995) ldquoReversible-Jump Markov Change Monte Carlo BayesianModel Determinationrdquo Biometrika 82 711ndash732

Grosbois V and Thompson P M (2005) ldquoNorth Atlantic Climate VariationInfluences Survival in Adult Fulmarsrdquo Oikos 109 273ndash290

Haase P (2000) ldquoSocial Organisation Behaviour and Population Parametersof Bottlenose Dolphins in Doubtful Sound Fiordlandrdquo unpublished mastersdissertation University of Otago Dept of Marine Science

Hammond P (1986) ldquoEstimating the Size of Naturally Marked Whale Pop-ulations Using CapturendashRecapture Techniquesrdquo Report of the InternationalWhaling Commission Special Issue 8 253ndash282

(1990) ldquoHeterogeneity in the Gulf of Maine Estimating HumpbackWhale Population Size When Capture Probabilities Are not Equalrdquo in In-dividual Recognition of Cetaceans Use of Photo-Identification and OtherTechniques to Estimate Population Parameters eds P S Hammond S A

Mizroch and G P Donovan Cambridge UK International Whaling Com-mission pp 135ndash153

Huggins R (1989) ldquoOn the Statistical Analysis of Capture Experimentsrdquo Bio-metrika 76 133ndash140

Joyce G G and Dorsey E M (1990) ldquoA Feasibility Study on the Useof Photo-Identification Techniques for Southern Hemisphere Minke WhaleStock Assessmentrdquo in Individual Recognition of Cetaceans Use of Photo-Identification and Other Techniques to Estimate Population Parameters edsP S Hammond S A Mizroch and G P Donovan Cambridge UK Inter-national Whaling Commission pp 419ndash423

King R and Brooks S (2001) ldquoOn the Bayesian Analysis of PopulationSizerdquo Biometrika 88 317ndash336

Lebreton J-D Burnham K P Clobert J and Anderson S R (1992)ldquoModeling Survival and Testing Biological Hypotheses Using Marked An-imals A Unified Approach With Case Studiesrdquo Ecological Monographs 6267ndash118

Lee S-M Huang L-H H and Ou S-T (2004) ldquoBand Recovery Model In-ference With Heterogeoneous Survival Ratesrdquo Statistica Sinica 14 513ndash531

Link W A and Barker R J (2005) ldquoModeling Association Among De-mographic Parameters in Analysis of Open Population CapturendashRecaptureDatardquo Biometrics 61 46ndash54

Mann J and Smuts B (1999) ldquoBehavioral Development in Wild BottlenoseDolphin Newborns (Tursiops sp)rdquo Behaviour 136 529ndash566

McCullagh P (1980) ldquoRegression Models for Ordinal Datardquo Journal of theRoyal Statistical Society Ser B 42 109ndash142

Otis D L Burnham K P White G C and Anderson D R (1978) ldquoStatis-tical Inference From Capture Data on Closed Animal Populationsrdquo WildlifeMonographs 62 135

Pledger S (2000) ldquoUnified Maximum Likelihood Estimates for ClosedCapturendashRecapture Models Using Mixturesrdquo Biometrics 56 434ndash442

Pledger S and Schwarz C J (2002) ldquoModelling Heterogeneity of Survivalin Band-Recovery Data Using Mixturesrdquo Journal of Applied Statistics 29315ndash327

Pledger S Pollock K H and Norris J L (2003) ldquoOpen CapturendashRecaptureModels With Heterogeneity I CormackndashJollyndashSeber Modelrdquo Ecology 59786ndash794

Pollock K H Nichols J D Brownie C and Hines J E (1990) ldquoStatisticalInference for CapturendashRecapture Samplingrdquo Wildlife Monographs 107 1ndash97

Read A Urian K W Wilson B and Waples D (2003) ldquoAbundance andStock Identity of Bottlenose Dolphins in the Bayes Sounds and Estuaries ofNorth Carolinardquo Marine Mammal Science 19 59ndash73

Saacutenchez-Mariacuten F J (2000) ldquoAutomatic Recognition of Biological ShapesWith and Without Representations of Shaperdquo Artificial Intelligence in Medi-cine 18 173ndash186

Saunders-Reed C A Hammond P S Grellier K and Thompson P M(1999) ldquoDevelopment of a Population Model for Bottlenose Dolphinsrdquo Re-search Survey and Monitoring Report 156 Scottish Natural Heritage

Scottish Natural Heritage (1995) Natura 2000 A Guide to the 1992 EC Habi-tats Directive in Scotlandrsquos Marine Environment Perth UK Scottish Nat-ural Heritage

Seber G (2002) The Estimation of Animal Abundance Caldwell NJ Black-burn Press

Stevick P T Palsboslashll P J Smith T D Bravington M V and Ham-mond P S (2001) ldquoErrors in Identification Using Natural Markings RatesSources and Effects on CapturendashRecapture Estimates of Abundancerdquo Cana-dian Journal of Fisheries and Aquatic Science 58 1861ndash1870

Taylor B L Martinez M Gerrodette T Barlow J and Hrovat Y N (2007)ldquoLessons From Monitoring Trends in Abundance of Marine Mammalsrdquo Ma-rine Mammal Science 23 157ndash175

Wells R S and Scott M D (1990) ldquoEstimating Bottlenose Dolphin Popu-lation Parameters From Individual Identification and CapturendashRelease Tech-niquesrdquo in Individual Recognition of Cetaceans Use of Photo-Identificationand Other Techniques to Estimate Population Parameters eds P S Ham-mond S A Mizroch and G P Donovan Cambridge UK InternationalWhaling Commission pp 407ndash415

Whitehead H (2001a) ldquoAnalysis of Animal Movement Using Opportunis-tic Individual Identifications Application to Sperm Whalesrdquo Ecology 821417ndash1432

(2001b) ldquoDirect Estimation of Within-Group Heterogeneity in Photo-Identification of Sperm Whalesrdquo Marine Mammal Science 17 718ndash728

Williams B K Nichols J D and Conroy M J (2001) Analysis and Man-agement of Animal Populations San Diego Academic Press

Wilson B Hammond P S and Thompson P M (1999) ldquoEstimating Size andAssessing Trends in a Coastal Bottlenose Dolphin Populationrdquo EcologicalApplications 9 288ndash300

Wilson B Reid R J Grellier K Thompson P M and Hammond P S(2004) ldquoConsidering the Temporal When Managing the Spatial A Popula-

Corkrey et al Bayesian Recapture Population Model 957

model could be extended to include multiple sites and yearsby using a capturendashrecapture history that includes site informa-tion and a yearndashsite term in the capture probability expressionBy doing this we suggest that geographic areas with more datacan be used to strengthen estimates from poorly sampled areasThis also may account for some individual variation currentlyidentified as individual capture heterogeneity because some in-dividuals may be more closely associated with some areas thanothers and may attenuate a possible confounding of migrationwith survival

