Survival and spatial fidelity of mouflons: The effect of location, age, and sex

Survival and Spatial Fidelity of Mou ons TheEffect of Location Age and Sex

Ruth KING and Stephen P BROOKS

This articleis motivatedbya seriesof dataona populationofmou ons (Ovis gmelini) inthe CarouxndashEspinouse massif and focuses upon discriminatingbetween competingbiolog-ical hypothesescorrespondingto the dependenceof any or all of the populationparametersupon either sex location or age We show how we can analyze the data using a Bayesianapproachwhere we are able to take into account prior information obtained via a previousradio-taggingstudy We consider the ArnasonndashSchwarz model together with its submodelsto describe the data Ef ciently exploringmodel space using reversible jump Markov chainMonte Carlo methodology we are able to calculate model-averaged estimates of parame-ters of interest which incorporate both parameter and model uncertainty In addition wequantitatively compare different biological hypotheses by calculating their correspondingposterior probabilities In particular we show that survival rates tend to remain constantwith some evidence to suggest a slight senescent declineWe also provide evidence to sug-gest that movement around the habitat is largely the same for both sexes up until age 4when the males appear to extend their migration range venturing further from the main ock in search of better grazing

Key Words ArnasonndashSchwarz Bayesian analysis Capturendashrecapture Markov chainMonte Carlo Model averagingModel discrimination

1 INTRODUCTION

The study and analysis of many key wildlife species has long been considered a vitalpart of managing and maintaining both local and global ecosystems and in particular inunderstanding the impact that human activity such as agricultural practices and pollutionhave upon the environment Most studies involve recording the capture and subsequentrecapture of animals in their natural habitat (Lebreton Burnham Clobert and Anderson1992 Pollock 1991) Models range from the simple LincolnndashPetersen model for estimat-

Ruth King is a Lecturer at the School of Mathematics and Statistics in the University of St Andrews CREEMThe Observatory Buchanan Gardens St Andrews Fife KY16 9LZ UK (E-mail ruthmcsst-andacuk) SteveBrooks is a Reader at the Statistical Laboratory in the University of Cambridge Wilberforce Road CambridgeCB3 0WB UK (E-mail stevestatslabcamacuk)

creg 2003 American Statistical Association and the International Biometric SocietyJournal of Agricultural Biological and Environmental Statistics Volume 8 Number 4 Pages 486ndash513DOI 1011981085711032570

486

SURVIVAL AND SPATIAL FIDELITY OF MOUFLONS 487

ing population size to more complex open population studies in which survival recoveryrecapture and possibly migration rates are of primary interest (Dupuis 1995 Schwarz andSeber 1998 Hestbeck Nichols and Malecki 1991)

One advantageof the Bayesian approach to statistical analysis for these kinds of studiesis the prevalence of usable expert knowledge which can be incorporated into the analysiseither to estimate population size (Castledine 1981 Smith 1988 Underhill 1990 Georgeand Robert 1992 Madigan and York 1997 King and Brooks 2001) or survival probabilities(Janz 1980 Link and Cam 2000 Brooks Catchpole and Morgan 2000a Vounatsou andSmith 1995 Brooks Catchpole Morgan and Barry 2000c) A second advantage of theBayesian approach is the availability of powerful new computational techniques capable oftackling the kinds of models that fully re ect the complexity of the dynamics driving thepopulations under study (Brooks 1998 Gamerman 1997 Brooks Catchpole and Morgan2000a) As the restrictions traditionally imposed by analytic tractability are removed weare faced with a new problem in exploring the enormous number of competing modelsavailable and in understanding what each tells us about the population

In the context of recoveryrecapture studies where we typically have three primaryparameters of interest (survival recapture and recovery rates) allowing for the possibilityof say age dependencealone gives rise to a huge class of competing models all of which tellus something different about the underlying dynamics of the population In the presenceof location information and covariate information such as sex the class of biologicallyplausible competing models quickly becomes enormous

Traditionallyin the presence of moderate classes of competingmodels two approachesdominate The rst is to ignore model selection altogether and to take the ldquoglobal modelrdquoin which each parameter depends upon everything In this case we obtain reasonable pa-rameter estimates in general but unless the associated dataset is large these estimates cansuffer from poor resolution Another drawback is that no structural understanding of theunderlying population dynamics is obtained and this is often a more important goal thanthat of parameter estimation as it is through the structure that we begin to learn about the in-teractions within the system that are vital for predictive inference for example The secondapproach is to use model selection techniques and these have proved popular both in gen-eral and in the context of recoveryrecapture data (Burnham and Anderson 1998) Becauseexhaustive comparisons are usually impossible in practice traditional approaches to thisproblem generally adopt a sequential approach starting with the ldquoglobalrdquo model in whichall parameters are fully age and year dependent and then successively combining years orages either randomly or on the basis of biological arguments Statistical tests such as thelikelihoodratio test (Lebreton et al 1992) and AIC criteria (Anderson Burnham and White1994) are then used to determine whether or not an improvement is gained See also An-derson Burnham and White (1998) Buckland Burnham and Augustin (1997) BurnhamAnderson and White (1994 1995a) Burnham White and Anderson (1995b) and Norrisand Pollock (1997) for example However sequential approaches are not guaranteed toobtain the ldquobestrdquo model and often highly plausible models remain completely unexplored

488 R KING AND S P BROOKS

Figure 1 Map of the NationalFaunaReserve indicatingthe division of the study area into three mutually exclusiveregions Figure reproduced from Dupuis et al (2002)

11 DATA

This article investigates the effect of location age and sex on the survival and spatial delity of a population of mou ons (Ovis gmelini) a species of small wild sheep locatedin the National Fauna Reserve on the southern part of the Massif Central The reserve isdivided into three separate regions on the basis of a small initial study in which 53 sheeptaken from three separate groups within the populationwere radio-tagged and their annualranges monitored There was of course some overlap between these ranges but from thesedata three exhaustive yet mutually exclusive regions were obtained Figure 1 provides amap of the reserve and the three subregions used

The study itself involved trapping young mou ons and marking those not previouslymarked In fact it is only possible to determine the age of animals aged 2 years or less andso animals which are older than this on initial capture remain (perhaps rather wastefully)unmarked and are not included in the studySubsequent recapture events are in fact resight-ings obtained from predetermined observation events and chance sightings both occurringduring the period from June to August each year If an animal is recovered dead in anyparticular year its location is not recorded

The data consist of individualcapture histories for a total of 281 mou ons (136 femalesand 145 males) recording the age and location at rst capture the ages at which subsequentrecaptures occur the location of any subsequent recaptures and the age at death if ananimal is recovered The data are provided by Dupuis Badia Maublanc and Bon (2002)who analyzed the data but did not consider the problem of model determination choosinginstead a single model upon which to base this analysis Given these data our model iscomprised of four primary parameters of interest survival recapture recovery (of dead


individuals) and migration Each of these may depend upon the age sex andor locationof the animal For example it has previously been suggested that the movement of animalsoutside the rutting season (October to December) differs between the sexes particularlyfor older animals (LePendu Maublanc Briederman and Dubois 1996 Cransac GerardMaublanc and Oeoin 1998 Dubois Bon Maublanc and Cransac 1994 Dubois et al1996) Females tend to be sedentary throughout life while the range for males increasesfrom youth to essentially the entire reserve by age 3 Thus there is some evidence a priori

to suggest some level of both sex and age dependence on migration Similarly we mightexpect survival rates to be lower both for very old and very young individualsso that survivalmay well depend at least on age

Computer packages such as MARK and MSSURVIV can perform classical analy-ses of such multisite recoveryrecapture data However allowing for dependence of eachof the four parameters on the three factors (age sex and location) we have a very largenumber of competing models each of which tells us something slightly different about thedynamics of this populationTo learn about the underlyingdynamicswe must discriminatebetween these models Within the classical framework only a limited number of modelscan feasibly be compared since each model needs to be considered individually in orderto perform likelihood ratio tests or calculate the corresponding AIC statistic Such modelsto be considered are usually based on biological understanding of the population Withinthe Bayesian framework we discriminate between competing models by estimating the(Bayesian posterior model) probabilities (Gamerman 1997) associated with each We shallshow that these can be obtained via a single Markov chain even for large classes of modelsusing reversible jump Markov chain Monte Carlo This Markov chain then automaticallyexplores the models which describe the data well (given the correspondingpriors) The pos-terior model probabilities that are obtained can as we shall see be used to either eliminateor draw attention to speci c population dynamics and through model averaging (Hoet-ing Madigan Raftery and Volinsky 1999) produce parameter estimates that re ect bothparameter and model uncertainty (see eg Brooks et al 2000a)

12 BAYESIAN APPROACH

The Bayesian approach begins by obtaining a prior distribution for the models andassociated parameters on the basis of expert opinion collected independently of the datax These beliefs are then updated by the data to obtain a posterior distribution ordm (microm mjx)

for the model m and associated parameter vector microm conditioning on the data observedBayesrsquo theorem implies that

ordm (microm mjx) Lm(xjmicrom)p(micromjm)p(m) (11)

where Lm(xjmicrom) denotes the likelihood function under model m evaluated at microm andp(micromjm) and p(m) denote the prior distributions on parameter and model space respec-tively These posterior distributions are typically high dimensional and complex and donot lend themselves to analytic study Thus posterior inference is commonly presented in


the form of posterior means variances and credible intervals for parameters of interestIn addition of particular interest may be the individual model probabilities obtained bymarginalizationof the joint posterior Formally if we let m = (m1 mk) denote the setof possiblemodels the correspondingposterior model probability for model mi is thereforegiven by

ordm (mijx) =L(xjmi)p(mi)Pk

i = 1 L(xjmi)p(mi)

where

L(xjmi) =

ZLmi (xjmicromi )p(micromi jmi)dmicromi

Thus we can quantitativelydiscriminatebetween different models by calculatingthe poste-rior model probability of each model Often Bayes factors are used to compare competingmodels which are simply the ratios of the posterior odds for the corresponding modelsKass and Raftery (1995) suggested that Bayes factors in the range (0ndash3) are ldquonot worth morethan a bare mentionrdquo while those in the range (3ndash20) represent ldquopositiverdquo (or ldquosubstantialrdquo)evidence in favour of one model over the other

These posteriormodel probabilitiesalso allow us to calculatemodel-averagedestimatesof parameters of interestThe model-averagingapproachobtainsa singleparameter estimatebased on all plausible models by weighting each according to their corresponding posteriormodel probabilityand so allows us to incorporatemodel uncertainty into our estimate of theparameter Formally the posterior model-averaged distribution of some parameter vectorreg common to all models mi i = 1 k is given by

ordm (regjx) =

kX

i = 1

ordm (regjx mi) ordm (mijx)

For further details see for example Hoeting et al (1999) and Madigan and Raftery (1994)All of these posterior summaries are most conveniently obtained using Markov chain

Monte Carlo (MCMC) and reversible jump (RJ) MCMC methods (Gelfand and Smith1990 Brooks 1998 Green 1995 King and Brooks 2002) Essentially these methods allowus to sample from the joint posterior distribution over both the parameter and model spaceWe can then use the corresponding sample from the posterior distribution to estimate thesummary statistics of the posterior distribution For example we can estimate the posteriormodel probabilitiesby simply recording the proportionof the time the RJMCMC simulationspends in each model and use these as an estimate of the corresponding posterior modelprobabilities

This article illustrates how we may perform a Bayesian analysis of multisite recap-turerecovery data In particular we discuss how RJMCMC methods may be used to an-alyze the mou on dataset where we are primarily interested in describing the underlyingdynamics of the populationby discriminating between competing hypothesesWe begin inSection 2 by introducing the class of plausible models appropriate to the mou on data andestablish the notation required to describe the different models We then describe the form


of the priors that we shall use in Section 3 and combine these with the likelihood functionto form the posterior distributionof interest We brie y discuss the MCMC techniquesusedto explore this posterior distribution and then present the results of our analysis in Section4 Finally we conclude with some discussion

2 PARAMETERS MODELS AND NOTATION

The data here are best described by the ArnasonndashSchwarz (AS) model which is welldiscussed in the literature (Schwarz Schweigert and Arnason 1993 Brownie et al 1993Dupuis 1995) Essentially the AS model assumes Markovian transitions between distinctregions within the study area so that the movement of an animal is independent of itsprevious migration history The model is based upon the assumption that an animal couldin theory move from any region in the study area to any other (ie the habitat is irreducible)and that individualsbehave independentlyof one anotherAs an extensionof the JollyndashSebermodel (Jolly 1965 Seber 1965) the AS model also assumes that individuals observed arerepresentative of the whole population and that marking animals (marks are assumed tobe permanent) has no appreciable affect on their subsequent behaviour (no trap af nity forexample)

If we assume that captures occur for animals aged t 2 T = f0 T iexcl 1g thatanimals are recaptured in regions r 2 R = f1 Rg and that each animal is assigneda sex c 2 C = fm fg (with the obvious notation for males and females) then our modelcomprises the following parameters

iquest ct(r) = Pr(an animal of sex c in location r at age t survives until age t + 1)

para ct(r) = Pr(an animal of sex c in location r at age t that dies before age t + 1

is recovered dead before age t + 1)

pct + 1(r) = Pr(an animal of sex c in location r at age t + 1 is recaptured at this time)

and

Aacutect (r s) = Pr(an animal of sex c in location r at age t moves to location s 2 R

by age t + 1 given that it is alive at age t + 1)

The above model describes the most general (or global) AS model where each of theparameters are all dependent upon sex age and location Dupuis et al (2002) analyzed themou on data using this globalmodel and ignored the issue of model choiceHowever manysubmodels provide plausible descriptions of the mou on population and discriminatingbetween them provides us with meaningful insights into the dynamics of the populationFor example the hypothesis that recapture dependsonly upon location implies that pc

t(r) =

p(r) for all t 2 T c 2 C and r 2 R This is equivalent to a restriction of (c t r) fromC poundT poundR to fc curren gpoundft curren gpoundR for the recovery parameters where c curren and t curren denote single sexand age values common to all sexes and ages We therefore reduce the number of recaptureparameters from 2T R to R Discrimination between competing hypotheses of this sort canbe achieved by exploring competing groupings of sex age and location for each parameter


and estimating the correspondingposteriorprobabilitiesassociatedwith eachAlternativelythese posterior model probabilities can be used as a basis for model-averaged predictiveinference as discussed in Section 12

Clearly any model is uniquelydescribed by the restrictions that we place on the param-eters in terms of the groups or sets of ages location and sex that have common parametervalues Often within classical analyses the models that are considered are of a simpleform For example models either with full or no age-dependenceThis restriction to simplemodels is largely due to the computational demands of exhaustive enumeration of muchlarger classes of models However more complex models are becoming increasing popular(see eg Catchpole et al 2000 where the population under study exhibits different rst-year adult and senior survival rates) Discriminating between such models provides nerresolution as to the underlying dynamics of the population and we consider quite generalforms here In order to specify a model we therefore need to describe the groupings of agelocation and sex for each of the four model parameters

Consider the survival parameters We let n denote the number of sets of ages for thesurvival parameters and let mT denote the sets of ages where each set Ti sup3 T de nes agroup of ages with common survival rate that is if t1 t2 2 Ti then iquest c

t1(r) = iquest c

t2(r) for all

c 2 C and r 2 R (Note that we drop any notational dependence on iquest as it is clear from thecontext) Thus mT = fT1 Tng Clearly the Ti must be both mutually exclusive andexhaustive of T so that [n

i = 1Ti = T and if t 2 Ti then t =2 Tj for i 6= j In the context ofthe mou on population it seems sensible to impose the condition that if t1 t2 2 Ti thenfor c 2 C iquest c

t1(r) = iquest c

t2(r) for all r 2 R that is the survival rates are common across all

regions (and sexes) for ages t1 and t2 This clearly imposes a dependence structure in whichthe location is conditional on the age structure in that the location dependence at some timet1 where t1 2 Ti is the same for all t2 2 Ti by de nition

Having de ned a grouping for ages we next de ne the grouping for locations takinginto account this conditional dependence on age We let mRi denote the groupings oflocations for ages in the set Ti Because R = f1 2 3g there are only ve distinct locationgroupings and we let mRi = fR1i Rniig where ni micro 5 denotes the number oflocation groups associated with ages in Ti Clearly for all i = 1 n [ni

l = 1Rli = Rand if r 2 Rli then r =2 Rji for j 6= l We then set mR = fmR1 mRng Finally wede ne the vector mC which denotes whether or not there is a sex-effect within each groupof ages This re ects the prior belief that though sex-effects may differ between ages thelocation of the animal does not affect this sex-dependence structure So that for animals of agiven age there is no difference in terms of the presence or absence of a sex-effect betweenlocations

The marginal model for survival is then denoted by m iquest = fmT mS mCg Usinganalogous notation for the recapture recovery and migration parameters (dropping the de-pendence on location for migration because grouping locations for migration is intrinsicallystrata-dependent in this context) the overall model is denoted by m = m iquest =mp=mpara =mAacute

We note that this constructiondescribes a class which is not exhaustiveof all submodelsof the AS model For examplewithin a particularage group the survival rate at each location


must either all be sex-dependentor all independentof sex Thus this constructioneffectivelyimposes a partial prior structure on model space in which all models not within the classdescribed have prior probability zero These restrictions are clearly sensible in the contextof the mou on analysis (as we shall discuss in Section 322) However other restrictions(or none at all) may be more appropriate in other contexts see King and Brooks (2002) forfurther discussion

3 BAYESIAN ANALYSIS OF THE DATA

In order to undertake the analysis we begin by constructing the likelihood and thenspecifying priors for both the model parameters and for the models themselves In this waywe obtain the posterior distribution given in (11) which we can explore via MCMC

31 THE LIKELIHOOD FUNCTION

Previous analyses based upon the AS model use auxiliary variable techniques to writedown a so-called ldquocompleterdquo likelihood function (Dupuis 1995 King and Brooks 2002and Dupuis et al 2002) These take advantage of the fact that if the location of everyanimal is known at all times then the likelihood takes a simple analytic form Becausethe location of unobserved animals is unknown auxiliary variable methods may be usedto impute these missing locations Though this simpli es the likelihood expression theanalysis (whether Bayesian or classical) is complicated by the need to impute these missingvaluesHowever King and Brooks(2003)derivedan explicitanalyticform for the likelihoodof the observed data which removes the need to add auxiliary variables to indicate thelocation of unobserved animals and we apply this approach here An explicit expressionfor the likelihoodfor multisite data where there are both live recaptures and dead recoveriesis given in Appendix A

32 PRIORS

Having obtained the likelihood we now need to assign priors both to the model pa-rameters and to the models themselves Generally these may be obtained from experts whohave some prior knowledge concerning the species under study Such speci cations maybe given directly in the form of statistical distributions but more commonly it is in theform of means andor intervals for the parameters from which prior distributions can beformed (see eg OrsquoHagan 1998) For this dataset we have no such prior information forthe recoveries and recaptures but there is independent information available on the survivaland migration rates obtained via a previous radio-tagging study which we can incorporateinto the analysis


321 Parameter Priors

Data are available from an initial radio-taggingstudy (Dupuis et al 2002) of both maleand female mou ons from which we may formulate prior beliefs concerning the survivaland migration probabilities Here the location of each animal is always known followingtagging as is the age at death However this study provides no additional informationrelating to the recapture and recovery parameters of the subsequentcapturendashrecapture studythat we wish to analyze here Thus we place independent standard Uniform priors on eachof the capture and recovery parameters within each possible model

For the survival parameters we take independentBeta priors with parameters obtainedfrom the radio-tagging study Similarly due to the sum-to-one constraint we take indepen-dent Dirichlet priors for the rows of the transition matrix with parameter values based uponthe radio-tagging study These informative priors are obtained as follows

Let nct(r s) denote the number of animals in the radio-tagging study that are of sex

c in location r 2 R at age t and in location s 2 R [ fyg at age t + 1 (These data areprovided in Dupuis et al 2002 table 2) Then nc

t(r y) is the number of animals of sex cand in location r at age t that die before age t + 1 We also let nc

t(r cent) =P

s2 R nct(r s)

denote the number of animals of sex c in location r at age t that survive until age t + 1Then for a given set of ages Th and set of regions Rlh we place a Beta( not c(l h) shy c(l h))

prior on the associated survival parameter where

not c(l h) = 1 +X

t 2 Th

X

r 2 Rlh

nct(r cent)

shy c(l h) = 1 +X

t 2 Th

X

r 2 Rlh

nct(r y)

These are essentially the posterior marginal distributions obtained from an analysis of theradio-taggeddata with a standard Uniform prior In this case the locationsare always knownand there is no recovery or recapture so the posterior marginals simplify to this standardform

Similarly the priors for the migration parameters for time group Th and migratingfrom region r 2 R are taken to be Dir(ec

h(r 1) ech(r 2) ec

h(r 3)) where

ech(r s) = 1 +

X

t2 Th

nct(r s)

for s 2 RA similar approach is used to derive the prior distributions for the different parameters

when no sex-effect is included in the model (essentially we also sum over c 2 C )As an illustration the model-averaged prior mean and 95 highest posterior density

interval (HPDI) for the survival and migration rates are presented in Figures 2 and 3respectively (Recall that we take Uniform priors for the recovery and recapture rates) Thesurvival rates are clearly similar across the different areas though the wide HPDIs suggesta large amount of prior uncertainty For the migration rates the priors generally appearto be more informative for younger mou ons (represented by the smaller variances) and


(a) (b)

Figure 2 The prior mean survival rates for each of the different regions with corresponding 95 HPDIs (repre-sented as vertical lines) where the dotted line (cent cent cent) represents area 1 the solid line (mdashndash) area 2 and the dashedline (- - -) area 3 for (a) the males and (b) the females

become increasingly vague as age increases The exception to this appears to be for malesin area 3 where there is generally a large amount of prior uncertainty for all ages thoughthis decreases for ages 3ndash4

322 Model Priors

Our prior knowledge about the mou on population suggests that age is the dominantfactor in describing the dynamics of the population in terms of the survival recapturerecovery and migration parameters We expect that movement rates for example maychange with age and though at any age movement may differ between areas animalswithin a group of ages (eg young animals) will possess the same spatial dynamics Wehold similar a priori beliefs about the dependence upon sex which is why in Section 2 weconstruct our class of plausible models by conditioning the sex and location dependenceon the sets of ages This essentially means that for a given parameter (survival recaptureor recovery) and any set of ages the parameter values for these ages have a common valuefor each sex and location (although the parameter values may differ between the locationsandor sex) Thus any dependence structure for the sex andor location is common for allages in the same set We shall further assume that mou ons of similar ages possess similarbehavioral characteristics and therefore we shall wish to combine only consecutivegroupsof ages This then de nes our class of plausible models

In terms of prior speci cation we shall discriminate between models that lie withinour class of plausible models described above and those that do not To the latter we assignprobability zero For the remainder we begin by specifying the prior probabilities for eachof the (marginal) models relating to each of the sets of parameters (ie survival recaptureetc) For a given set of parameters we place an equal prior probability for each possible setof ages that is for each possible set of change points on the ages This re ects the belief


Fig

ure

3

The

pri

or m

ean

mig

ratio

n ra

tes

for

each

reg

ion

for

the

mal

es (

figur

es (

a)ndash(

c))

and

fem

ales

(fig

ures

(d)

ndash(f)

) to

eac

h of

the

poss

ible

are

as w

ith

corr

espo

ndin

g 95

H

PD

Is (

repr

esen

ted

as v

erti

cal l

ines

) w

here

the

dotte

d lin

e (

)

rep

rese

nts

area

1 t

he s

olid

line

(mdash

mdashmdash

) ar

ea 2

and

the

dash

ed li

ne(-

- -

) ar

ea 3

(a)

(b)

(c) (f

)(e

)(d

)


that each possible age-dependence structure of the parameters is equally likely Withoutloss of generality suppose that we consider the model with k sets of ages (so that thereare k iexcl 1 change points) and consider the lth age set for l 2 [1 k] We have no priorinformation relating to the dependence of the parameters on location (given the age set)and so place a at prior over the dependence of the parameter on each set of locationsFinally irrespective of the sets of locations with common parameter values we place anequal prior probability of a sex-effect or no sex-effect for the age set l This prior structureis then true for all l 2 [1 k] The overall prior for the model is then simply the productof the priors for the marginal models for each of the sets of parameters as we assume thepriors are independentWe note that this prior is not at over the different plausible modelsand that a at prior would imply that models with a larger number of sets of ages for theparameters are more likely a priori This does not seem appropriate here For more generalmodel structures and alternative prior speci cations see King and Brooks (2002)

Combining the priors in Section 32 with the likelihood from Section 31 via Bayesrsquoformula given in (11) we obtain a posterior distribution for our models and their associatedparameters Inference is obtained in the form of posterior summary statistics such as pos-terior means and variances of parameters of interest Analytic evaluation of these statisticsis impossible and so MCMC methods (Brooks 1998) are used to draw samples from theposterior and statistics of interest are then estimated empirically from this sample Fulldetails of the (RJ)MCMC simulation scheme are provided in Appendix B

4 RESULTS

We ran the MCMC procedure outlined above for a total of ten million iterations withthe rst million discarded as burn-in Independent repetitions of the procedure producesessentially identical results and coupled with the standard diagnostic procedures (Brooksand Roberts 1998 Cowles and Carlin 1996) we accept that the results have convergedFor simplicity we concentrate on the marginal models for each of the sets of parameters(integrating out the other parameters) and begin by examining the survival rates

41 SURVIVAL PROBABILITIES

Table 1 provides the model-averaged posterior probability of a sex-effect for each ageand each of the four parameters for brevityalthoughwe shalldiscusseach in thecorrespond-ing sections relating to the parameters Because each parameter is either sex-dependent ornot probabilitiesbelow 025 or greater than 075 would be considered ldquosigni cantrdquo in termsof the corresponding Bayes factor as discussed in Section 12 Thus there is fairly strongevidence that the survival rates are largely independent of sex (Bayes factors between 3 and9 for ages 0ndash5) The higher posterior probabilities of a sex effect for the survival rates forolder mou ons is probably due to there being fewer mou ons observed at older ages Thisresults in there being less information contained in the data so that the posterior becomesincreasingly dominated by the prior for the oldest mou ons


Table 1 The PosteriorProbabilitythat each of the Parametersare Dependenton the Sex of the Mouonfor each Age

Posterior probability of sex effect

Age Survival Recapture Recovery Migration

0 010 ndash 061 0151 010 062 062 0372 011 063 063 0503 012 070 061 0564 016 070 062 0705 024 070 061 0716 032 068 051 0707 049 062 049 0698 ndash 059 ndash ndash

The age groupings with posterior support greater than 5 are provided in Table 2 Themost probablemodel a posteriori identi es the survivalas age-independentwith a posteriorprobability of 0244 However there is some evidence to suggest that survival rates maychange in later years (as demonstrated by the second-most probable model)

The posterior support for models with little age dependencecan be seen in the posteriormodel-averaged estimates of the survival probabilitieswhich are given in Figure 4 There issome evidenceof a decline in survival for older mou ons but the survival rates are generallyvery similar across the different locationsThis is largelydue to thehighposteriorprobabilityfor survival being independent of location this probability is gt 45 for mou ons aged0ndash5 and approximately 30 for mou ons aged 6ndash7 and has largest posterior support forall ages The decline in survival with age appears to begin around age 4 at which point thesurvival rates steadily decrease We note that the width of the 95 HPDIs monotonicallyincreases with age but are generally similar between strata and sexes This increase ismost probably related to fewer mou ons being observed at older ages thus reducing theamount of information contained in the data and increasing our uncertaintyrelating to theseparameters

We note also that the posterior for the survival rate looks quite different from the priorplotted in Figure 2 This suggests that the data contains substantial information on theseparameters and is able to draw the posterior away from the prior beliefs expressed

The posteriorsummary statisticplots for themales and females are also very similar this

Table 2 The Marginal Models for the Sets of Ages for the Survival Parameters with Posterior Supportgt 5

Posterior modelSurvival model probability

T1 = f01234567g 024T1 = f01234g T2 = f567g 012T1 = f0123456g T2 = f7g 009T1 = f012345g T2 = f67g 008T1 = f0123g T2 = f4567g 007T1 = f012345g T2 = f6g T3 = f7g 006T1 = f01234g T2 = f56g T3 = f7g 005


(a) (b)

Figure 4 The posterior mean survival rates or each of the different regions with corresponding 95 HPDIs(represented as vertical lines) where the dotted line (cent cent cent) represents area 1 the solid line (mdashndash) area 2 and thedashed line (- - -) area 3 for (a) the males and (b) the females

is not unexpecteddue to the fairly large posterior probabilities that males and females have acommon survival rate over the different ages Overall there appears to be little evidence ofdependence of survival upon either location or sex but some evidence to support a declinein old age This appears to correspond with a change in the movement dynamic at aroundthe same age as we shall see later