We use a separate data set and a negative-binomial model toinflate the estimate of the proportion of nicked animals in thesamples Another study (Da Silva Rodrigues Leite and Mi-lan 2003) obtained an abundance estimate for bowhead whalesusing an inflation factor derived from the numbers of good andbad photographs of marked and unmarked whales Our analysisavoids the lack of independence between good photographs anddistinctively marked animals in the photographs that might in-validate the estimate of variance in the inflation factor Otherstudies avoid this issue by including only separately catego-rized good-quality photographs (eg Wilson et al 1999 ReadUrian Wilson and Waples 2003) However independent es-timates of inflation when available such as in our study arepreferred

We do not know of other cetacean studies that combinephoto-identification series of the left and right sides but ananalogous approach is that of log-linear capturendashrecapturemodeling (King and Brooks 2001) in which animals are identi-fied from separate lists Some photo-identification studies mightuse information from both sides in the process of identifyinganimals (Joyce and Dorsey 1990) A more robust approach isto perform the identification on each side separately We haveshown how these separate data can then be combined

A major advantage to our modeling approach is its extensibil-ity Within the Bayesian framework models can be comparedby such methods as reversible-jump Markov chain Monte Carlotechniques (Green 1995) Model extensions include extendingthe population model considering an individual-level approachThis would allow incorporation of the heterogeneity into thispart of the model and also modeling of the unobserved popu-lation More detailed modeling of heterogeneity could be ex-plored by including hyperpriors for the variances of the recap-ture model terms For convenience we analyze the data on anannual basis Additional information would be available if theanalyses were conducted on a finer time scale such as monthsor by individual trip This would allow the model to be extendedto include seasonal effects For example dolphins are observedmore frequently in summer months than in winter months andthis tendency could be included in an extended model by chang-ing the capture probability to be logit(π) = δt + εj + ζd whereδt t isin [1 12] is a monthly capture coefficient This maybe of use in modeling fine-scale movements The response vari-able itself also can be generalized Although we are careful toonly include data from high-quality photographs and nicked an-imals there is still the possibility of some identification errorsparticularly for more subtly marked individuals A possible ex-tension of the model could allow identification to be assigneddifferent levels of certainty for example by incorporating anintermediate state between nonrecapture and recapture that we

term ldquopossible recapturerdquo This could be done by replacing thelogit function logit(ν) = log(ν(1 minus ν)) where ν correspondsto the probability of recapture by a proportional odds function(McCullagh 1980) This technique has previously found appli-cation in a study of photoidentification used to model photo-graphic quality scores and individual distinctiveness of hump-back whale tail flukes (Friday Smith Stevick and Allen 2000)In this approach the probability of ldquopossible recapturerdquo is givenby ν1 = π1 the probability of certain recapture is given byν2 = π1 + π2 and includes the possible recapture category andthe probability of no capture is given by ν0 = 1 minus π1 + π2There are two proportional odds functions corresponding topossible recapture and certain recapture which are logit(ν1) =log(π1(π2 +π3)) and logit(ν2) = log((π1 +π2)π3) We thencould extend the recapture model using logit(ν1) = α+βj +γk1

and logit(ν2) = α + βj + γk2 This proportional odds approachmay be particularly useful in situations where automatic match-ing algorithms (Saacutenchez-Mariacuten 2000) provide an estimate ofthe likelihood of a match which could then be incorporatedinto abundance estimate models

APPENDIX DERIVATION OF POSTERIORS

Here we show how we derive the posterior conditional distributionsfor the model parameters and describe how each is updated within theMCMC algorithm The full joint distribution has the form shown in(A1) In (A1) the first two lines correspond to the population modelthe third line corresponds to the inflation model the fourth line cor-responds to the observation model the fifth line corresponds to therecapture model and the remaining lines correspond to priors

π = P(B1|ω1N0W0)P (S1|αN0)

(Sec 31)

timesSprod

j=2

P(Bj |Sjminus1Bjminus1Wjminus1ωj )

times P(Sj |Sjminus1Bjminus1 αβjminus1)

(Sec 31)

timesSprod

j=1

P(Wj |Sj Bj υj )

Tjprod

i=1

P(tnji |υj twji )

(Sec 32)

timesSprod

j=1

P(nLj|δL εjLSj Bj )P (nRj

|δR εjRSj Bj )

(Sec 33)

timesSprod

j=1

P(Hdj |αβj γd δL εjL ζdL δR εjR ζdR)

(Sec 34)

times P(N0|λN)P (W0|N0 υ0)P (α δL δRλN υ0)

(Sec 41)

timesSprod

j=1

P(βj εjL εjRωj υj )

nprod

d=1

P(γd)P (ζdL)P (ζdR)

(Sec 41) (A1)

958 Journal of the American Statistical Association September 2008

For each parameter we obtain the corresponding posterior condi-tional distribution by extracting all terms containing that parameterfrom the full joint distribution When the conditional distributions havea standard form we use the Gibbs sampler (Brooks 1998) to update theparameters otherwise we use MetropolisndashHastings updates (Chib andGreenberg 1995)

Within a MetropolisndashHastings update we propose a new valuefor a parameter x and denote the proposed value by xprime To decidewhether to accept the proposed value we calculate an acceptance ra-tio α = min(1A) where A is the acceptance ratio The acceptanceratio is given by A = π(xprime)q(x)π(x)q(xprime) where π(x) is the poste-rior probability of x and q(x) is the probability of the proposed valueWe accept the new value xprime with probability α otherwise we leavethe value of x unchanged (See Chib and Greenberg 1995 for an exam-ple)

Normal proposal distributions are used for continuous parameterscentered around the current parameter value Discrete uniform pro-posals are used for discrete variables each centered around the currentparameter values The discrete proposals also require constraints to en-sure that the proposed values are valid For example a proposed valuefor N0 must be less than the current value of W0 We provide a detaileddescription of each update next

Population Model

The population model has parameters ωj λ Bj Sj and N0 andare updated using Gibbs or MetropolisndashHastings samplers The condi-tional posterior distributions for ωj and λN can be derived as

ωj |Bj Sjminus1Bjminus1

sim Beta(Bj + 0238Wjminus1 minus Sjminus1 minus Bjminus1 minus Bj + 1)

and

λN |N0 sim Gam(76 + N0101)

These are individually updated using the Gibbs sampler during eachiteration of the Markov chain

The parameters Bj Sj and N0 are updated by the MetropolisndashHastings algorithm using acceptance ratios calculated as described ear-lier We use uniform proposals centered on the current parameter valueand limited to the range of possible values

For the Sj rsquos we use three alternative proposals

Sprimej=1 sim Unif

(max

(0 S1 minus 5 S2 minus B1max(n1Ln1R) minus N1

2 minus B1)

min(S1 + 5N0W1 minus B1W1 minus B1 minus B2

W1 minus B1 minus 1))