42 RECAPTURE PROBABILITIES

Considering Table 1 the recapture rates appear to generally favor a sex-effect but theresults are far from conclusive (Bayes factors between 1 and 24 in favor of an effect) Inaddition they do not appear to highlight any signi cant differences between the ages ofthe animals in terms of the presence or absence of a sex-effect Table 3 provides the aposteriori most probable groups of ages Clearly no single model dominates the posteriorwith the most probable model having a Bayes factor of between 15 to 2 in relation to theother models identi ed in Table 3 However all the models identi ed have a distinctly largercapture probability for young mou ons (aged 1ndash2) There appears to be some uncertaintyas to the dependence of the recapture rates upon location (not tabulated) For ages 1 and2 the grouping of locations with largest posterior support is f1 3g f2g with posteriorprobability approximately 47 for both ages This is in contrast to all older mou ons withhighest posterior support going to the model in which the recapture rate is independent oflocation that is the model f1 2 3g This probability is at least 45 for mou ons aged 3ndash6and 30 for mou ons aged 7ndash8

The posterior mean recapture probabilities are very similar between the male andfemale populations as can be seen in Figure 5 (We note that the prior mean recapture rateis equal to 05 because we place a at Uniform prior over the recapture rates in each model)


Table 3 The Marginal Models for the Sets of Ages for the Capture Parameters with Posterior Proba-bility gt 5

Recapture model Posterior probability

T1 = f12g T2 = f345678g 015T1 = f12g T2 = f3456g T3 = f7g T4 = f8g 010T1 = f12g T2 = f34567g T3 = f8g 009T1 = f12g T2 = f3456g T3 = f78g 008

The corresponding HPDIs appear to be fairly wide re ecting both parameter and modeluncertainty We also note that the variance of the estimates (as with survival) generallyincrease as age increases this is again due to there being less information concerning oldermou ons as fewer are observed

In summary recapture rates in region 2 appear to be generally higher than in regions 1and 3 In addition there is some evidence to suggest that recaptures are higher for youngermou on Both of which may tie in with the mou ons use of the habitat over time and theinteraction with the data-collectionprocess For example as we shall see later on there area larger proportion of mou ons in region 2 than any other and we suspect that recaptureeffort is focused in this area There is also some very mild evidence to suggest a sex-effectat all ages but the model averaged parameter estimates appear to differ very little betweenthe sexes and we suspect that with more data this effect may disappear

43 RECOVERY PROBABILITIES

As for the recapture rates the evidence for the presence of a sex effect is inconclusive(see Table 1) We also note that all possible age groupings were visited within the Markov

(a) (b)

Figure 5 The posterior mean recapture rates for each of the different regions with corresponding 95 HPDIs(represented as vertical lines) where the dotted line (cent cent cent) represents area 1 the solid line (mdashndash) area 2 and thedashed line (- - -) area 3 for (a) the males and (b) the females


(a) (b)

Figure 6 The posterior mean recovery rates for each of the different regions with corresponding 95 HPDIs(represented as vertical lines) where the dotted line (cent cent cent) represents area 1 the solid line (mdashndash) area 2 and thedashed line (- - -) area 3 for (a) the males and (b) the females

chain (a total of 128) with the posterior appearing to be fairly at across the different sets ofages There appears to be a large amount of model uncertainty relating to the age structureof the recovery rates with several models having (relatively) small posterior support butcomparable Bayes factors However the models with largest posterior support all appearto have distinct recovery rates for the oldest mou ons In particular the model with largestposterior support of 7 is T1 = f0 1 2 3 4 5g T2 = f6g T3 = f7g though this isvirtually indistinguishable from the model T1 = f0 1 2 3 4 5 6g T2 = f7g The onlyother model with posterior support greater than 5 is T1 = f0 1 2 3 4 5g T2 = f6 7gwhich has a posterior model probability of 6 Thus we note that the models identi ed ashaving the largest posterior support are all close neighbors of each other and suggest thatrecoveries may change (increase) in later years

Though not tabulated the posterior probabilities associated with the grouping of loca-

tions suggest that there is no dependence of recoveries on location The location-

independence model has largest posterior support for ages 0ndash5 with a posterior support

of 55 for the youngest mou ons monotonicallydecreasing to 33 for the mou ons aged

5 For mou ons aged 6ndash7 the posterior support is fairly evenly spread over the different

location groupings

The posterior model-averaged estimates for the recovery probabilities are given in

Figure 6 The mean estimates of the recovery probabilities are very similar between the

two sexes There also appears to be a general increase in the mean and standard deviationof the recovery probabilities with age with a sharp increase at age 6 consistent with the

age-dependence models identi ed in the analysis

In summary there is some evidence to suggest that recovery rates increase in later life

and although independent of location in early life tend to be lower in region 2 for older

mou ons It is interesting to note that while recovery rates appear to remain fairly static

in region 2 as the mou on age the recovery rates appear to increase in regions 1 and 3


However we note that the correspondingHPDIs are fairly wide for these parameters (again

increasing with age) which re ects the fact that few old animals are recovered within the

study and so we have less information concerning these parameters

44 MIGRATION PROBABILITIES

Table 1 suggests that there is a general increase in the probability of a sex-effect with

age for the migration probabilities There is positive evidence (Bayes factor of 567) to

suggest that there is no difference between the sexes for the youngest mou on and some

evidence to suggest that there may be some differences in older animals This would also

suggest that the migration rates possess some age-dependence between the youngest and

the older mou ons The most probable marginal models for the age-dependence structures

are given in Table 4 In specifying the models we gain most understanding by looking

simultaneously at the age groupings and their dependence upon gender We denote the

existence of a sex-effect by the inclusion of an ldquoSrdquo in the set of ages whose absencedenotes that the parameters are independent of sex

It can be clearly seen from Table 4 that the majority of models identi ed with largest

posteriorsupport(a totalof 5 out of the7 modelswith posteriorsupportgreater than5)have

no sex-effect for young mou ons but do include a sex-effect for older mou ons However

the age where the sex of the animal begins to affect movement is not easily extracted though

the majority of models identi ed suggest that the change occurs before age 4 This may

not be unsurprising if individualsldquomaturerdquo at different ages with respect to their migratory

behaviorand there appears to be evidenceof this in practice We also note that if we consider

only the male dataset the model T1 = f0 1 2 3g T2 = f4 5 6 7g has the largest posterior

support (of 31) which re ects the behavioral beliefs of the experts (see Section 1) who

suggested that male mou ons extend their migration range as they mature Similarly for the

female dataset the age-independence model has 57 posterior support Again this agrees

with ecologists who suggest that females tend to be more sedentary and exhibit the same

migration patterns throughout their entire lives

The corresponding model-averaged posterior means for the migration probabilities

for the male and female populations are given in Figure 7 Clearly the animals appear

to be largely sedentary with little movement between the different regions However the

Table 4 The Models for the Migration Probabilities with Posterior Support Greater than 5

Migration model Posterior probability

T1 = f0g T2 = f1234567 Sg 016T1 = f01234567g 012T1 = f0123g T2 = f4567Sg 011T1 = f01234567 Sg 009T1 = f01g T2 = f234567 Sg 008T1 = f0123456g T2 = f7 Sg 006T1 = f0123456g T2 = f7g 006


Fig

ure

7

The

pos

teri

or m

ean

mig

ratio

n ra

tes

for

each

reg

ion

for

the

mal

es (

figur

es (

a)ndash(

c))

and

fem

ales

(fig

ures

(d)

ndash(f)

) to

eac

h of

the

pos

sibl

e ar

eas

with

cor

resp

ondi

ng 9

5 H

PD

Is (

repr

esen

ted

as v

erti

cal l

ines

) w

here

the

dotte

d li

ne (

) re

pres

ents

are

a 1

the

soli

d li

ne (

mdashmdash

mdash) a

rea

2 a

nd th

e da

shed

line

(- -

-)

area

3

(a)

(b)

(c)

(f)

(e)

(d)


probability of emigrating from each of the areas becomes markedly higher for males with

increasing age We also note that the mean probability of staying in each of the regions is

consistentlyhigher for females than males This once again af rms the expertsrsquo knowledge

of the species For both the male and female populationsthe probability of emigrating from

region 3 is larger than that for emigrating from either of the other two areas We also note

that the probability of migrating between areas 1 and 3 is lower than the other possible

migrationsThis may be due to the geography of the study region with stratum 2 separating

the other two areas so that migrating between areas 1 and 3 involves traveling the greatest

distanceand (usually)passing throughstratum2 (see Figure1) Similarmigrationalpatterns

are present in the prior means (see Figure 3) though these are generally less smooth than the

correspondingposteriorplotsFinallywe note that once again there appears to be increasing

uncertainty in the parameters with increasing age demonstrated by wider HPDIs

In summary there is strong evidence to suggest that movement is independent of sex

for the youngest mou on (no doubt because infants will follow their mother regardless of

sex) There is then some evidence to suggest that migration patterns diverge between the

sexes at around age 3ndash4 when males extend their migration range while females remain

largely sedentary Examining the transition probabilities themselves there appears to be a

greater af nity for region 2 among the population than the other regions We can assess

this more directly by considering the proportion of animals in each of the three regions

which may help explain the location-dependence observed for the recovery and recapture

parameters

Information of this sort is easily obtained via the posterior samples of the migration

parameters For each transition matrix ordf = fAacute(r s) r s 2 Rg sampled from the posterior

we can calculate the associated stationary distribution as the solution (in terms of the R-

dimensional vector y) to

yT ordf = yT (41)

If element yr of y denotes the proportion of individuals in region r 2 R then imposing the

constraint thatP

r 2 R yr = 1 we can solve (41) to nd the stationary proportions y The

posterior distribution of y can then be obtained by averaging over samples of ordf

The corresponding mean proportion of individuals in each area is given in Figure 8

together with 95 HPDIs for each age and sex It is clear that area 2 is more popular

than the other two regions generally containing more than half the population We can see

that the proportion of females in each area is similar across ages agreeing with our earlier

observationsHowever the males appear to spread out more with an increasing proportion

of males in region 3 as their age increases This is consistent with the theory that mature

mou on both become more mobile (ie move around the habitat more) and that they seek

out new grazing areas away from the main grazing area in region 2 Further study is needed

to investigate this dynamic in more detail


(a) (b)

Figure 8 The posterior mean of the stationary distribution of the proportion of individuals within each regionwith corresponding 95 HPDIs (represented as vertical lines) where the dotted line (cent cent cent) represents area 1 thesolid line (mdashndash) area 2 and the dashed line (- - -) area 3 for (a) the males and (b) the females

5 DISCUSSION

This article describes the Bayesian analysis of data on a population of mou ons viaRJMCMC using the ArnasonndashSchwarz model to describe movement around the habitatUsing the Bayesian approach we are able to incorporate prior information from an indepen-dent radio-tagging study providing further information on both the movement and survivalof the mou ons In additionwe are able to quantitativelycompare different models whichrepresent competing biological hypotheses using posterior model probabilities calculatedvia the reversible jump algorithm

We consider the analysis of the mou on data in detail and discover several interestingpopulation dynamics by considering this issue of model choice First of all survival ratesappear to be fairly constant over most ages althoughthere is some evidence that it decreasesas animals mature This may be due in large part to senescence with a decline starting ataround age 5 but the movement dynamic for males at least appears to change at around thesame time and so there may well be a link between this change in behavior and the survivalrate However the fact that this decrease is common to both males and females suggeststhat senescence effects dominatehere There is strong evidence to suggest that movement isindependent of sex for the youngest mou on (no doubt as the young follow the movementof their mother whatever their sex) There is also evidence to suggest that migration patternsdiverge between the sexes at around age 4 when males extend their migration range whilefemales remain largely sedentaryOverall animals appear to spend the most amount of timein region 2 which is central to the habitat At this stage it is not clear whether this is simplydue to the fact that to move from region 1 to region 3 an animal would normally crossregion 2 or because the habitat is better suited to the mou on in that region Certainly thispreference does not appear to change with age which challenges the hypothesis that region


2 might contain the primary breeding ground Finally it is interesting to note that malestend to make less use of region 1 than females and that they make increasing use of region3 as they mature This may be due to the fact that region 3 is less heavily grazed and themore mobile males perceive the improved grazing as dominating the drawbacks of beinglocated further from the main herd

The Bayesian analysis presented in this article provides a valuable insight into thedynamics of the mou on population under study that is not obtained through the analysisof any single model Interesting extensions would be to include covariate information tohelp differentiate between age and time-dependent variation In addition in the context ofother applications alternative parametric forms for the parameters might be included Forexample if survival follows a senescent decline we might consider modeling that declineexplicitly by imposing a parametric form on survival and its relationship to age This wouldprovide a more parsimonious representation and would be valuable in terms of predictionOf course prediction is also possible within the framework described here by taking themodel parameters to be random rather than xed effects Extensions along these lines arethe focus of current research

APPENDIXES

A LIKELIHOOD

King and Brooks (2003) showed that the likelihood can be expressed in the followingform

L(xjmicro) =Y

c2 C

Y

r 2 R

2

4TY

t = 0

TY

j = t

Agrave c(tj)(r)vc

(tj)(r)T iexcl1Y

t = 0

tY

l = 0

Y

s2 ROc

(lt)(r s)nc(lt)(rs)

poundT iexcl1Y

t= 0

tY

l = 0

Dc(lt)(r)dc

(lt)(r)

(A1)

where each of these terms are de ned as followsThis likelihood can be seen as a generalization of Catchpole Freeman Morgan and

Harris (1998) for the simple recoveryndashrecapture context (ie no migration) We leti) vc

(tj)(r) denote the number of animals of sex c that are observed for the last timein location r 2 R aged t such that (if they survive) they will be aged j para t at the nal capture time of the study

ii) nc(lt)(r s) denote the number of animals of sex c that are observed in location r 2 R

at age l and next observed alive in location s 2 R at age t + 1 andiii) dc

(lt)(r) denote the number of animals of sex c recovered dead between ages t andt + 1 that are last observed alive at age l in location r 2 R


For convenience we adopt a vector notation for each of the suf cient statistics so thatv = fvc

t (r) c 2 C r 2 R j para t t 2 T g with similar de nitions for n and dThe likelihood is a product of the following recursive formulas which arise when we

integrate out the auxiliary variables commonly used for the AS model

1 Let Agrave c(tj)(r) denote the probability that an animal of sex c is seen for the last time

at age t = 0 T in location r 2 R given that (if it survives) it will be of age j micro T atthe end of the study Then