Sprime2lejltJ sim Unif

(max

(0 Sj minus 5 Sj+1 minus Bj max(njLnjR) minus Bj

2 minus Bj

)

min(Sj + 5 Sjminus1 + Bjminus1Wj minus Bj Wj minus Bj minus Bj+1

Wj minus Bj minus 1))

and

Sprimej=J sim Unif

(max

(0 SJ minus 5max(nJLnJR) minus BJ 2 minus NJ

)

min(SJ minus 5 Sjminus1 + Bjminus1Wj minus Bj Wj minus Bj minus 1))

For Bj we use three alternative proposals

B prime1 sim Unif

(max

(0B1 minus 15 S2 minus S1max(n1Ln1R) minus S1

2 minus S1)

min(B1 + 15W0 minus N0W1 minus S1W1 minus S1 minus B2

W1 minus S1 minus 1))

B prime2lejltJ sim Unif

(max

(0Bj minus 15 Sj+1 minus Sj max(njLnjR) minus Sj

2 minus Sj

)

min(Bj + 15Wjminus1 minus Sjminus1 minus Bjminus1Wj minus Sj

Wj minus Sj minus Bj+1Wj minus Sj minus 1))

and

B primeJ sim Unif

(max

(0BJ minus 15max(nJLnJR) minus SJ 2 minus SJ

)

min(BJ + 15WJminus1 minus SJminus1 minus BJminus1

WJ minus SJ WJ minus SJ minus 1))

For N0 we use

N prime0 sim Unif

(max(5N0 minus 15 S1)min(N0 + 15W0 minus B1W0 minus 1)

)

Inflation Model

The inflation model parameters Wj and υj are updated usingMetropolisndashHastings and Gibbs updates The conditional posterior dis-tributions for υj can be derived as

υ0|N0W0 sim Beta(N0 + 1W0 minus N0 + 1)

and

υjgt0|Sj Bj Wj tnji twji sim Beta

(

Sj + Bj +T (j)sum

i=1

tnji + 1

Wj minus Sj minus Bj +T (j)sum

i=1

twji minus tnji + 1

)

These are updated separately using the Gibbs sampler on each itera-tion

The remaining parameter Wj has a nonstandard conditional pos-terior distribution We use larger updates than in the population modelbecause the initial total population is expected to be larger in size

W prime0 sim Unif

(max(0W0 minus 15N0 + 1N0 + B1)W0 + 15

)

W prime1lejltJ sim Unif

(max(0Wj minus 15 Sj + Bj + 1 Sj + Bj + Bj+1)

Wj + 15)

and

W primeJ sim Unif

(max(0WJ minus 15 SJ + BJ + 1)WJ + 15

)

Recapture Model

For the recapture model we have parameters α βj γd δL εjLζdL δR εjR and ζdR These all have nonstandard conditional pos-terior distributions and we use the MetropolisndashHastings algorithm toupdate the parameters on each iteration We use individual normal dis-tribution proposals centered on the current parameter values and withvariance equal to the prior variance for example

αprime sim N(α185)

[Received June 2005 Revised May 2007]

Corkrey et al Bayesian Recapture Population Model 959

REFERENCES

Brooks S P (1998) ldquoMarkov Chain Monte Carlo Method and Its ApplicationrdquoThe Statistician 47 69ndash100

Brooks S P and Roberts G O (1998) ldquoConvergence Assessment Techniquesfor Markov Chain Monte Carlordquo Statistics and Computing 8 319ndash335

Buckland S T (1982) ldquoA CapturendashRecapture Survival Analysisrdquo Journal ofAnimal Ecology 51 833ndash847

(1990) ldquoEstimation of Survival Rates From Sightings of IndividuallyIdentifiable Whalesrdquo in Individual Recognition of Cetaceans Use of Photo-Identification and Other Techniques to Estimate Population Parameters edsP S Hammond S A Mizroch and G P Donovan Cambridge UK Inter-national Whaling Commission pp 149ndash153

Burnham K and Overton W (1979) ldquoRobust Estimation of Population SizeWhen Capture Probabilities Vary Among Animalsrdquo Ecology 60 927ndash936

Carothers A (1979) ldquoQuantifying Unequal Catchability and Its Effect on Sur-vival Estimates in an Actual Populationrdquo Journal of Animal Ecology 48863ndash869

Chao A (1987) ldquoEstimating the Population Size for CapturendashRecapture DataWith Unequal Catchabilityrdquo Biometrics 43 783ndash791

Chao A Lee S-M and Jeng S-L (1992) ldquoEstimating Population Size forCapturendashRecapture Data When Capture Probabilities Vary by Time and Indi-vidual Animalrdquo Biometrics 48 201ndash216

Chib S and Greenberg E (1995) ldquoUnderstanding the MetropolisndashHastingsAlgorithmrdquo The American Statistician 49 327ndash335

Cormack R (1972) ldquoThe Logic of CapturendashRecapture Experimentsrdquo Biomet-rics 28 337ndash343

Coull B A and Agresti A (1999) ldquoThe Use of Mixed Logit Models to Re-flect Heterogeneity in CapturendashRecapture Studiesrdquo Biometrics 55 294ndash301

Da Silva C Q Rodrigues J Leite J G and Milan L A (2003) ldquoBayesianEstimation of the Size of a Closed Population Using Photo-ID With Part ofthe Population Uncatchablerdquo Communications in Statistics 32 677ndash696

Dorazio R M and Royle J A (2003) ldquoMixture Models for Estimating theSize of a Closed Population When Capture Rates Vary Among IndividualsrdquoBiometrics 59 351ndash364

Durban J Elston D Ellifrit D Dickson E Hammon P and Thompson P(2005) ldquoMultisite Mark-Recapture for Cetaceans Population Estimates WithBayesian Model Averagingrdquo Marine Mammal Science 21 80ndash92

Fienberg S E Johnson M S and Junker B W (1999) ldquoClassical Multi-level and Bayesian Approaches to Population Size Estimation Using MultipleListsrdquo Journal of the Royal Statistical Society Ser A 162 383ndash405

Forcada J and Aguilar A (2000) ldquoUse of Photographic Identification inCapturendashRecapture Studies of Mediterranean Monk Sealsrdquo Marine MammalScience 16 767ndash793

Franklin A B Anderson D R and Burnham K P (2002) ldquoEstimation ofLong-Term Trends and Variation in Avian Survival Probabilities Using Ran-dom Effects Modelsrdquo Journal of Applied Statistics 29 267ndash287

Friday N Smith T D Stevick P T and Allen J (2000) ldquoMeasure-ment of Photographic Quality and Individual Distinctiveness for the Photo-graphic Identification of Humpback Whales Megaptera novaeangliaerdquo Ma-rine Mammal Science 16 355ndash374

Gaspar R (2003) ldquoStatus of the Resident Bottlenose Dolphin Population inthe Sado Estuary Past Present and Futurerdquo unpublished doctoral dissertationUniversity of St Andrews School of Biology

Gelfand A E and Sahu S K (1999) ldquoIdentifiability Improper Priors andGibbs Sampling for Generalized Linear Modelsrdquo Journal of the AmericanStatistical Association 94 247ndash253

Gerrodette T (1987) ldquoA Power Analysis for Detecting Trendsrdquo Ecology 681364ndash1372

Goodman D (2004) ldquoMethods for Joint Inference From Multiple Data Sourcesfor Improved Estimates of Population Size and Survival Ratesrdquo Marine Mam-mal Science 20 401ndash423

Green P J (1995) ldquoReversible-Jump Markov Change Monte Carlo BayesianModel Determinationrdquo Biometrika 82 711ndash732

Grosbois V and Thompson P M (2005) ldquoNorth Atlantic Climate VariationInfluences Survival in Adult Fulmarsrdquo Oikos 109 273ndash290

Haase P (2000) ldquoSocial Organisation Behaviour and Population Parametersof Bottlenose Dolphins in Doubtful Sound Fiordlandrdquo unpublished mastersdissertation University of Otago Dept of Marine Science

Hammond P (1986) ldquoEstimating the Size of Naturally Marked Whale Pop-ulations Using CapturendashRecapture Techniquesrdquo Report of the InternationalWhaling Commission Special Issue 8 253ndash282

(1990) ldquoHeterogeneity in the Gulf of Maine Estimating HumpbackWhale Population Size When Capture Probabilities Are not Equalrdquo in In-dividual Recognition of Cetaceans Use of Photo-Identification and OtherTechniques to Estimate Population Parameters eds P S Hammond S A

Mizroch and G P Donovan Cambridge UK International Whaling Com-mission pp 135ndash153

Huggins R (1989) ldquoOn the Statistical Analysis of Capture Experimentsrdquo Bio-metrika 76 133ndash140

Joyce G G and Dorsey E M (1990) ldquoA Feasibility Study on the Useof Photo-Identification Techniques for Southern Hemisphere Minke WhaleStock Assessmentrdquo in Individual Recognition of Cetaceans Use of Photo-Identification and Other Techniques to Estimate Population Parameters edsP S Hammond S A Mizroch and G P Donovan Cambridge UK Inter-national Whaling Commission pp 419ndash423

King R and Brooks S (2001) ldquoOn the Bayesian Analysis of PopulationSizerdquo Biometrika 88 317ndash336

Lebreton J-D Burnham K P Clobert J and Anderson S R (1992)ldquoModeling Survival and Testing Biological Hypotheses Using Marked An-imals A Unified Approach With Case Studiesrdquo Ecological Monographs 6267ndash118

Lee S-M Huang L-H H and Ou S-T (2004) ldquoBand Recovery Model In-ference With Heterogeoneous Survival Ratesrdquo Statistica Sinica 14 513ndash531

Link W A and Barker R J (2005) ldquoModeling Association Among De-mographic Parameters in Analysis of Open Population CapturendashRecaptureDatardquo Biometrics 61 46ndash54

Mann J and Smuts B (1999) ldquoBehavioral Development in Wild BottlenoseDolphin Newborns (Tursiops sp)rdquo Behaviour 136 529ndash566

McCullagh P (1980) ldquoRegression Models for Ordinal Datardquo Journal of theRoyal Statistical Society Ser B 42 109ndash142

Otis D L Burnham K P White G C and Anderson D R (1978) ldquoStatis-tical Inference From Capture Data on Closed Animal Populationsrdquo WildlifeMonographs 62 135

Pledger S (2000) ldquoUnified Maximum Likelihood Estimates for ClosedCapturendashRecapture Models Using Mixturesrdquo Biometrics 56 434ndash442

Pledger S and Schwarz C J (2002) ldquoModelling Heterogeneity of Survivalin Band-Recovery Data Using Mixturesrdquo Journal of Applied Statistics 29315ndash327

Pledger S Pollock K H and Norris J L (2003) ldquoOpen CapturendashRecaptureModels With Heterogeneity I CormackndashJollyndashSeber Modelrdquo Ecology 59786ndash794

Pollock K H Nichols J D Brownie C and Hines J E (1990) ldquoStatisticalInference for CapturendashRecapture Samplingrdquo Wildlife Monographs 107 1ndash97

Read A Urian K W Wilson B and Waples D (2003) ldquoAbundance andStock Identity of Bottlenose Dolphins in the Bayes Sounds and Estuaries ofNorth Carolinardquo Marine Mammal Science 19 59ndash73

Saacutenchez-Mariacuten F J (2000) ldquoAutomatic Recognition of Biological ShapesWith and Without Representations of Shaperdquo Artificial Intelligence in Medi-cine 18 173ndash186

Saunders-Reed C A Hammond P S Grellier K and Thompson P M(1999) ldquoDevelopment of a Population Model for Bottlenose Dolphinsrdquo Re-search Survey and Monitoring Report 156 Scottish Natural Heritage

Scottish Natural Heritage (1995) Natura 2000 A Guide to the 1992 EC Habi-tats Directive in Scotlandrsquos Marine Environment Perth UK Scottish Nat-ural Heritage

Seber G (2002) The Estimation of Animal Abundance Caldwell NJ Black-burn Press

Stevick P T Palsboslashll P J Smith T D Bravington M V and Ham-mond P S (2001) ldquoErrors in Identification Using Natural Markings RatesSources and Effects on CapturendashRecapture Estimates of Abundancerdquo Cana-dian Journal of Fisheries and Aquatic Science 58 1861ndash1870

Taylor B L Martinez M Gerrodette T Barlow J and Hrovat Y N (2007)ldquoLessons From Monitoring Trends in Abundance of Marine Mammalsrdquo Ma-rine Mammal Science 23 157ndash175

Wells R S and Scott M D (1990) ldquoEstimating Bottlenose Dolphin Popu-lation Parameters From Individual Identification and CapturendashRelease Tech-niquesrdquo in Individual Recognition of Cetaceans Use of Photo-Identificationand Other Techniques to Estimate Population Parameters eds P S Ham-mond S A Mizroch and G P Donovan Cambridge UK InternationalWhaling Commission pp 407ndash415

Whitehead H (2001a) ldquoAnalysis of Animal Movement Using Opportunis-tic Individual Identifications Application to Sperm Whalesrdquo Ecology 821417ndash1432

(2001b) ldquoDirect Estimation of Within-Group Heterogeneity in Photo-Identification of Sperm Whalesrdquo Marine Mammal Science 17 718ndash728

Williams B K Nichols J D and Conroy M J (2001) Analysis and Man-agement of Animal Populations San Diego Academic Press

Wilson B Hammond P S and Thompson P M (1999) ldquoEstimating Size andAssessing Trends in a Coastal Bottlenose Dolphin Populationrdquo EcologicalApplications 9 288ndash300

Wilson B Reid R J Grellier K Thompson P M and Hammond P S(2004) ldquoConsidering the Temporal When Managing the Spatial A Popula-

958 Journal of the American Statistical Association September 2008

For each parameter we obtain the corresponding posterior condi-tional distribution by extracting all terms containing that parameterfrom the full joint distribution When the conditional distributions havea standard form we use the Gibbs sampler (Brooks 1998) to update theparameters otherwise we use MetropolisndashHastings updates (Chib andGreenberg 1995)