1 iexcl Agrave c(tj)(r) = iquest c

t(r)

Atilde

1 iexclX

s2 RAacutec

t (r s)(1 iexcl pct+ 1(s)) Agrave c

(t + 1j)(s)

+ (1 iexcl iquest ct(r)) para c

t(r)

where

Agrave c(tt)(r) = 1 8r 2 R

2 Let Oc(lt)(r s) denote the probability that an animal of sex c observed in location

r 2 R at age l 2 T remains unobserved until it is subsequently resighted in location s 2 Rat age t + 1 0 micro l micro t micro T iexcl 1 Then

Oc(lt)(r s) = pt+ 1(s)Qc

(lt)(r s)

where Qc(lt)(r s) denotes the probability that an animal of sex c migrates from region

r 2 R at age l 2 T to location s 2 R at age t + 1 and is unobserved between those agesand is given by

Qc(lt)(r s) =

(iquest c

l (r)Aacutecl (r s) l = t

iquest cl (r)

Pk 2 R (1 iexcl pc

l + 1(k))Aacutecl (r k)Qc

(l + 1t)(k s) l lt t

3 Let Dc(lt)(r) denote the probabilitythat an animalof sex c is recovereddeadbetween

ages t and t + 1 given that it is last observed at age l micro t in location r Then

Dc(lt)(r) =

((1 iexcl iquest c

t(r)) para ct(r) l = tP

s 2 R (1 iexcl iquest ct(s)) para c

t(s)(1 iexcl pct(s))Qc

(ltiexcl1)(r s) l lt t

The proofs of the form for these recursive formulae together with a derivation of thelikelihood in (A1) are given in King and Brooks (2003)

B MCMC SIMULATION

Observations from the posterior are obtained via MCMC methods (Brooks 1998) Weconsider two distinct forms of MCMC mechanism The rst allows transitions within amodel updating the parameters the second updates the model a move which typicallyinvolves adding or deleting parameters and therefore altering the dimension of the statespace We brie y discuss each in turn


B1 WITHIN-MODEL MOVES

Within each iterationof the chain we update each of the model parameters in turn usinga random walk MetropolisndashHastings step For the survival probabilities we cycle througheach set of ages and location Suppose that we propose to update the survival parametersfor the set of ages Th and location set Rlh Then if there is a sex-effect on the parameterswe propose to update the survival rates of the males and then the females separatelyOtherwise if there is no sex-effect we propose to update the male and female parametersusing the same procedure simultaneously For example suppose that there is a sex-effecton the parameters then for c = m (ie updating the survival rates of the males) t 2 Th

and r 2 Rlh we propose new parameter values

iquest0 ct (r) = iquest c

t(r) + u

where u is randomly drawn from some distribution For simplicity we set u sup1 U [iexcl deg deg ]where deg is chosen via pilot tuning We then accept the proposed move with the standardMetropolisndashHastings acceptance probability (Chib and Greenberg 1995 Brooks 1998)Any moves where the proposed parameters are outside the interval [0 1] are automaticallyrejected We then repeat the procedure for the females We use an analogous procedure forthe recapture and recovery probabilities A value of deg = 01 for each of the parametersappears to work well in practice

For the migration probabilitieswe need to retain the sum to unity constraint and so werequire a different updatingprocedure than for the other parameters We cycle through eachset of ages and propose to update each of the migration probabilities using the followingalgorithm For the set of ages Th we set s1 = 1 and randomly choose s2 2 Rnfs1g Thenas before if there is a sex-effect on the migration probabilities relating to the set of ages Thwe update the male and female parameters individually Else if there is no sex-effect wepropose to update both the male and female parameters using the same procedure Supposethat the parameters are independent of the sex of an animal Then for each c 2 C and agest 2 Th we propose new parameter values

Aacute0 ct (r s1) = Aacutec

t (r s1) + u


t (r s2) iexcl u

where u is drawn from some distribution In practice we set u sup1 U [ iexcl 01 01] chosen viapilot tuningAs before we reject any moves where the proposed migration probabilities arenot in the interval [0 1] We then repeat this procedure for s1 = 2 R Of course thereare many other updating schemes that could have been used However this scheme appearsto work well in that it is rapidly mixing and exploresparameter space with reasonablespeed

B2 BETWEEN-MODEL MOVES

Between-model moves are trans-dimensional in that they involve adding or deletingparameters from the model We therefore require the more general transition scheme knownas reversible jump MCMC (Green 1995)


The reversible jump procedure can be viewed as an extension of the MetropolisndashHastings algorithmwhich allows moves between different dimensionsBrie y when mov-ing from a lower to a higher dimensional model we propose values for the new parametersfrom some (essentially arbitrary) proposal distribution This move is then accepted withsome probability as with the MetropolisndashHastings updating scheme The reader is referredto Green (1995) and Richardson and Green (1997) for further general discussion

In the context of our mou on data between-model moves involve updating the groupsof ages locationand sex for the model parameters As an illustrationof the updatingschemewe consider in detail the updating of the sex-dependence on the parameters

Within each iteration of the Markov chain we cycle through each group of ages andlocation for each of the parameters and propose to add or remove the sex-dependence Thesurvival recapture and recovery probabilities have essentially the same updating schemewhile the migration probabilitiesneed to be considered separately due to their sum to unityconstraint

Initially we consider the set of survival probabilities and suppose that there is no sex-effect on a given set of ages Th Then within the reversible jump move we propose toadd a sex-effect by splitting the current parameters into two sets one for males and one forfemales We let M denote the current model with no sex-effect and correspondingparametervector micro and let micro0 denote the proposed set of parameters in model M 0 with a sex-effectRecall that the superscripts on the parameters relate to the set of sexes upon which theprobabilities are dependentm for males and f for females For the current model we notethat as there is no sex-effect on the parameters iquest m

t (r) = iquest ft (r) for all t 2 Th and r 2 R

Then for each age t 2 Th and region r 2 R we de ne the new survival probabilities forthe males and females to be

iquest0 mt (r) = iquest m

t (r) + deg t(r)

iquest0 ft (r) = iquest f

t (r) iexcl deg t(r)

Here we set deg t(r) = deg lh for all t 2 Th and r 2 Rlh where deg lh sup1 N (0 frac14 2) for some xedfrac14 2 chosen via pilot tuning For convenience we set sup2 = f deg lh l = 1 nhg where nh

denotes the number of location groups associated with the set of ages Th For this examplefrac14 2 = 001 appears to work well Elementary algebra gives the Jacobian jJ j = 2nh Thecorresponding acceptance probability is simply min(1 A1) for

A1 =L(micro0jx)p(micro0)p(M 0)jJ jL(microjx)p(micro)q(sup2)p(M )

where p(M ) denotes the prior for model M For the reverse where we propose to removethe sex-effect for the ages t 2 Th to retain the reversibility of the move we set

iquest0 mt (r) = iquest

0 ft (r) =

12

( iquest mt (r) + iquest f

t (r))

for each t 2 Th and r 2 R Then the corresponding acceptance probability for remov-ing sex-dependence on the parameters is simply the minimum of one and the reciprocal


of the above expression We use an analogous procedure for the recapture and recoveryprobabilities but use a different algorithm for the migration probabilities

In order to considermoving between migrationmodelswe need to ensure that we retainthe sum to unity constraint We cycle through each group of ages proposing to update thesex-effect on the migration probabilities in the following way Suppose that we propose tomove from (M micro) to (M 0 micro0) which involves increasing the dimension of the model byadding a sex-effect to a given parameter We propose to add a sex-effect to the migrationprobabilities in set Th so that in the current model we have Aacutem

t (r s) = Aacuteft (r s) for all

t 2 Th and r s 2 R Then for each t 2 Th and location r 2 R we set

Aacute0 mt (r s) = Aacutem

t (r s) + (r s)

Aacute0 ft (r s) = Aacutef

t (r s)

with probability 12 else we set


t (r s)


t (r s) + (r s)

for s = 1 R iexcl 1 where (r s) sup1 N (0 frac14 2) for some xed frac14 2 chosen via pilot tuningIn practice frac14 2 = 001 appears to work well in this example To ensure the sum to unityconstraint on the migration probabilities for each t 2 Th and r 2 R we set

Aacute0 mt (r R) = 1 iexcl

Riexcl1X

s = 1

Aacute0 mt (r s) Aacute

0 ft (r R) = 1 iexcl

Riexcl1X

s = 1

Aacute0 ft (r s)

The corresponding acceptance probability for proposing to add a sex-effect to themigration probabilities is then

A2 =L(micro0jx)p(micro0)p(M 0)

L(microjx)p(micro)q()p(M )

where = ((r s) r 2 R s = 1 R iexcl 1) and because the Jacobian is simply oneTo retain the reversibility of the move when we propose to remove the sex-effect from

the given set of ages Th we use the following procedure For each r s 2 R and t 2 Thwith probability 1

2 we set

Aacute0 mt (r s) = Aacute

0 ft (r s) = Aacutem

t (r s)

else we set


0 ft (r s) = Aacutef

t (r s)

Then the corresponding acceptance probability is again simply the minimum of unity andthe reciprocal of the above expression

Similar schemes can be used to update the groups of ages and locations see King andBrooks (2002) for further discussion


The MCMC simulation therefore involves sequentiallyupdating rst the current modelparameters and then the model itselfwithineach iterationThis providesus with a dependentsample from the joint posterior distribution from which we can derive empirical estimatesof posterior quantities of interest

An implementationof this algorithm in Fortran is available from the journal Web site

ACKNOWLEDGMENTSWe are very grateful to Jerome Dupuis with whom we have enjoyed many very helpful discussions with

regard to the analysis presented here We would also like to thank two anonymous referees and the AE for theirhelpful comments on an earlier draft of the article The authors gratefully acknowledge the nancial support ofthe EPSRC

[Received April 2002 Revised December 2002]

REFERENCES

Anderson D R Burnham K P and White G C (1994) ldquoAIC Model Selection in Overdispersed Capture-Recapture Datardquo Ecology 75 1780ndash1793

(1998) ldquoComparison of AIC and CAIC for Model Selection and Statistical Inference from CapturendashRecapture Studiesrdquo Applied Statistics 25 263ndash282

Brooks S P (1998) ldquoMarkov Chain Monte Carlo Method and its Applicationrdquo The Statistician 47 69ndash100

Brooks S P Catchpole E A and Morgan B J T (2000a) ldquoBayesian Animal Survival Estimationrdquo StatisticalScience 15 357ndash376

Brooks S P Catchpole E A Morgan B J T and Barry S C (2000b) ldquoOn the Bayesian Analysis of Ring-Recovery Datardquo Biometrics 56 951ndash956

Brooks S P and Roberts G O (1998) ldquoDiagnosing Convergence of Markov Chain Monte Carlo AlgorithmsrdquoStatistics and Computing 8 319ndash335

Brownie C Hines J E Nichols J D Pollock K H and Hestbeck J B (1993) ldquoCapturendashRecapture Studiesfor Multiple Strata including Non-Markovian Transition Probabilitiesrdquo Biometrics 49 1173ndash1187

Buckland S T Burnham K P and Augustin N H (1997) ldquoModel Selection An Integral Part of InferencerdquoBiometrics 53 603ndash618

Burnham K P and Anderson D R (1998) Model Selection and Inference A Practical Information-TheoreticApproach Springer New York

Burnham K P Anderson D R and White G C (1994) ldquoEvaluation of the KullbackndashLeibler Discrepancy forModel Selection in Open Population CapturendashRecapture Modelsrdquo Biometrical Journal 36 299ndash315

(1995a) ldquoSelection Among Open Population CapturendashRecapture Models when Capture-recapture areHeterogeneousrdquo Journal of Applied Statistics 22 611ndash624

Burnham K P White G C and AndersonD R (1995b) ldquoModel Selection Strategy in the Analysis of CapturendashRecapture Datardquo Biometrics 51 888ndash898

Castledine B J (1981) ldquoA Bayesian Analysis of Multiple-Recapture Sampling for a Closed PopulationrdquoBiometrika 67 197ndash210

Catchpole E A Freeman S N Morgan B J T and Harris M P (1998)ldquoIntegrated RecoveryRecapture DataAnalysisrdquo Biometrics 54 33ndash46


Catchpole E A Morgan B J T Coulson T N Freeman S N and Albon S D (2000) ldquoFactors In uencingSoay Sheep Survivalrdquo Journal of the Royal Statistical Society Series CmdashApplied Statistics 49 453ndash472

Chib S and Greenberg E (1995) ldquoUnderstanding the MetropolisndashHastings Algorithmrdquo The American Statis-tician 49 327ndash335

Cowles M K and Carlin B P (1996) ldquoMarkov Chain Monte Carlo Convergence Diagnostics A ComparativeReviewrdquo Journal of the American Statistical Association 91 883ndash904

Cransac N Gerard J F Maublanc M L and Oeoin D (1998) ldquoAn Example of Segregation Between Ageand Sex Classes Only Weakly Related to Habitat Use in Mou on Sheep (Ovis gmelini)rdquo Journal of Zoology244 371ndash378

Dubois M Bon R Maublanc M L and Cransac N (1994) ldquoDispersal Patterns of Corsican Mou on EwesImportance of Age and Proximate In uencesrdquo Applied Animal Behaviour Science 42 29ndash40

Dubois M Khazraie K Guilhem C Maublanc M L and LePendu Y (1996) ldquoPhilopatry in Mou on RamsDuring the Rutting Season Psycho-ethological Determinism and Functional Consequencesrdquo BehaviouralProcesses 35 93ndash100

Dupuis J A (1995) ldquoBayesian Estimation of Movement and Survival Probabilities from CapturendashRecaptureDatardquo Biometrika 82 761ndash772

Dupuis J A Badia J Maublanc M and Bon R (2002) ldquoSurvival and Spatial Fidelity of Mou ons (Ovisgmelini) A Bayesian Analysis of an Age-dependent CapturendashRecapture Modelrdquo Journal of AgriculturalBiological and Environmental Statistics 7 277ndash298

Gamerman D (1997) Markov Chain Monte Carlo Stochastic Simulation for Bayesian Inference New YorkChapman and Hall

Gelfand A E and Smith A F M (1990) ldquoSampling Based Approaches to Calculating Marginal DensitiesrdquoJournal of the American Statistical Association 85 398ndash409

George E I and Robert C P (1992) ldquoCapturendashRecapture Estimation via Gibbs Samplingrdquo Biometrika 79677ndash683

Green P J (1995) ldquoReversible Jump MCMC Computation and Bayesian Model Determinationrdquo Biometrika 82711ndash732