Within a MetropolisndashHastings update we propose a new valuefor a parameter x and denote the proposed value by xprime To decidewhether to accept the proposed value we calculate an acceptance ra-tio α = min(1A) where A is the acceptance ratio The acceptanceratio is given by A = π(xprime)q(x)π(x)q(xprime) where π(x) is the poste-rior probability of x and q(x) is the probability of the proposed valueWe accept the new value xprime with probability α otherwise we leavethe value of x unchanged (See Chib and Greenberg 1995 for an exam-ple)

Normal proposal distributions are used for continuous parameterscentered around the current parameter value Discrete uniform pro-posals are used for discrete variables each centered around the currentparameter values The discrete proposals also require constraints to en-sure that the proposed values are valid For example a proposed valuefor N0 must be less than the current value of W0 We provide a detaileddescription of each update next

Population Model

The population model has parameters ωj λ Bj Sj and N0 andare updated using Gibbs or MetropolisndashHastings samplers The condi-tional posterior distributions for ωj and λN can be derived as

ωj |Bj Sjminus1Bjminus1

sim Beta(Bj + 0238Wjminus1 minus Sjminus1 minus Bjminus1 minus Bj + 1)

and

λN |N0 sim Gam(76 + N0101)

These are individually updated using the Gibbs sampler during eachiteration of the Markov chain

The parameters Bj Sj and N0 are updated by the MetropolisndashHastings algorithm using acceptance ratios calculated as described ear-lier We use uniform proposals centered on the current parameter valueand limited to the range of possible values

For the Sj rsquos we use three alternative proposals

Sprimej=1 sim Unif

(max

(0 S1 minus 5 S2 minus B1max(n1Ln1R) minus N1

2 minus B1)

min(S1 + 5N0W1 minus B1W1 minus B1 minus B2

W1 minus B1 minus 1))

Sprime2lejltJ sim Unif

(max

(0 Sj minus 5 Sj+1 minus Bj max(njLnjR) minus Bj

2 minus Bj

)

min(Sj + 5 Sjminus1 + Bjminus1Wj minus Bj Wj minus Bj minus Bj+1

Wj minus Bj minus 1))

and

Sprimej=J sim Unif

(max

(0 SJ minus 5max(nJLnJR) minus BJ 2 minus NJ

)

min(SJ minus 5 Sjminus1 + Bjminus1Wj minus Bj Wj minus Bj minus 1))

For Bj we use three alternative proposals

B prime1 sim Unif

(max

(0B1 minus 15 S2 minus S1max(n1Ln1R) minus S1

2 minus S1)

min(B1 + 15W0 minus N0W1 minus S1W1 minus S1 minus B2

W1 minus S1 minus 1))

B prime2lejltJ sim Unif

(max

(0Bj minus 15 Sj+1 minus Sj max(njLnjR) minus Sj

2 minus Sj

)

min(Bj + 15Wjminus1 minus Sjminus1 minus Bjminus1Wj minus Sj

Wj minus Sj minus Bj+1Wj minus Sj minus 1))

and

B primeJ sim Unif

(max

(0BJ minus 15max(nJLnJR) minus SJ 2 minus SJ

)

min(BJ + 15WJminus1 minus SJminus1 minus BJminus1

WJ minus SJ WJ minus SJ minus 1))

For N0 we use

N prime0 sim Unif

(max(5N0 minus 15 S1)min(N0 + 15W0 minus B1W0 minus 1)

)

Inflation Model

The inflation model parameters Wj and υj are updated usingMetropolisndashHastings and Gibbs updates The conditional posterior dis-tributions for υj can be derived as

υ0|N0W0 sim Beta(N0 + 1W0 minus N0 + 1)

and

υjgt0|Sj Bj Wj tnji twji sim Beta

(

Sj + Bj +T (j)sum

i=1

tnji + 1

Wj minus Sj minus Bj +T (j)sum

i=1

twji minus tnji + 1

)

These are updated separately using the Gibbs sampler on each itera-tion

The remaining parameter Wj has a nonstandard conditional pos-terior distribution We use larger updates than in the population modelbecause the initial total population is expected to be larger in size

W prime0 sim Unif

(max(0W0 minus 15N0 + 1N0 + B1)W0 + 15

)

W prime1lejltJ sim Unif

(max(0Wj minus 15 Sj + Bj + 1 Sj + Bj + Bj+1)

Wj + 15)

and

W primeJ sim Unif

(max(0WJ minus 15 SJ + BJ + 1)WJ + 15

)

Recapture Model

For the recapture model we have parameters α βj γd δL εjLζdL δR εjR and ζdR These all have nonstandard conditional pos-terior distributions and we use the MetropolisndashHastings algorithm toupdate the parameters on each iteration We use individual normal dis-tribution proposals centered on the current parameter values and withvariance equal to the prior variance for example

αprime sim N(α185)

[Received June 2005 Revised May 2007]

Corkrey et al Bayesian Recapture Population Model 959

REFERENCES

Brooks S P (1998) ldquoMarkov Chain Monte Carlo Method and Its ApplicationrdquoThe Statistician 47 69ndash100

Brooks S P and Roberts G O (1998) ldquoConvergence Assessment Techniquesfor Markov Chain Monte Carlordquo Statistics and Computing 8 319ndash335

Buckland S T (1982) ldquoA CapturendashRecapture Survival Analysisrdquo Journal ofAnimal Ecology 51 833ndash847

(1990) ldquoEstimation of Survival Rates From Sightings of IndividuallyIdentifiable Whalesrdquo in Individual Recognition of Cetaceans Use of Photo-Identification and Other Techniques to Estimate Population Parameters edsP S Hammond S A Mizroch and G P Donovan Cambridge UK Inter-national Whaling Commission pp 149ndash153

Burnham K and Overton W (1979) ldquoRobust Estimation of Population SizeWhen Capture Probabilities Vary Among Animalsrdquo Ecology 60 927ndash936

Carothers A (1979) ldquoQuantifying Unequal Catchability and Its Effect on Sur-vival Estimates in an Actual Populationrdquo Journal of Animal Ecology 48863ndash869

Chao A (1987) ldquoEstimating the Population Size for CapturendashRecapture DataWith Unequal Catchabilityrdquo Biometrics 43 783ndash791

Chao A Lee S-M and Jeng S-L (1992) ldquoEstimating Population Size forCapturendashRecapture Data When Capture Probabilities Vary by Time and Indi-vidual Animalrdquo Biometrics 48 201ndash216

Chib S and Greenberg E (1995) ldquoUnderstanding the MetropolisndashHastingsAlgorithmrdquo The American Statistician 49 327ndash335

Cormack R (1972) ldquoThe Logic of CapturendashRecapture Experimentsrdquo Biomet-rics 28 337ndash343