Hestbeck J B Nichols J D and Malecki R A (1991) ldquoEstimation of Movement and Site Fidelity usingMark-Resight Data on Wintering Canada Geeserdquo Ecology 72 523ndash533

Hoeting J A Madigan D Raftery A E and VolinskyC T (1999) ldquoBayesian Model Averaging A TutorialrdquoStatistical Science 14 382ndash401

Janz R J (1980) ldquoPrior Knowledge and Ornithologyrdquo in Statistics in Ornithology eds B J T Morgan and P MNorth Berlin Springer-Verlag pp 303ndash310

JollyG M (1965)ldquoExplicitEstimates from CapturendashRecapture Data withbothDeath and Immigration-StochasticModelrdquo Biometrika 52 225ndash247

Kass R E and Raftery A E (1995) ldquoBayes Factorsrdquo Journal of the American Statistical Association 90773ndash795

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

(2002) ldquoBayesian Model Discrimination for Multiple Strata CapturendashRecapture Datardquo Biometrika 89785ndash806

(2003) ldquoA Note on Closed Form Likelihoods for Arnason-Schwarz Modelsrdquo Biometrika 90 435ndash444

LebretonJ-DBurnhamK P Clobert J andAndersonD R (1992)ldquoModelingSurvivaland testing BiologicalHypotheses using Marked Animals A Uni ed Approach with Case Studiesrdquo Ecological Monographs 6267ndash118

LePenduY Maublanc M L Briedermann L and DuboisM (1996)ldquoSpatial Structure and Activity in Groupsof Mediterranean Mou on (Ovis gmelini) A Comparative Studyrdquo Applied Animal Behaviour Science 46201ndash216

Link W A and Cam E (2000) ldquoOf BUGS and Birds An Introduction to Markov Chain Monte Carlordquounpublished manuscript


Madigan D and Raftery A E (1994) ldquoModel Selection and Accounting for Model Uncertainty in GraphicalModels Using Occamrsquos Windowrdquo Journal of the American Statistical Association 89 1535ndash1547

Madigan D and York J C (1997) ldquoBayesian Methods for Estimation of the Size of a Closed PopulationrdquoBiometrika 84 19ndash31

Norris J L and PollockK H (1997) ldquoIncludingModel Uncertainty in Estimating Variances in MultipleCaptureStudiesrdquo Environmental and Ecological Statistics 3 235ndash244

OrsquoHagan A O (1998) ldquoEliciting Expert Beliefs in Substantial Practical Applicationsrdquo The Statistician 4721ndash36

Pollock K H (1991) ldquoModelling Capture Recapture and Removal Statistics for Estimation of DemographicParameters for Fish and Wildlife Populations Past Present and Futurerdquo Journal of the American StatisticalAssociation 86 225ndash238

Richardson S and Green P J (1997) ldquoOn Bayesian Analysis of Mixtures with an Unknown Number of Com-ponentsrdquo Journal of the Royal Statistical Society Series B 59 731ndash792

Schwarz C G Schweigert J F and Arnason A N (1993) ldquoEstimating Migration Rates using Tag-RecoveryDatardquo Biometrics 49 177ndash193

Schwarz C J and Seber G A F (1998) ldquoA Review of Estimating Animal Abundance IIIrdquo Statistical Science14 427ndash456

Seber G A F (1965) ldquoA Note on the Multiple-Recapture Censusrdquo Biometrika 52 249ndash259

Smith P J (1988) ldquoBayesian Methods for CapturendashRecapture Surveysrdquo Biometrics 44 1177ndash1189

Underhill L G (1990) ldquoBayesian Estimation of the Size of a Closed Populationrdquo The Ring 13 235ndash254

Vounatsou P and Smith A F M (1995) ldquoBayesian Analysis of Ring-Recovery Data via Markov Chain MonteCarlordquo Biometrics 51 687ndash708


ing population size to more complex open population studies in which survival recoveryrecapture and possibly migration rates are of primary interest (Dupuis 1995 Schwarz andSeber 1998 Hestbeck Nichols and Malecki 1991)

One advantageof the Bayesian approach to statistical analysis for these kinds of studiesis the prevalence of usable expert knowledge which can be incorporated into the analysiseither to estimate population size (Castledine 1981 Smith 1988 Underhill 1990 Georgeand Robert 1992 Madigan and York 1997 King and Brooks 2001) or survival probabilities(Janz 1980 Link and Cam 2000 Brooks Catchpole and Morgan 2000a Vounatsou andSmith 1995 Brooks Catchpole Morgan and Barry 2000c) A second advantage of theBayesian approach is the availability of powerful new computational techniques capable oftackling the kinds of models that fully re ect the complexity of the dynamics driving thepopulations under study (Brooks 1998 Gamerman 1997 Brooks Catchpole and Morgan2000a) As the restrictions traditionally imposed by analytic tractability are removed weare faced with a new problem in exploring the enormous number of competing modelsavailable and in understanding what each tells us about the population

In the context of recoveryrecapture studies where we typically have three primaryparameters of interest (survival recapture and recovery rates) allowing for the possibilityof say age dependencealone gives rise to a huge class of competing models all of which tellus something different about the underlying dynamics of the population In the presenceof location information and covariate information such as sex the class of biologicallyplausible competing models quickly becomes enormous

Traditionallyin the presence of moderate classes of competingmodels two approachesdominate The rst is to ignore model selection altogether and to take the ldquoglobal modelrdquoin which each parameter depends upon everything In this case we obtain reasonable pa-rameter estimates in general but unless the associated dataset is large these estimates cansuffer from poor resolution Another drawback is that no structural understanding of theunderlying population dynamics is obtained and this is often a more important goal thanthat of parameter estimation as it is through the structure that we begin to learn about the in-teractions within the system that are vital for predictive inference for example The secondapproach is to use model selection techniques and these have proved popular both in gen-eral and in the context of recoveryrecapture data (Burnham and Anderson 1998) Becauseexhaustive comparisons are usually impossible in practice traditional approaches to thisproblem generally adopt a sequential approach starting with the ldquoglobalrdquo model in whichall parameters are fully age and year dependent and then successively combining years orages either randomly or on the basis of biological arguments Statistical tests such as thelikelihoodratio test (Lebreton et al 1992) and AIC criteria (Anderson Burnham and White1994) are then used to determine whether or not an improvement is gained See also An-derson Burnham and White (1998) Buckland Burnham and Augustin (1997) BurnhamAnderson and White (1994 1995a) Burnham White and Anderson (1995b) and Norrisand Pollock (1997) for example However sequential approaches are not guaranteed toobtain the ldquobestrdquo model and often highly plausible models remain completely unexplored



11 DATA
















i = 1 L(xjmi)p(mi)

where

L(xjmi) =




ordm (regjx) =

kX

i = 1














and




t(r) =






t1(r) = iquest c

t2(r) for all



t1(r) = iquest c













32 PRIORS









t(r cent) =P

s2 R nct(r s)



not c(l h) = 1 +X

t 2 Th

X

r 2 Rlh

nct(r cent)

shy c(l h) = 1 +X

t 2 Th

X

r 2 Rlh

nct(r y)



h(r 1) ech(r 2) ec

h(r 3)) where

ech(r s) = 1 +

X

t2 Th

nct(r s)





(a) (b)



322 Model Priors




Fig

ure

3

The

pri

or m

ean

mig

ratio

n ra

tes

for

each

reg

ion

for

the

mal

es (

figur

es (

a)ndash(

c))

and

fem

ales

(fig

ures

(d)

ndash(f)

) to

eac

h of

the

poss

ible

are

as w

ith

corr

espo

ndin

g 95

H

PD

Is (

repr

esen

ted

as v

erti

cal l

ines

) w

here

the

dotte

d lin

e (

)

rep

rese

nts

area

1 t

he s

olid

line

(mdash

mdashmdash

) ar

ea 2

and

the

dash

ed li

ne(-

- -

) ar

ea 3

(a)

(b)

(c) (f

)(e

)(d

)




4 RESULTS

















(a) (b)














(a) (b)



(a) (b)








location groupings











































Fig

ure

7

The

pos

teri

or m

ean

mig

ratio

n ra

tes

for

each

reg

ion

for

the

mal

es (

figur

es (

a)ndash(

c))

and

fem

ales

(fig

ures

(d)

ndash(f)

) to

eac

h of

the

pos

sibl

e ar

eas

with

cor

resp

ondi

ng 9

5 H

PD

Is (

repr

esen

ted

as v

erti

cal l

ines

) w

here

the

dotte

d li

ne (

) re

pres

ents

are

a 1

the

soli

d li

ne (

mdashmdash

mdash) a

rea

2 a

nd th

e da

shed

line

(- -

-)

area

3

(a)

(b)

(c)

(f)

(e)

(d)






















parameters





yT ordf = yT (41)


constraint thatP













(a) (b)


5 DISCUSSION






APPENDIXES

A LIKELIHOOD


L(xjmicro) =Y

c2 C

Y

r 2 R

2

4TY

t = 0

TY

j = t

Agrave c(tj)(r)vc

(tj)(r)T iexcl1Y

t = 0

tY

l = 0

Y

s2 ROc

(lt)(r s)nc(lt)(rs)

poundT iexcl1Y

t= 0

tY

l = 0

Dc(lt)(r)dc

(lt)(r)

(A1)














t(r)

Atilde

1 iexclX

s2 RAacutec


(t + 1j)(s)


t(r)

where





(lt)(r s)



Qc(lt)(r s) =

(iquest c


iquest cl (r)

Pk 2 R (1 iexcl pc





Dc(lt)(r) =

((1 iexcl iquest c






B MCMC SIMULATION







t(r) + u




t (r s1) + u


t (r s2) iexcl u












t (r) + deg t(r)








0 ft (r) =

12


t (r))








t (r s) + (r s)


t (r s)



t (r s)


t (r s) + (r s)



Riexcl1X

s = 1



Riexcl1X

s = 1

Aacute0 ft (r s)






2 we set



t (r s)

else we set



t (r s)









REFERENCES






















































11 DATA
















i = 1 L(xjmi)p(mi)

where

L(xjmi) =




ordm (regjx) =

kX

i = 1














and




t(r) =






t1(r) = iquest c

t2(r) for all



t1(r) = iquest c













32 PRIORS









t(r cent) =P

s2 R nct(r s)



not c(l h) = 1 +X

t 2 Th

X

r 2 Rlh

nct(r cent)

shy c(l h) = 1 +X

t 2 Th

X

r 2 Rlh

nct(r y)



h(r 1) ech(r 2) ec

h(r 3)) where

ech(r s) = 1 +

X

t2 Th

nct(r s)





(a) (b)



322 Model Priors




Fig

ure

3

The

pri

or m

ean

mig

ratio

n ra

tes

for

each

reg

ion

for

the

mal

es (

figur

es (

a)ndash(

c))

and

fem

ales

(fig

ures

(d)

ndash(f)

) to

eac

h of

the

poss

ible

are

as w

ith

corr

espo

ndin

g 95

H

PD

Is (

repr

esen

ted

as v

erti

cal l

ines

) w

here

the

dotte

d lin

e (

)

rep

rese

nts

area

1 t

he s

olid

line

(mdash

mdashmdash

) ar

ea 2

and

the

dash

ed li

ne(-

- -

) ar

ea 3

(a)

(b)

(c) (f

)(e

)(d

)




4 RESULTS

















(a) (b)














(a) (b)



(a) (b)








location groupings











































Fig

ure

7

The

pos

teri

or m

ean

mig

ratio

n ra

tes

for

each

reg

ion

for

the

mal

es (

figur

es (

a)ndash(

c))

and

fem

ales

(fig

ures

(d)

ndash(f)

) to

eac

h of

the

pos

sibl

e ar

eas

with

cor

resp

ondi

ng 9

5 H

PD

Is (

repr

esen

ted

as v

erti

cal l

ines

) w

here

the

dotte

d li

ne (

) re

pres

ents

are

a 1

the

soli

d li

ne (

mdashmdash

mdash) a

rea

2 a

nd th

e da

shed

line

(- -

-)

area

3

(a)

(b)

(c)

(f)

(e)

(d)






















parameters





yT ordf = yT (41)


constraint thatP













(a) (b)


5 DISCUSSION






APPENDIXES

A LIKELIHOOD


L(xjmicro) =Y

c2 C

Y

r 2 R

2

4TY

t = 0

TY

j = t

Agrave c(tj)(r)vc

(tj)(r)T iexcl1Y

t = 0

tY

l = 0

Y

s2 ROc

(lt)(r s)nc(lt)(rs)

poundT iexcl1Y

t= 0

tY

l = 0

Dc(lt)(r)dc

(lt)(r)

(A1)














t(r)

Atilde

1 iexclX

s2 RAacutec


(t + 1j)(s)


t(r)

where





(lt)(r s)



Qc(lt)(r s) =

(iquest c


iquest cl (r)

Pk 2 R (1 iexcl pc





Dc(lt)(r) =

((1 iexcl iquest c






B MCMC SIMULATION







t(r) + u




t (r s1) + u


t (r s2) iexcl u












t (r) + deg t(r)








0 ft (r) =

12


t (r))








t (r s) + (r s)


t (r s)



t (r s)


t (r s) + (r s)



Riexcl1X

s = 1



Riexcl1X

s = 1

Aacute0 ft (r s)






2 we set



t (r s)

else we set



t (r s)









REFERENCES
































































i = 1 L(xjmi)p(mi)

where

L(xjmi) =




ordm (regjx) =

kX

i = 1














and




t(r) =






t1(r) = iquest c

t2(r) for all



t1(r) = iquest c













32 PRIORS









t(r cent) =P

s2 R nct(r s)



not c(l h) = 1 +X

t 2 Th

X

r 2 Rlh

nct(r cent)

shy c(l h) = 1 +X

t 2 Th

X

r 2 Rlh

nct(r y)



h(r 1) ech(r 2) ec

h(r 3)) where

ech(r s) = 1 +

X

t2 Th

nct(r s)





(a) (b)



322 Model Priors




Fig

ure

3

The

pri

or m

ean

mig

ratio

n ra

tes

for

each

reg

ion

for

the

mal

es (

figur

es (

a)ndash(

c))

and

fem

ales

(fig

ures

(d)

ndash(f)

) to

eac

h of

the

poss

ible

are

as w

ith

corr

espo

ndin

g 95

H

PD

Is (

repr

esen

ted

as v

erti

cal l

ines

) w

here

the

dotte

d lin

e (

)

rep

rese

nts

area

1 t

he s

olid

line

(mdash

mdashmdash

) ar

ea 2

and

the

dash

ed li

ne(-

- -

) ar

ea 3

(a)