Coull B A and Agresti A (1999) ldquoThe Use of Mixed Logit Models to Re-flect Heterogeneity in CapturendashRecapture Studiesrdquo Biometrics 55 294ndash301

Da Silva C Q Rodrigues J Leite J G and Milan L A (2003) ldquoBayesianEstimation of the Size of a Closed Population Using Photo-ID With Part ofthe Population Uncatchablerdquo Communications in Statistics 32 677ndash696

Dorazio R M and Royle J A (2003) ldquoMixture Models for Estimating theSize of a Closed Population When Capture Rates Vary Among IndividualsrdquoBiometrics 59 351ndash364

Durban J Elston D Ellifrit D Dickson E Hammon P and Thompson P(2005) ldquoMultisite Mark-Recapture for Cetaceans Population Estimates WithBayesian Model Averagingrdquo Marine Mammal Science 21 80ndash92

Fienberg S E Johnson M S and Junker B W (1999) ldquoClassical Multi-level and Bayesian Approaches to Population Size Estimation Using MultipleListsrdquo Journal of the Royal Statistical Society Ser A 162 383ndash405

Forcada J and Aguilar A (2000) ldquoUse of Photographic Identification inCapturendashRecapture Studies of Mediterranean Monk Sealsrdquo Marine MammalScience 16 767ndash793

Franklin A B Anderson D R and Burnham K P (2002) ldquoEstimation ofLong-Term Trends and Variation in Avian Survival Probabilities Using Ran-dom Effects Modelsrdquo Journal of Applied Statistics 29 267ndash287

Friday N Smith T D Stevick P T and Allen J (2000) ldquoMeasure-ment of Photographic Quality and Individual Distinctiveness for the Photo-graphic Identification of Humpback Whales Megaptera novaeangliaerdquo Ma-rine Mammal Science 16 355ndash374

Gaspar R (2003) ldquoStatus of the Resident Bottlenose Dolphin Population inthe Sado Estuary Past Present and Futurerdquo unpublished doctoral dissertationUniversity of St Andrews School of Biology

Gelfand A E and Sahu S K (1999) ldquoIdentifiability Improper Priors andGibbs Sampling for Generalized Linear Modelsrdquo Journal of the AmericanStatistical Association 94 247ndash253

Gerrodette T (1987) ldquoA Power Analysis for Detecting Trendsrdquo Ecology 681364ndash1372

Goodman D (2004) ldquoMethods for Joint Inference From Multiple Data Sourcesfor Improved Estimates of Population Size and Survival Ratesrdquo Marine Mam-mal Science 20 401ndash423

Green P J (1995) ldquoReversible-Jump Markov Change Monte Carlo BayesianModel Determinationrdquo Biometrika 82 711ndash732

Grosbois V and Thompson P M (2005) ldquoNorth Atlantic Climate VariationInfluences Survival in Adult Fulmarsrdquo Oikos 109 273ndash290

Haase P (2000) ldquoSocial Organisation Behaviour and Population Parametersof Bottlenose Dolphins in Doubtful Sound Fiordlandrdquo unpublished mastersdissertation University of Otago Dept of Marine Science

Hammond P (1986) ldquoEstimating the Size of Naturally Marked Whale Pop-ulations Using CapturendashRecapture Techniquesrdquo Report of the InternationalWhaling Commission Special Issue 8 253ndash282

(1990) ldquoHeterogeneity in the Gulf of Maine Estimating HumpbackWhale Population Size When Capture Probabilities Are not Equalrdquo in In-dividual Recognition of Cetaceans Use of Photo-Identification and OtherTechniques to Estimate Population Parameters eds P S Hammond S A

Mizroch and G P Donovan Cambridge UK International Whaling Com-mission pp 135ndash153

Huggins R (1989) ldquoOn the Statistical Analysis of Capture Experimentsrdquo Bio-metrika 76 133ndash140

Joyce G G and Dorsey E M (1990) ldquoA Feasibility Study on the Useof Photo-Identification Techniques for Southern Hemisphere Minke WhaleStock Assessmentrdquo in Individual Recognition of Cetaceans Use of Photo-Identification and Other Techniques to Estimate Population Parameters edsP S Hammond S A Mizroch and G P Donovan Cambridge UK Inter-national Whaling Commission pp 419ndash423

King R and Brooks S (2001) ldquoOn the Bayesian Analysis of PopulationSizerdquo Biometrika 88 317ndash336

Lebreton J-D Burnham K P Clobert J and Anderson S R (1992)ldquoModeling Survival and Testing Biological Hypotheses Using Marked An-imals A Unified Approach With Case Studiesrdquo Ecological Monographs 6267ndash118

Lee S-M Huang L-H H and Ou S-T (2004) ldquoBand Recovery Model In-ference With Heterogeoneous Survival Ratesrdquo Statistica Sinica 14 513ndash531

Link W A and Barker R J (2005) ldquoModeling Association Among De-mographic Parameters in Analysis of Open Population CapturendashRecaptureDatardquo Biometrics 61 46ndash54

Mann J and Smuts B (1999) ldquoBehavioral Development in Wild BottlenoseDolphin Newborns (Tursiops sp)rdquo Behaviour 136 529ndash566

McCullagh P (1980) ldquoRegression Models for Ordinal Datardquo Journal of theRoyal Statistical Society Ser B 42 109ndash142

Otis D L Burnham K P White G C and Anderson D R (1978) ldquoStatis-tical Inference From Capture Data on Closed Animal Populationsrdquo WildlifeMonographs 62 135

Pledger S (2000) ldquoUnified Maximum Likelihood Estimates for ClosedCapturendashRecapture Models Using Mixturesrdquo Biometrics 56 434ndash442

Pledger S and Schwarz C J (2002) ldquoModelling Heterogeneity of Survivalin Band-Recovery Data Using Mixturesrdquo Journal of Applied Statistics 29315ndash327

Pledger S Pollock K H and Norris J L (2003) ldquoOpen CapturendashRecaptureModels With Heterogeneity I CormackndashJollyndashSeber Modelrdquo Ecology 59786ndash794

Pollock K H Nichols J D Brownie C and Hines J E (1990) ldquoStatisticalInference for CapturendashRecapture Samplingrdquo Wildlife Monographs 107 1ndash97

Read A Urian K W Wilson B and Waples D (2003) ldquoAbundance andStock Identity of Bottlenose Dolphins in the Bayes Sounds and Estuaries ofNorth Carolinardquo Marine Mammal Science 19 59ndash73

Saacutenchez-Mariacuten F J (2000) ldquoAutomatic Recognition of Biological ShapesWith and Without Representations of Shaperdquo Artificial Intelligence in Medi-cine 18 173ndash186

Saunders-Reed C A Hammond P S Grellier K and Thompson P M(1999) ldquoDevelopment of a Population Model for Bottlenose Dolphinsrdquo Re-search Survey and Monitoring Report 156 Scottish Natural Heritage

Scottish Natural Heritage (1995) Natura 2000 A Guide to the 1992 EC Habi-tats Directive in Scotlandrsquos Marine Environment Perth UK Scottish Nat-ural Heritage

Seber G (2002) The Estimation of Animal Abundance Caldwell NJ Black-burn Press

Stevick P T Palsboslashll P J Smith T D Bravington M V and Ham-mond P S (2001) ldquoErrors in Identification Using Natural Markings RatesSources and Effects on CapturendashRecapture Estimates of Abundancerdquo Cana-dian Journal of Fisheries and Aquatic Science 58 1861ndash1870

Taylor B L Martinez M Gerrodette T Barlow J and Hrovat Y N (2007)ldquoLessons From Monitoring Trends in Abundance of Marine Mammalsrdquo Ma-rine Mammal Science 23 157ndash175

Wells R S and Scott M D (1990) ldquoEstimating Bottlenose Dolphin Popu-lation Parameters From Individual Identification and CapturendashRelease Tech-niquesrdquo in Individual Recognition of Cetaceans Use of Photo-Identificationand Other Techniques to Estimate Population Parameters eds P S Ham-mond S A Mizroch and G P Donovan Cambridge UK InternationalWhaling Commission pp 407ndash415

Whitehead H (2001a) ldquoAnalysis of Animal Movement Using Opportunis-tic Individual Identifications Application to Sperm Whalesrdquo Ecology 821417ndash1432

(2001b) ldquoDirect Estimation of Within-Group Heterogeneity in Photo-Identification of Sperm Whalesrdquo Marine Mammal Science 17 718ndash728

Williams B K Nichols J D and Conroy M J (2001) Analysis and Man-agement of Animal Populations San Diego Academic Press

Wilson B Hammond P S and Thompson P M (1999) ldquoEstimating Size andAssessing Trends in a Coastal Bottlenose Dolphin Populationrdquo EcologicalApplications 9 288ndash300

Wilson B Reid R J Grellier K Thompson P M and Hammond P S(2004) ldquoConsidering the Temporal When Managing the Spatial A Popula-

Corkrey et al Bayesian Recapture Population Model 959

REFERENCES

Brooks S P (1998) ldquoMarkov Chain Monte Carlo Method and Its ApplicationrdquoThe Statistician 47 69ndash100

Brooks S P and Roberts G O (1998) ldquoConvergence Assessment Techniquesfor Markov Chain Monte Carlordquo Statistics and Computing 8 319ndash335

Buckland S T (1982) ldquoA CapturendashRecapture Survival Analysisrdquo Journal ofAnimal Ecology 51 833ndash847

(1990) ldquoEstimation of Survival Rates From Sightings of IndividuallyIdentifiable Whalesrdquo in Individual Recognition of Cetaceans Use of Photo-Identification and Other Techniques to Estimate Population Parameters edsP S Hammond S A Mizroch and G P Donovan Cambridge UK Inter-national Whaling Commission pp 149ndash153

Burnham K and Overton W (1979) ldquoRobust Estimation of Population SizeWhen Capture Probabilities Vary Among Animalsrdquo Ecology 60 927ndash936

Carothers A (1979) ldquoQuantifying Unequal Catchability and Its Effect on Sur-vival Estimates in an Actual Populationrdquo Journal of Animal Ecology 48863ndash869

Chao A (1987) ldquoEstimating the Population Size for CapturendashRecapture DataWith Unequal Catchabilityrdquo Biometrics 43 783ndash791

Chao A Lee S-M and Jeng S-L (1992) ldquoEstimating Population Size forCapturendashRecapture Data When Capture Probabilities Vary by Time and Indi-vidual Animalrdquo Biometrics 48 201ndash216

Chib S and Greenberg E (1995) ldquoUnderstanding the MetropolisndashHastingsAlgorithmrdquo The American Statistician 49 327ndash335

Cormack R (1972) ldquoThe Logic of CapturendashRecapture Experimentsrdquo Biomet-rics 28 337ndash343

Coull B A and Agresti A (1999) ldquoThe Use of Mixed Logit Models to Re-flect Heterogeneity in CapturendashRecapture Studiesrdquo Biometrics 55 294ndash301

Da Silva C Q Rodrigues J Leite J G and Milan L A (2003) ldquoBayesianEstimation of the Size of a Closed Population Using Photo-ID With Part ofthe Population Uncatchablerdquo Communications in Statistics 32 677ndash696

Dorazio R M and Royle J A (2003) ldquoMixture Models for Estimating theSize of a Closed Population When Capture Rates Vary Among IndividualsrdquoBiometrics 59 351ndash364

Durban J Elston D Ellifrit D Dickson E Hammon P and Thompson P(2005) ldquoMultisite Mark-Recapture for Cetaceans Population Estimates WithBayesian Model Averagingrdquo Marine Mammal Science 21 80ndash92

Fienberg S E Johnson M S and Junker B W (1999) ldquoClassical Multi-level and Bayesian Approaches to Population Size Estimation Using MultipleListsrdquo Journal of the Royal Statistical Society Ser A 162 383ndash405

Forcada J and Aguilar A (2000) ldquoUse of Photographic Identification inCapturendashRecapture Studies of Mediterranean Monk Sealsrdquo Marine MammalScience 16 767ndash793

Franklin A B Anderson D R and Burnham K P (2002) ldquoEstimation ofLong-Term Trends and Variation in Avian Survival Probabilities Using Ran-dom Effects Modelsrdquo Journal of Applied Statistics 29 267ndash287

Friday N Smith T D Stevick P T and Allen J (2000) ldquoMeasure-ment of Photographic Quality and Individual Distinctiveness for the Photo-graphic Identification of Humpback Whales Megaptera novaeangliaerdquo Ma-rine Mammal Science 16 355ndash374

Gaspar R (2003) ldquoStatus of the Resident Bottlenose Dolphin Population inthe Sado Estuary Past Present and Futurerdquo unpublished doctoral dissertationUniversity of St Andrews School of Biology

Gelfand A E and Sahu S K (1999) ldquoIdentifiability Improper Priors andGibbs Sampling for Generalized Linear Modelsrdquo Journal of the AmericanStatistical Association 94 247ndash253

Gerrodette T (1987) ldquoA Power Analysis for Detecting Trendsrdquo Ecology 681364ndash1372

Goodman D (2004) ldquoMethods for Joint Inference From Multiple Data Sourcesfor Improved Estimates of Population Size and Survival Ratesrdquo Marine Mam-mal Science 20 401ndash423

Green P J (1995) ldquoReversible-Jump Markov Change Monte Carlo BayesianModel Determinationrdquo Biometrika 82 711ndash732

Grosbois V and Thompson P M (2005) ldquoNorth Atlantic Climate VariationInfluences Survival in Adult Fulmarsrdquo Oikos 109 273ndash290

Haase P (2000) ldquoSocial Organisation Behaviour and Population Parametersof Bottlenose Dolphins in Doubtful Sound Fiordlandrdquo unpublished mastersdissertation University of Otago Dept of Marine Science