(b)

(c) (f

)(e

)(d

)




4 RESULTS

















(a) (b)














(a) (b)



(a) (b)








location groupings











































Fig

ure

7

The

pos

teri

or m

ean

mig

ratio

n ra

tes

for

each

reg

ion

for

the

mal

es (

figur

es (

a)ndash(

c))

and

fem

ales

(fig

ures

(d)

ndash(f)

) to

eac

h of

the

pos

sibl

e ar

eas

with

cor

resp

ondi

ng 9

5 H

PD

Is (

repr

esen

ted

as v

erti

cal l

ines

) w

here

the

dotte

d li

ne (

) re

pres

ents

are

a 1

the

soli

d li

ne (

mdashmdash

mdash) a

rea

2 a

nd th

e da

shed

line

(- -

-)

area

3

(a)

(b)

(c)

(f)

(e)

(d)






















parameters





yT ordf = yT (41)


constraint thatP













(a) (b)


5 DISCUSSION






APPENDIXES

A LIKELIHOOD


L(xjmicro) =Y

c2 C

Y

r 2 R

2

4TY

t = 0

TY

j = t

Agrave c(tj)(r)vc

(tj)(r)T iexcl1Y

t = 0

tY

l = 0

Y

s2 ROc

(lt)(r s)nc(lt)(rs)

poundT iexcl1Y

t= 0

tY

l = 0

Dc(lt)(r)dc

(lt)(r)

(A1)














t(r)

Atilde

1 iexclX

s2 RAacutec


(t + 1j)(s)


t(r)

where





(lt)(r s)



Qc(lt)(r s) =

(iquest c


iquest cl (r)

Pk 2 R (1 iexcl pc





Dc(lt)(r) =

((1 iexcl iquest c






B MCMC SIMULATION







t(r) + u




t (r s1) + u


t (r s2) iexcl u












t (r) + deg t(r)








0 ft (r) =

12


t (r))








t (r s) + (r s)


t (r s)



t (r s)


t (r s) + (r s)



Riexcl1X

s = 1



Riexcl1X

s = 1

Aacute0 ft (r s)






2 we set



t (r s)

else we set



t (r s)









REFERENCES























































i = 1 L(xjmi)p(mi)

where

L(xjmi) =




ordm (regjx) =

kX

i = 1














and




t(r) =






t1(r) = iquest c

t2(r) for all



t1(r) = iquest c













32 PRIORS









t(r cent) =P

s2 R nct(r s)



not c(l h) = 1 +X

t 2 Th

X

r 2 Rlh

nct(r cent)

shy c(l h) = 1 +X

t 2 Th

X

r 2 Rlh

nct(r y)



h(r 1) ech(r 2) ec

h(r 3)) where

ech(r s) = 1 +

X

t2 Th

nct(r s)





(a) (b)



322 Model Priors




Fig

ure

3

The

pri

or m

ean

mig

ratio

n ra

tes

for

each

reg

ion

for

the

mal

es (

figur

es (

a)ndash(

c))

and

fem

ales

(fig

ures

(d)

ndash(f)

) to

eac

h of

the

poss

ible

are

as w

ith

corr

espo

ndin

g 95

H

PD

Is (

repr

esen

ted

as v

erti

cal l

ines

) w

here

the

dotte

d lin

e (

)

rep

rese

nts

area

1 t

he s

olid

line

(mdash

mdashmdash

) ar

ea 2

and

the

dash

ed li

ne(-

- -

) ar

ea 3

(a)

(b)

(c) (f

)(e

)(d

)




4 RESULTS

















(a) (b)














(a) (b)



(a) (b)








location groupings











































Fig

ure

7

The

pos

teri

or m

ean

mig

ratio

n ra

tes

for

each

reg

ion

for

the

mal

es (

figur

es (

a)ndash(

c))

and

fem

ales

(fig

ures

(d)

ndash(f)

) to

eac

h of

the

pos

sibl

e ar

eas

with

cor

resp

ondi

ng 9

5 H

PD

Is (

repr

esen

ted

as v

erti

cal l

ines

) w

here

the

dotte

d li

ne (

) re

pres

ents

are

a 1

the

soli

d li

ne (

mdashmdash

mdash) a

rea

2 a

nd th

e da

shed

line

(- -

-)

area

3

(a)

(b)

(c)

(f)

(e)

(d)






















parameters





yT ordf = yT (41)


constraint thatP













(a) (b)


5 DISCUSSION






APPENDIXES

A LIKELIHOOD


L(xjmicro) =Y

c2 C

Y

r 2 R

2

4TY

t = 0

TY

j = t

Agrave c(tj)(r)vc

(tj)(r)T iexcl1Y

t = 0

tY

l = 0

Y

s2 ROc

(lt)(r s)nc(lt)(rs)

poundT iexcl1Y

t= 0

tY

l = 0

Dc(lt)(r)dc

(lt)(r)

(A1)














t(r)

Atilde

1 iexclX

s2 RAacutec


(t + 1j)(s)


t(r)

where





(lt)(r s)



Qc(lt)(r s) =

(iquest c


iquest cl (r)

Pk 2 R (1 iexcl pc





Dc(lt)(r) =

((1 iexcl iquest c






B MCMC SIMULATION







t(r) + u




t (r s1) + u


t (r s2) iexcl u












t (r) + deg t(r)








0 ft (r) =

12


t (r))








t (r s) + (r s)


t (r s)



t (r s)


t (r s) + (r s)



Riexcl1X

s = 1



Riexcl1X

s = 1

Aacute0 ft (r s)






2 we set



t (r s)

else we set



t (r s)









REFERENCES





























































and




t(r) =






t1(r) = iquest c

t2(r) for all



t1(r) = iquest c













32 PRIORS









t(r cent) =P

s2 R nct(r s)



not c(l h) = 1 +X

t 2 Th

X

r 2 Rlh

nct(r cent)

shy c(l h) = 1 +X

t 2 Th

X

r 2 Rlh

nct(r y)



h(r 1) ech(r 2) ec

h(r 3)) where

ech(r s) = 1 +

X

t2 Th

nct(r s)





(a) (b)



322 Model Priors




Fig

ure

3

The

pri

or m

ean

mig

ratio

n ra

tes

for

each

reg

ion

for

the

mal

es (

figur

es (

a)ndash(

c))

and

fem

ales

(fig

ures

(d)

ndash(f)

) to

eac

h of

the

poss

ible

are

as w

ith

corr

espo

ndin

g 95

H

PD

Is (

repr

esen

ted

as v

erti

cal l

ines

) w

here

the

dotte

d lin

e (

)

rep

rese

nts

area

1 t

he s

olid

line

(mdash

mdashmdash

) ar

ea 2

and

the

dash

ed li

ne(-

- -

) ar

ea 3

(a)

(b)

(c) (f

)(e

)(d

)




4 RESULTS

















(a) (b)














(a) (b)



(a) (b)








location groupings











































Fig

ure

7

The

pos

teri

or m

ean

mig

ratio

n ra

tes

for

each

reg

ion

for

the

mal

es (

figur

es (

a)ndash(

c))

and

fem

ales

(fig

ures

(d)

ndash(f)

) to

eac

h of

the

pos

sibl

e ar

eas

with

cor

resp

ondi

ng 9

5 H

PD

Is (

repr

esen

ted

as v

erti

cal l

ines

) w

here

the

dotte

d li

ne (

) re

pres

ents

are

a 1

the

soli

d li

ne (

mdashmdash

mdash) a

rea

2 a

nd th

e da

shed

line

(- -

-)

area

3

(a)

(b)

(c)

(f)

(e)

(d)






















parameters





yT ordf = yT (41)


constraint thatP













(a) (b)


5 DISCUSSION






APPENDIXES

A LIKELIHOOD


L(xjmicro) =Y

c2 C

Y

r 2 R

2

4TY

t = 0

TY

j = t

Agrave c(tj)(r)vc

(tj)(r)T iexcl1Y

t = 0

tY

l = 0

Y

s2 ROc

(lt)(r s)nc(lt)(rs)

poundT iexcl1Y

t= 0

tY

l = 0

Dc(lt)(r)dc

(lt)(r)

(A1)














t(r)

Atilde

1 iexclX

s2 RAacutec


(t + 1j)(s)


t(r)

where





(lt)(r s)



Qc(lt)(r s) =

(iquest c


iquest cl (r)

Pk 2 R (1 iexcl pc





Dc(lt)(r) =

((1 iexcl iquest c






B MCMC SIMULATION







t(r) + u




t (r s1) + u


t (r s2) iexcl u












t (r) + deg t(r)








0 ft (r) =

12


t (r))








t (r s) + (r s)


t (r s)



t (r s)


t (r s) + (r s)



Riexcl1X

s = 1



Riexcl1X

s = 1

Aacute0 ft (r s)






2 we set



t (r s)

else we set



t (r s)









REFERENCES
























































t1(r) = iquest c

t2(r) for all



t1(r) = iquest c













32 PRIORS









t(r cent) =P

s2 R nct(r s)



not c(l h) = 1 +X

t 2 Th

X

r 2 Rlh

nct(r cent)

shy c(l h) = 1 +X

t 2 Th

X

r 2 Rlh

nct(r y)



h(r 1) ech(r 2) ec

h(r 3)) where

ech(r s) = 1 +

X

t2 Th

nct(r s)





(a) (b)



322 Model Priors




Fig

ure

3

The

pri

or m

ean

mig

ratio

n ra

tes

for

each

reg

ion

for

the

mal

es (

figur

es (

a)ndash(

c))

and

fem

ales

(fig

ures

(d)

ndash(f)

) to

eac

h of

the

poss

ible

are

as w

ith

corr

espo

ndin

g 95

H

PD

Is (

repr

esen

ted

as v

erti

cal l

ines

) w

here

the

dotte

d lin

e (

)

rep

rese

nts

area

1 t

he s

olid

line

(mdash

mdashmdash

) ar

ea 2

and

the

dash

ed li

ne(-

- -

) ar

ea 3

(a)

(b)

(c) (f

)(e

)(d

)




4 RESULTS

















(a) (b)














(a) (b)



(a) (b)








location groupings











































Fig

ure

7

The

pos

teri

or m

ean

mig

ratio

n ra

tes

for

each

reg

ion

for

the

mal

es (

figur

es (

a)ndash(

c))

and

fem

ales

(fig

ures

(d)

ndash(f)

) to

eac

h of

the

pos

sibl

e ar

eas

with

cor

resp

ondi

ng 9

5 H

PD

Is (

repr

esen

ted

as v

erti

cal l

ines

) w

here

the

dotte

d li

ne (

) re

pres

ents

are

a 1

the

soli

d li

ne (

mdashmdash

mdash) a

rea

2 a

nd th

e da

shed

line

(- -

-)

area

3

(a)

(b)

(c)

(f)

(e)

(d)






















parameters





yT ordf = yT (41)


constraint thatP













(a) (b)


5 DISCUSSION






APPENDIXES

A LIKELIHOOD


L(xjmicro) =Y

c2 C

Y

r 2 R

2

4TY

t = 0

TY

j = t

Agrave c(tj)(r)vc

(tj)(r)T iexcl1Y

t = 0

tY

l = 0

Y

s2 ROc

(lt)(r s)nc(lt)(rs)

poundT iexcl1Y

t= 0

tY

l = 0

Dc(lt)(r)dc

(lt)(r)

(A1)














t(r)

Atilde

1 iexclX

s2 RAacutec


(t + 1j)(s)


t(r)

where





(lt)(r s)



Qc(lt)(r s) =

(iquest c


iquest cl (r)

Pk 2 R (1 iexcl pc





Dc(lt)(r) =

((1 iexcl iquest c






B MCMC SIMULATION







t(r) + u




t (r s1) + u


t (r s2) iexcl u












t (r) + deg t(r)








0 ft (r) =

12


t (r))








t (r s) + (r s)


t (r s)



t (r s)


t (r s) + (r s)



Riexcl1X

s = 1



Riexcl1X

s = 1

Aacute0 ft (r s)






2 we set



t (r s)

else we set



t (r s)









REFERENCES


























































32 PRIORS









t(r cent) =P

s2 R nct(r s)



not c(l h) = 1 +X

t 2 Th

X

r 2 Rlh

nct(r cent)

shy c(l h) = 1 +X

t 2 Th

X

r 2 Rlh

nct(r y)



h(r 1) ech(r 2) ec

h(r 3)) where

ech(r s) = 1 +

X

t2 Th

nct(r s)





(a) (b)



322 Model Priors




Fig

ure

3

The

pri

or m

ean

mig

ratio

n ra

tes

for

each

reg

ion

for

the

mal

es (

figur

es (

a)ndash(

c))

and

fem

ales

(fig

ures

(d)

ndash(f)

) to

eac

h of

the

poss

ible

are

as w

ith

corr

espo

ndin

g 95

H

PD

Is (

repr

esen

ted

as v

erti

cal l

ines

) w

here

the

dotte

d lin

e (

)

rep

rese

nts

area

1 t

he s

olid

line

(mdash

mdashmdash

) ar

ea 2

and

the

dash

ed li

ne(-

- -

) ar

ea 3

(a)

(b)

(c) (f

)(e

)(d

)




4 RESULTS

















(a) (b)














(a) (b)



(a) (b)








location groupings











































Fig

ure

7

The

pos

teri

or m

ean

mig

ratio

n ra

tes

for

each

reg

ion

for

the

mal

es (

figur

es (

a)ndash(

c))

and

fem

ales

(fig

ures

(d)

ndash(f)

) to

eac

h of

the

pos

sibl

e ar

eas

with

cor

resp

ondi

ng 9

5 H

PD

Is (

repr

esen

ted

as v

erti

cal l

ines

) w

here

the

dotte

d li

ne (

) re

pres

ents

are

a 1

the

soli

d li

ne (

mdashmdash

mdash) a

rea

2 a

nd th

e da

shed

line

(- -

-)

area

3

(a)

(b)

(c)

(f)

(e)

(d)






















parameters





yT ordf = yT (41)


constraint thatP













(a) (b)


5 DISCUSSION






APPENDIXES

A LIKELIHOOD


L(xjmicro) =Y

c2 C

Y

r 2 R

2

4TY

t = 0

TY

j = t

Agrave c(tj)(r)vc

(tj)(r)T iexcl1Y

t = 0

tY

l = 0

Y

s2 ROc

(lt)(r s)nc(lt)(rs)

poundT iexcl1Y

t= 0

tY

l = 0

Dc(lt)(r)dc

(lt)(r)

(A1)














t(r)

Atilde

1 iexclX

s2 RAacutec


(t + 1j)(s)


t(r)

where





(lt)(r s)



Qc(lt)(r s) =

(iquest c


iquest cl (r)

Pk 2 R (1 iexcl pc





Dc(lt)(r) =

((1 iexcl iquest c






B MCMC SIMULATION







t(r) + u




t (r s1) + u


t (r s2) iexcl u












t (r) + deg t(r)








0 ft (r) =

12


t (r))








t (r s) + (r s)


t (r s)



t (r s)


t (r s) + (r s)



Riexcl1X

s = 1



Riexcl1X

s = 1

Aacute0 ft (r s)






2 we set



t (r s)

else we set



t (r s)









REFERENCES



























































t(r cent) =P

s2 R nct(r s)



not c(l h) = 1 +X

t 2 Th

X

r 2 Rlh

nct(r cent)

shy c(l h) = 1 +X

t 2 Th

X

r 2 Rlh

nct(r y)



h(r 1) ech(r 2) ec

h(r 3)) where

ech(r s) = 1 +

X

t2 Th

nct(r s)





(a) (b)



322 Model Priors




Fig

ure

3

The

pri

or m

ean

mig

ratio

n ra

tes

for

each

reg

ion

for

the

mal

es (

figur

es (

a)ndash(

c))

and

fem

ales

(fig

ures

(d)

ndash(f)

) to

eac

h of

the

poss

ible

are

as w

ith

corr

espo

ndin

g 95

H

PD

Is (

repr

esen

ted

as v

erti

cal l

ines

) w

here

the

dotte

d lin

e (

)

rep

rese

nts

area

1 t

he s

olid

line

(mdash

mdashmdash

) ar

ea 2

and

the

dash

ed li

ne(-

- -

) ar

ea 3

(a)

(b)

(c) (f

)(e

)(d

)




4 RESULTS

















(a) (b)














(a) (b)



(a) (b)








location groupings











































Fig

ure

7

The

pos

teri

or m

ean

mig

ratio

n ra

tes

for

each

reg

ion

for

the

mal

es (

figur

es (

a)ndash(

c))

and

fem

ales

(fig

ures

(d)

ndash(f)

) to

eac

h of

the

pos

sibl

e ar

eas

with

cor

resp

ondi

ng 9

5 H

PD

Is (

repr

esen

ted

as v

erti

cal l

ines

) w

here

the

dotte

d li

ne (

) re

pres

ents

are

a 1

the

soli

d li

ne (

mdashmdash

mdash) a

rea

2 a

nd th

e da

shed

line

(- -

-)

area

3

(a)

(b)

(c)

(f)

(e)

(d)






















parameters





yT ordf = yT (41)


constraint thatP













(a) (b)


5 DISCUSSION






APPENDIXES

A LIKELIHOOD


L(xjmicro) =Y

c2 C

Y

r 2 R

2

4TY

t = 0

TY

j = t

Agrave c(tj)(r)vc

(tj)(r)T iexcl1Y

t = 0

tY

l = 0

Y

s2 ROc

(lt)(r s)nc(lt)(rs)

poundT iexcl1Y

t= 0

tY

l = 0

Dc(lt)(r)dc

(lt)(r)

(A1)














t(r)

Atilde

1 iexclX

s2 RAacutec


(t + 1j)(s)


t(r)

where





(lt)(r s)



Qc(lt)(r s) =

(iquest c


iquest cl (r)

Pk 2 R (1 iexcl pc





Dc(lt)(r) =

((1 iexcl iquest c






B MCMC SIMULATION







t(r) + u




t (r s1) + u


t (r s2) iexcl u












t (r) + deg t(r)








0 ft (r) =

12


t (r))








t (r s) + (r s)


t (r s)



t (r s)


t (r s) + (r s)



Riexcl1X

s = 1



Riexcl1X

s = 1

Aacute0 ft (r s)






2 we set



t (r s)

else we set



t (r s)









REFERENCES





















































(a) (b)



322 Model Priors




Fig

ure

3

The

pri

or m

ean

mig

ratio

n ra

tes

for

each

reg

ion

for

the

mal

es (

figur

es (

a)ndash(

c))

and

fem

ales

(fig

ures

(d)

ndash(f)

) to

eac

h of

the

poss

ible

are

as w

ith

corr

espo

ndin

g 95

H

PD

Is (

repr

esen

ted

as v

erti

cal l

ines

) w

here

the

dotte

d lin

e (

)

rep

rese

nts

area

1 t

he s

olid

line

(mdash

mdashmdash

) ar

ea 2

and

the

dash

ed li

ne(-

- -

) ar

ea 3

(a)

(b)

(c) (f

)(e

)(d

)




4 RESULTS

















(a) (b)














(a) (b)



(a) (b)








location groupings











































Fig

ure

7

The

pos

teri

or m

ean

mig

ratio

n ra

tes

for

each

reg

ion

for

the

mal

es (

figur

es (

a)ndash(

c))

and

fem

ales

(fig

ures

(d)

ndash(f)

) to

eac

h of

the

pos

sibl

e ar

eas

with

cor

resp

ondi

ng 9

5 H

PD

Is (

repr

esen

ted

as v

erti

cal l

ines

) w

here

the

dotte

d li

ne (

) re

pres

ents

are

a 1

the

soli

d li

ne (

mdashmdash

mdash) a

rea

2 a

nd th

e da

shed

line

(- -

-)

area

3

(a)

(b)

(c)

(f)

(e)

(d)






















parameters





yT ordf = yT (41)


constraint thatP













(a) (b)


5 DISCUSSION






APPENDIXES

A LIKELIHOOD


L(xjmicro) =Y

c2 C

Y

r 2 R

2

4TY

t = 0

TY

j = t

Agrave c(tj)(r)vc

(tj)(r)T iexcl1Y

t = 0

tY

l = 0

Y

s2 ROc

(lt)(r s)nc(lt)(rs)

poundT iexcl1Y

t= 0

tY

l = 0

Dc(lt)(r)dc

(lt)(r)

(A1)














t(r)

Atilde

1 iexclX

s2 RAacutec


(t + 1j)(s)


t(r)

where





(lt)(r s)



Qc(lt)(r s) =

(iquest c


iquest cl (r)

Pk 2 R (1 iexcl pc





Dc(lt)(r) =

((1 iexcl iquest c






B MCMC SIMULATION







t(r) + u




t (r s1) + u


t (r s2) iexcl u












t (r) + deg t(r)








0 ft (r) =

12


t (r))








t (r s) + (r s)


t (r s)



t (r s)


t (r s) + (r s)



Riexcl1X

s = 1



Riexcl1X

s = 1

Aacute0 ft (r s)






2 we set



t (r s)

else we set



t (r s)









REFERENCES





















































Fig

ure

3

The

pri

or m

ean

mig

ratio

n ra

tes

for

each

reg

ion

for

the

mal

es (

figur

es (

a)ndash(

c))

and

fem

ales

(fig

ures

(d)

ndash(f)

) to

eac

h of

the

poss

ible

are

as w

ith

corr

espo

ndin

g 95

H

PD

Is (

repr

esen

ted

as v

erti

cal l

ines

) w

here

the

dotte

d lin

e (

)

rep

rese

nts

area

1 t

he s

olid

line

(mdash

mdashmdash

) ar

ea 2

and

the

dash

ed li

ne(-

- -

) ar

ea 3

(a)

(b)

(c) (f

)(e

)(d

)




4 RESULTS

















(a) (b)














(a) (b)



(a) (b)








location groupings











































Fig

ure

7

The

pos

teri

or m

ean

mig

ratio

n ra

tes

for

each

reg

ion

for

the

mal

es (

figur

es (

a)ndash(

c))

and

fem

ales

(fig

ures

(d)

ndash(f)

) to

eac

h of

the

pos

sibl

e ar

eas

with

cor

resp

ondi

ng 9

5 H

PD

Is (

repr

esen

ted

as v

erti

cal l

ines

) w

here

the

dotte

d li

ne (

) re

pres

ents

are

a 1

the

soli

d li

ne (

mdashmdash

mdash) a

rea

2 a

nd th

e da

shed

line

(- -

-)

area

3

(a)

(b)

(c)

(f)

(e)

(d)






















parameters





yT ordf = yT (41)


constraint thatP













(a) (b)


5 DISCUSSION






APPENDIXES

A LIKELIHOOD


L(xjmicro) =Y

c2 C

Y

r 2 R

2

4TY

t = 0

TY

j = t

Agrave c(tj)(r)vc

(tj)(r)T iexcl1Y

t = 0

tY

l = 0

Y

s2 ROc

(lt)(r s)nc(lt)(rs)

poundT iexcl1Y

t= 0

tY

l = 0

Dc(lt)(r)dc

(lt)(r)

(A1)














t(r)

Atilde

1 iexclX

s2 RAacutec


(t + 1j)(s)


t(r)

where





(lt)(r s)



Qc(lt)(r s) =

(iquest c


iquest cl (r)

Pk 2 R (1 iexcl pc





Dc(lt)(r) =

((1 iexcl iquest c






B MCMC SIMULATION







t(r) + u




t (r s1) + u


t (r s2) iexcl u












t (r) + deg t(r)








0 ft (r) =

12


t (r))








t (r s) + (r s)


t (r s)



t (r s)


t (r s) + (r s)



Riexcl1X

s = 1



Riexcl1X

s = 1

Aacute0 ft (r s)






2 we set



t (r s)

else we set



t (r s)









REFERENCES























































4 RESULTS

















(a) (b)














(a) (b)



(a) (b)








location groupings











































Fig

ure

7

The

pos

teri

or m

ean

mig

ratio

n ra

tes

for

each

reg

ion

for

the

mal

es (

figur

es (

a)ndash(

c))

and

fem

ales

(fig

ures

(d)

ndash(f)

) to

eac

h of

the

pos

sibl

e ar

eas

with

cor

resp

ondi

ng 9

5 H

PD

Is (

repr

esen

ted

as v

erti

cal l

ines

) w

here

the

dotte

d li

ne (

) re

pres

ents

are

a 1

the

soli

d li

ne (

mdashmdash

mdash) a

rea

2 a

nd th

e da

shed

line

(- -

-)

area

3

(a)

(b)

(c)

(f)

(e)

(d)






















parameters





yT ordf = yT (41)


constraint thatP













(a) (b)


5 DISCUSSION






APPENDIXES

A LIKELIHOOD


L(xjmicro) =Y

c2 C

Y

r 2 R

2

4TY

t = 0

TY

j = t

Agrave c(tj)(r)vc

(tj)(r)T iexcl1Y

t = 0

tY

l = 0

Y

s2 ROc

(lt)(r s)nc(lt)(rs)

poundT iexcl1Y

t= 0

tY

l = 0

Dc(lt)(r)dc

(lt)(r)

(A1)














t(r)

Atilde

1 iexclX

s2 RAacutec


(t + 1j)(s)


t(r)

where





(lt)(r s)



Qc(lt)(r s) =

(iquest c


iquest cl (r)

Pk 2 R (1 iexcl pc





Dc(lt)(r) =

((1 iexcl iquest c






B MCMC SIMULATION







t(r) + u




t (r s1) + u


t (r s2) iexcl u












t (r) + deg t(r)








0 ft (r) =

12


t (r))








t (r s) + (r s)


t (r s)



t (r s)


t (r s) + (r s)



Riexcl1X

s = 1



Riexcl1X

s = 1

Aacute0 ft (r s)






2 we set



t (r s)

else we set



t (r s)









REFERENCES

































































(a) (b)














(a) (b)



(a) (b)








location groupings











































Fig

ure

7

The

pos

teri

or m

ean

mig

ratio

n ra

tes

for

each

reg

ion

for

the

mal

es (

figur

es (

a)ndash(

c))

and

fem

ales

(fig

ures

(d)

ndash(f)

) to

eac

h of

the

pos

sibl

e ar

eas

with

cor

resp

ondi

ng 9

5 H

PD

Is (

repr

esen

ted

as v

erti

cal l

ines

) w

here

the

dotte

d li

ne (

) re

pres

ents

are

a 1

the

soli

d li

ne (

mdashmdash

mdash) a

rea

2 a

nd th

e da

shed

line

(- -

-)

area

3

(a)

(b)

(c)

(f)

(e)

(d)






















parameters





yT ordf = yT (41)


constraint thatP













(a) (b)


5 DISCUSSION






APPENDIXES

A LIKELIHOOD


L(xjmicro) =Y

c2 C

Y

r 2 R

2

4TY

t = 0

TY

j = t

Agrave c(tj)(r)vc

(tj)(r)T iexcl1Y

t = 0

tY

l = 0

Y

s2 ROc

(lt)(r s)nc(lt)(rs)

poundT iexcl1Y

t= 0

tY

l = 0

Dc(lt)(r)dc

(lt)(r)

(A1)














t(r)

Atilde

1 iexclX

s2 RAacutec


(t + 1j)(s)


t(r)

where





(lt)(r s)



Qc(lt)(r s) =

(iquest c


iquest cl (r)

Pk 2 R (1 iexcl pc





Dc(lt)(r) =

((1 iexcl iquest c






B MCMC SIMULATION







t(r) + u




t (r s1) + u


t (r s2) iexcl u












t (r) + deg t(r)








0 ft (r) =

12


t (r))








t (r s) + (r s)


t (r s)



t (r s)


t (r s) + (r s)



Riexcl1X

s = 1



Riexcl1X

s = 1

Aacute0 ft (r s)






2 we set



t (r s)

else we set



t (r s)









REFERENCES





















































(a) (b)














(a) (b)



(a) (b)








location groupings











































Fig

ure

7

The

pos

teri

or m

ean

mig

ratio

n ra

tes

for

each

reg

ion

for

the

mal

es (

figur

es (

a)ndash(

c))

and

fem

ales

(fig

ures

(d)

ndash(f)

) to

eac

h of

the

pos

sibl

e ar

eas

with

cor

resp

ondi

ng 9

5 H

PD

Is (

repr

esen

ted

as v

erti

cal l

ines

) w

here

the

dotte

d li

ne (

) re

pres

ents

are

a 1

the

soli

d li

ne (

mdashmdash

mdash) a

rea

2 a

nd th

e da

shed

line

(- -

-)

area

3

(a)

(b)

(c)

(f)

(e)

(d)






















parameters





yT ordf = yT (41)


constraint thatP













(a) (b)