Hammond P (1986) ldquoEstimating the Size of Naturally Marked Whale Pop-ulations Using CapturendashRecapture Techniquesrdquo Report of the InternationalWhaling Commission Special Issue 8 253ndash282

(1990) ldquoHeterogeneity in the Gulf of Maine Estimating HumpbackWhale Population Size When Capture Probabilities Are not Equalrdquo in In-dividual Recognition of Cetaceans Use of Photo-Identification and OtherTechniques to Estimate Population Parameters eds P S Hammond S A

Mizroch and G P Donovan Cambridge UK International Whaling Com-mission pp 135ndash153

Huggins R (1989) ldquoOn the Statistical Analysis of Capture Experimentsrdquo Bio-metrika 76 133ndash140

Joyce G G and Dorsey E M (1990) ldquoA Feasibility Study on the Useof Photo-Identification Techniques for Southern Hemisphere Minke WhaleStock Assessmentrdquo in Individual Recognition of Cetaceans Use of Photo-Identification and Other Techniques to Estimate Population Parameters edsP S Hammond S A Mizroch and G P Donovan Cambridge UK Inter-national Whaling Commission pp 419ndash423

King R and Brooks S (2001) ldquoOn the Bayesian Analysis of PopulationSizerdquo Biometrika 88 317ndash336

Lebreton J-D Burnham K P Clobert J and Anderson S R (1992)ldquoModeling Survival and Testing Biological Hypotheses Using Marked An-imals A Unified Approach With Case Studiesrdquo Ecological Monographs 6267ndash118

Lee S-M Huang L-H H and Ou S-T (2004) ldquoBand Recovery Model In-ference With Heterogeoneous Survival Ratesrdquo Statistica Sinica 14 513ndash531

Link W A and Barker R J (2005) ldquoModeling Association Among De-mographic Parameters in Analysis of Open Population CapturendashRecaptureDatardquo Biometrics 61 46ndash54

Mann J and Smuts B (1999) ldquoBehavioral Development in Wild BottlenoseDolphin Newborns (Tursiops sp)rdquo Behaviour 136 529ndash566

McCullagh P (1980) ldquoRegression Models for Ordinal Datardquo Journal of theRoyal Statistical Society Ser B 42 109ndash142

Otis D L Burnham K P White G C and Anderson D R (1978) ldquoStatis-tical Inference From Capture Data on Closed Animal Populationsrdquo WildlifeMonographs 62 135

Pledger S (2000) ldquoUnified Maximum Likelihood Estimates for ClosedCapturendashRecapture Models Using Mixturesrdquo Biometrics 56 434ndash442

Pledger S and Schwarz C J (2002) ldquoModelling Heterogeneity of Survivalin Band-Recovery Data Using Mixturesrdquo Journal of Applied Statistics 29315ndash327

Pledger S Pollock K H and Norris J L (2003) ldquoOpen CapturendashRecaptureModels With Heterogeneity I CormackndashJollyndashSeber Modelrdquo Ecology 59786ndash794

Pollock K H Nichols J D Brownie C and Hines J E (1990) ldquoStatisticalInference for CapturendashRecapture Samplingrdquo Wildlife Monographs 107 1ndash97

Read A Urian K W Wilson B and Waples D (2003) ldquoAbundance andStock Identity of Bottlenose Dolphins in the Bayes Sounds and Estuaries ofNorth Carolinardquo Marine Mammal Science 19 59ndash73

Saacutenchez-Mariacuten F J (2000) ldquoAutomatic Recognition of Biological ShapesWith and Without Representations of Shaperdquo Artificial Intelligence in Medi-cine 18 173ndash186

Saunders-Reed C A Hammond P S Grellier K and Thompson P M(1999) ldquoDevelopment of a Population Model for Bottlenose Dolphinsrdquo Re-search Survey and Monitoring Report 156 Scottish Natural Heritage

Scottish Natural Heritage (1995) Natura 2000 A Guide to the 1992 EC Habi-tats Directive in Scotlandrsquos Marine Environment Perth UK Scottish Nat-ural Heritage

Seber G (2002) The Estimation of Animal Abundance Caldwell NJ Black-burn Press

Stevick P T Palsboslashll P J Smith T D Bravington M V and Ham-mond P S (2001) ldquoErrors in Identification Using Natural Markings RatesSources and Effects on CapturendashRecapture Estimates of Abundancerdquo Cana-dian Journal of Fisheries and Aquatic Science 58 1861ndash1870

Taylor B L Martinez M Gerrodette T Barlow J and Hrovat Y N (2007)ldquoLessons From Monitoring Trends in Abundance of Marine Mammalsrdquo Ma-rine Mammal Science 23 157ndash175

Wells R S and Scott M D (1990) ldquoEstimating Bottlenose Dolphin Popu-lation Parameters From Individual Identification and CapturendashRelease Tech-niquesrdquo in Individual Recognition of Cetaceans Use of Photo-Identificationand Other Techniques to Estimate Population Parameters eds P S Ham-mond S A Mizroch and G P Donovan Cambridge UK InternationalWhaling Commission pp 407ndash415

Whitehead H (2001a) ldquoAnalysis of Animal Movement Using Opportunis-tic Individual Identifications Application to Sperm Whalesrdquo Ecology 821417ndash1432

(2001b) ldquoDirect Estimation of Within-Group Heterogeneity in Photo-Identification of Sperm Whalesrdquo Marine Mammal Science 17 718ndash728

Williams B K Nichols J D and Conroy M J (2001) Analysis and Man-agement of Animal Populations San Diego Academic Press

Wilson B Hammond P S and Thompson P M (1999) ldquoEstimating Size andAssessing Trends in a Coastal Bottlenose Dolphin Populationrdquo EcologicalApplications 9 288ndash300

Wilson B Reid R J Grellier K Thompson P M and Hammond P S(2004) ldquoConsidering the Temporal When Managing the Spatial A Popula-

960 Journal of the American Statistical Association September 2008

tion Range Expansion Impacts Protected Areas-Based Management for Bot-tlenose Dolphinsrdquo Animal Conservation 7 331ndash338

Wilson B Thompson P M and Hammond P S (1997) ldquoHabitat Use byBottlenose Dolphins Seasonal Distribution and Stratified Movement in theMoray Firth Scotlandrdquo Journal of Applied Ecology 34 1365ndash1374

Wilson D R B (1995) ldquoThe Ecology of Bottlenose Dolphins in the MorayFirth Scotland A Population at the Northern Extreme of the Speciesrsquos

Rangerdquo unpublished doctoral dissertation University of Aberdeen Schoolof Biological Sciences

Wuumlrsig B and Jefferson T A (1990) ldquoMethods for Photo-Identification forSmall Cetaceansrdquo in Individual Recognition of Cetaceans Use of Photo-Identification and Other Techniques to Estimate Population Parameters edsP S Hammond S A Mizroch and G P Donovan Cambridge UK Inter-national Whaling Commission pp 43ndash52