5 DISCUSSION






APPENDIXES

A LIKELIHOOD


L(xjmicro) =Y

c2 C

Y

r 2 R

2

4TY

t = 0

TY

j = t

Agrave c(tj)(r)vc

(tj)(r)T iexcl1Y

t = 0

tY

l = 0

Y

s2 ROc

(lt)(r s)nc(lt)(rs)

poundT iexcl1Y

t= 0

tY

l = 0

Dc(lt)(r)dc

(lt)(r)

(A1)














t(r)

Atilde

1 iexclX

s2 RAacutec


(t + 1j)(s)


t(r)

where





(lt)(r s)



Qc(lt)(r s) =

(iquest c


iquest cl (r)

Pk 2 R (1 iexcl pc





Dc(lt)(r) =

((1 iexcl iquest c






B MCMC SIMULATION







t(r) + u




t (r s1) + u


t (r s2) iexcl u












t (r) + deg t(r)








0 ft (r) =

12


t (r))








t (r s) + (r s)


t (r s)



t (r s)


t (r s) + (r s)



Riexcl1X

s = 1



Riexcl1X

s = 1

Aacute0 ft (r s)






2 we set



t (r s)

else we set



t (r s)









REFERENCES




























































(a) (b)



(a) (b)








location groupings











































Fig

ure

7

The

pos

teri

or m

ean

mig

ratio

n ra

tes

for

each

reg

ion

for

the

mal

es (

figur

es (

a)ndash(

c))

and

fem

ales

(fig

ures

(d)

ndash(f)

) to

eac

h of

the

pos

sibl

e ar

eas

with

cor

resp

ondi

ng 9

5 H

PD

Is (

repr

esen

ted

as v

erti

cal l

ines

) w

here

the

dotte

d li

ne (

) re

pres

ents

are

a 1

the

soli

d li

ne (

mdashmdash

mdash) a

rea

2 a

nd th

e da

shed

line

(- -

-)

area

3

(a)

(b)

(c)

(f)

(e)

(d)






















parameters





yT ordf = yT (41)


constraint thatP













(a) (b)


5 DISCUSSION






APPENDIXES

A LIKELIHOOD


L(xjmicro) =Y

c2 C

Y

r 2 R

2

4TY

t = 0

TY

j = t

Agrave c(tj)(r)vc

(tj)(r)T iexcl1Y

t = 0

tY

l = 0

Y

s2 ROc

(lt)(r s)nc(lt)(rs)

poundT iexcl1Y

t= 0

tY

l = 0

Dc(lt)(r)dc

(lt)(r)

(A1)














t(r)

Atilde

1 iexclX

s2 RAacutec


(t + 1j)(s)


t(r)

where





(lt)(r s)



Qc(lt)(r s) =

(iquest c


iquest cl (r)

Pk 2 R (1 iexcl pc





Dc(lt)(r) =

((1 iexcl iquest c






B MCMC SIMULATION







t(r) + u




t (r s1) + u


t (r s2) iexcl u












t (r) + deg t(r)








0 ft (r) =

12


t (r))








t (r s) + (r s)


t (r s)



t (r s)


t (r s) + (r s)



Riexcl1X

s = 1



Riexcl1X

s = 1

Aacute0 ft (r s)






2 we set



t (r s)

else we set



t (r s)









REFERENCES





















































(a) (b)








location groupings











































Fig

ure

7

The

pos

teri

or m

ean

mig

ratio

n ra

tes

for

each

reg

ion

for

the

mal

es (

figur

es (

a)ndash(

c))

and

fem

ales

(fig

ures

(d)

ndash(f)

) to

eac

h of

the

pos

sibl

e ar

eas

with

cor

resp

ondi

ng 9

5 H

PD

Is (

repr

esen

ted

as v

erti

cal l

ines

) w

here

the

dotte

d li

ne (

) re

pres

ents

are

a 1

the

soli

d li

ne (

mdashmdash

mdash) a

rea

2 a

nd th

e da

shed

line

(- -

-)

area

3

(a)

(b)

(c)

(f)

(e)

(d)






















parameters





yT ordf = yT (41)


constraint thatP













(a) (b)


5 DISCUSSION






APPENDIXES

A LIKELIHOOD


L(xjmicro) =Y

c2 C

Y

r 2 R

2

4TY

t = 0

TY

j = t

Agrave c(tj)(r)vc

(tj)(r)T iexcl1Y

t = 0

tY

l = 0

Y

s2 ROc

(lt)(r s)nc(lt)(rs)

poundT iexcl1Y

t= 0

tY

l = 0

Dc(lt)(r)dc

(lt)(r)

(A1)














t(r)

Atilde

1 iexclX

s2 RAacutec


(t + 1j)(s)


t(r)

where





(lt)(r s)



Qc(lt)(r s) =

(iquest c


iquest cl (r)

Pk 2 R (1 iexcl pc





Dc(lt)(r) =

((1 iexcl iquest c






B MCMC SIMULATION







t(r) + u




t (r s1) + u


t (r s2) iexcl u












t (r) + deg t(r)








0 ft (r) =

12


t (r))








t (r s) + (r s)


t (r s)



t (r s)


t (r s) + (r s)



Riexcl1X

s = 1



Riexcl1X

s = 1

Aacute0 ft (r s)






2 we set



t (r s)

else we set



t (r s)









REFERENCES






















































































Fig

ure

7

The

pos

teri

or m

ean

mig

ratio

n ra

tes

for

each

reg

ion

for

the

mal

es (

figur

es (

a)ndash(

c))

and

fem

ales

(fig

ures

(d)

ndash(f)

) to

eac

h of

the

pos

sibl

e ar

eas

with

cor

resp

ondi

ng 9

5 H

PD

Is (

repr

esen

ted

as v

erti

cal l

ines

) w

here

the

dotte

d li

ne (

) re

pres

ents

are

a 1

the

soli

d li

ne (

mdashmdash

mdash) a

rea

2 a

nd th

e da

shed

line

(- -

-)

area

3

(a)

(b)

(c)

(f)

(e)

(d)






















parameters





yT ordf = yT (41)


constraint thatP













(a) (b)


5 DISCUSSION






APPENDIXES

A LIKELIHOOD


L(xjmicro) =Y

c2 C

Y

r 2 R

2

4TY

t = 0

TY

j = t

Agrave c(tj)(r)vc

(tj)(r)T iexcl1Y

t = 0

tY

l = 0

Y

s2 ROc

(lt)(r s)nc(lt)(rs)

poundT iexcl1Y

t= 0

tY

l = 0

Dc(lt)(r)dc

(lt)(r)

(A1)














t(r)

Atilde

1 iexclX

s2 RAacutec


(t + 1j)(s)


t(r)

where





(lt)(r s)



Qc(lt)(r s) =

(iquest c


iquest cl (r)

Pk 2 R (1 iexcl pc





Dc(lt)(r) =

((1 iexcl iquest c






B MCMC SIMULATION







t(r) + u




t (r s1) + u


t (r s2) iexcl u












t (r) + deg t(r)








0 ft (r) =

12


t (r))








t (r s) + (r s)


t (r s)



t (r s)


t (r s) + (r s)



Riexcl1X

s = 1



Riexcl1X

s = 1

Aacute0 ft (r s)






2 we set



t (r s)

else we set



t (r s)









REFERENCES





















































Fig

ure

7

The

pos

teri

or m

ean

mig

ratio

n ra

tes

for

each

reg

ion

for

the

mal

es (

figur

es (

a)ndash(

c))

and

fem

ales

(fig

ures

(d)

ndash(f)

) to

eac

h of

the

pos

sibl

e ar

eas

with

cor

resp

ondi

ng 9

5 H

PD

Is (

repr

esen

ted

as v

erti

cal l

ines

) w

here

the

dotte

d li

ne (

) re

pres

ents

are

a 1

the

soli

d li

ne (

mdashmdash

mdash) a

rea

2 a

nd th

e da

shed

line

(- -

-)

area

3

(a)

(b)

(c)

(f)

(e)

(d)






















parameters





yT ordf = yT (41)


constraint thatP













(a) (b)


5 DISCUSSION






APPENDIXES

A LIKELIHOOD


L(xjmicro) =Y

c2 C

Y

r 2 R

2

4TY

t = 0

TY

j = t

Agrave c(tj)(r)vc

(tj)(r)T iexcl1Y

t = 0

tY

l = 0

Y

s2 ROc

(lt)(r s)nc(lt)(rs)

poundT iexcl1Y

t= 0

tY

l = 0

Dc(lt)(r)dc

(lt)(r)

(A1)














t(r)

Atilde

1 iexclX

s2 RAacutec


(t + 1j)(s)


t(r)

where





(lt)(r s)



Qc(lt)(r s) =

(iquest c


iquest cl (r)

Pk 2 R (1 iexcl pc





Dc(lt)(r) =

((1 iexcl iquest c






B MCMC SIMULATION







t(r) + u




t (r s1) + u


t (r s2) iexcl u












t (r) + deg t(r)








0 ft (r) =

12


t (r))








t (r s) + (r s)


t (r s)



t (r s)


t (r s) + (r s)



Riexcl1X

s = 1



Riexcl1X

s = 1

Aacute0 ft (r s)






2 we set



t (r s)

else we set



t (r s)









REFERENCES









































































parameters





yT ordf = yT (41)


constraint thatP













(a) (b)


5 DISCUSSION






APPENDIXES

A LIKELIHOOD


L(xjmicro) =Y

c2 C

Y

r 2 R

2

4TY

t = 0

TY

j = t

Agrave c(tj)(r)vc

(tj)(r)T iexcl1Y

t = 0

tY

l = 0

Y

s2 ROc

(lt)(r s)nc(lt)(rs)

poundT iexcl1Y

t= 0

tY

l = 0

Dc(lt)(r)dc

(lt)(r)

(A1)














t(r)

Atilde

1 iexclX

s2 RAacutec


(t + 1j)(s)


t(r)

where





(lt)(r s)



Qc(lt)(r s) =

(iquest c


iquest cl (r)

Pk 2 R (1 iexcl pc





Dc(lt)(r) =

((1 iexcl iquest c






B MCMC SIMULATION







t(r) + u




t (r s1) + u


t (r s2) iexcl u












t (r) + deg t(r)








0 ft (r) =

12


t (r))








t (r s) + (r s)


t (r s)



t (r s)


t (r s) + (r s)



Riexcl1X

s = 1



Riexcl1X

s = 1

Aacute0 ft (r s)






2 we set



t (r s)

else we set



t (r s)









REFERENCES





















































(a) (b)


5 DISCUSSION






APPENDIXES

A LIKELIHOOD


L(xjmicro) =Y

c2 C

Y

r 2 R

2

4TY

t = 0

TY

j = t

Agrave c(tj)(r)vc

(tj)(r)T iexcl1Y

t = 0

tY

l = 0

Y

s2 ROc

(lt)(r s)nc(lt)(rs)

poundT iexcl1Y

t= 0

tY

l = 0

Dc(lt)(r)dc

(lt)(r)

(A1)














t(r)

Atilde

1 iexclX

s2 RAacutec


(t + 1j)(s)


t(r)

where





(lt)(r s)



Qc(lt)(r s) =

(iquest c


iquest cl (r)

Pk 2 R (1 iexcl pc





Dc(lt)(r) =

((1 iexcl iquest c






B MCMC SIMULATION







t(r) + u




t (r s1) + u


t (r s2) iexcl u












t (r) + deg t(r)








0 ft (r) =

12


t (r))








t (r s) + (r s)


t (r s)



t (r s)


t (r s) + (r s)



Riexcl1X

s = 1



Riexcl1X

s = 1

Aacute0 ft (r s)






2 we set



t (r s)

else we set



t (r s)









REFERENCES























































APPENDIXES

A LIKELIHOOD


L(xjmicro) =Y

c2 C

Y

r 2 R

2

4TY

t = 0

TY

j = t

Agrave c(tj)(r)vc

(tj)(r)T iexcl1Y

t = 0

tY

l = 0

Y

s2 ROc

(lt)(r s)nc(lt)(rs)

poundT iexcl1Y

t= 0

tY

l = 0

Dc(lt)(r)dc

(lt)(r)

(A1)














t(r)

Atilde

1 iexclX

s2 RAacutec


(t + 1j)(s)


t(r)

where





(lt)(r s)



Qc(lt)(r s) =

(iquest c


iquest cl (r)

Pk 2 R (1 iexcl pc





Dc(lt)(r) =

((1 iexcl iquest c






B MCMC SIMULATION







t(r) + u




t (r s1) + u


t (r s2) iexcl u












t (r) + deg t(r)








0 ft (r) =

12


t (r))








t (r s) + (r s)


t (r s)



t (r s)


t (r s) + (r s)



Riexcl1X

s = 1



Riexcl1X

s = 1

Aacute0 ft (r s)






2 we set



t (r s)

else we set



t (r s)









REFERENCES



























































t(r)

Atilde

1 iexclX

s2 RAacutec


(t + 1j)(s)


t(r)

where





(lt)(r s)



Qc(lt)(r s) =

(iquest c


iquest cl (r)

Pk 2 R (1 iexcl pc





Dc(lt)(r) =

((1 iexcl iquest c






B MCMC SIMULATION







t(r) + u




t (r s1) + u


t (r s2) iexcl u












t (r) + deg t(r)








0 ft (r) =

12


t (r))








t (r s) + (r s)


t (r s)



t (r s)


t (r s) + (r s)



Riexcl1X

s = 1



Riexcl1X

s = 1

Aacute0 ft (r s)






2 we set



t (r s)

else we set



t (r s)









REFERENCES

























































t(r) + u




t (r s1) + u


t (r s2) iexcl u












t (r) + deg t(r)








0 ft (r) =

12


t (r))








t (r s) + (r s)


t (r s)



t (r s)


t (r s) + (r s)



Riexcl1X

s = 1



Riexcl1X

s = 1

Aacute0 ft (r s)






2 we set



t (r s)

else we set



t (r s)









REFERENCES




























































t (r) + deg t(r)








0 ft (r) =

12


t (r))








t (r s) + (r s)


t (r s)



t (r s)


t (r s) + (r s)



Riexcl1X

s = 1



Riexcl1X

s = 1

Aacute0 ft (r s)






2 we set



t (r s)

else we set



t (r s)









REFERENCES


























































t (r s) + (r s)


t (r s)



t (r s)


t (r s) + (r s)



Riexcl1X

s = 1



Riexcl1X

s = 1

Aacute0 ft (r s)






2 we set



t (r s)

else we set



t (r s)









REFERENCES


























































REFERENCES






































































































Documents

Survival and spatial fidelity of mouflons: The effect of location, age, and sex