population abundance estimation with ... -...

Research Article

Population Abundance Estimation WithHeterogeneous Encounter Probabilities UsingNumerical Integration

GARY C. WHITE,1 Department of Fish, Wildlife and Conservation Biology, Colorado State University, Ft. Collins, CO 80523, USA

EVAN G. COOCH, Department of Natural Resources, Cornell University, Ithaca, NY 14853, USA

ABSTRACT Estimation of population abundance is a common problem in wildlife ecology andmanagement. Capture-mark-reencounter (CMR) methods using marked animals are a standard approach,particularly in recent history with the development of innovative methods of marking using camera traps orDNA samples. However, estimates of abundance from multiple encounters of marked individuals are biasedlow when individual heterogeneity of encounter probabilities is not accounted for in the estimator. Weevaluated the operating characteristics of the Huggins logit-normal estimator through computer simulations,using Gaussian–Hermite quadrature to model individual encounter heterogeneity. We simulated individualencounter data following a factorial design with 2 levels of sampling occasions (t¼ 5, 10), 3 levels ofabundance (N¼ 100, 500, 1,000), 4 levels of median detection probabilities (p¼ 0.1, 0.2, 0.4, 0.6) for eachsampling occasion (on the probability scale), and 4 levels of individual heterogeneity (sp¼ 0, 0.5, 1, 2; on thelogit normal scale), resulting in a design space consisting of 96 simulation scenarios (2� 3� 4� 4). For eachscenario, we performed 1,000 simulations using the Huggins estimatorsMt,M0,MtRE, andM0RE, where theRE subscript corresponds to the random effects model. As expected, the Mt and M0 estimators were biasedwhen individual heterogeneity was present but unbiased for sp¼ 0 data. The estimators forMtRE andM0RE

were biased high for N¼ 100 and median p� 0.2 but showed little bias elsewhere. The bias is attributed tothe occasional sets of data that result in a low overall detection probability and a resulting highly skewedsampling distribution of N . This result is confirmed in that the median of the sampling distributions was onlyslightly biased high. The random effects estimators performed poorly for sp¼ 0 data, mainly because a loglink function forces the estimate of sp> 0. However, the Fletcher c statistic provided useful evidence toevaluate sp> 0, as did likelihood ratio tests of the null hypothesis sp¼ 0. Generally, confidence intervalcoverage of N appears close to the nominal 95% expected when the estimator is not biased. � 2017 TheWildlife Society.

KEY WORDS abundance estimation, capture-mark-reencounter, Huggins estimator, individual heterogeneity, Mh,numerical integration, Program MARK.

Estimation of abundance (i.e., no. of individuals in thepopulation over a sampling interval) is of interest in ecologyand conservation management, and the literature devoted tothe estimation of abundance based on (live) capture-mark-reencounter (CMR) data is large, and often technicallychallenging (Schwarz and Seber 1999, Borchers et al. 2002,Williams et al. 2002, Royle et al. 2013). However, under thebinomial assumption that encounter of any individual in apopulation is independent of the detection of any otherindividual, virtually all estimators of population abundance(N) for closed populations (where abundance is constant overthe sampling interval; no net movement in or out of the

population) have the following simple canonical form(Williams et al. 2002):

N ¼ n

p

where n is a count (related to the no. of unique individualscaught in some group at some point in time), and p is theprobability of encounter related to the count. The count n is aminimum estimate of population abundance, and is adjustedto account for the possibility that the probability of detectionduring any sampling event is <1 (i.e., to account for the factthat the count n generally does not include all individuals inthe sample); in most situations, some individuals present inthe population, and available for detection, go undetectedduring a sampling event, with probability 1� p.There are underlying challenges to the estimation of

abundance in the presence of heterogeneity of individual

Received: 19 August 2016; Accepted: 21 October 2016

1E-mail: [email protected]

The Journal of Wildlife Management 81(2):322–336; 2017; DOI: 10.1002/jwmg.21199

322 The Journal of Wildlife Management � 81(2)

capture probabilities. If one were to take a strict randomsample from a closed population (i.e., such that allindividuals have the same probability of being includedin the sample), one would expect the estimate of abundanceto be unbiased (although a particular estimator might leadto bias under some circumstances; Seber 1982, Borcherset al. 2002). In the case of individual heterogeneity in theprobability of encounter, where in the limit each individual iin the population has a unique encounter probability, pi,negative bias in estimates of abundance is expected(Burnham and Overton 1978, 1979; Seber 1982; Rosenberget al. 1995; Chao 2001). Individuals with high detectionprobabilities would tend to appear in the encounteredsample in greater proportion than they occur in thepopulation. This results in the estimated average encounterprobability of encountered individuals tending to be greaterthan the true population average encounter probability.Thus, the denominator in the canonical estimator forabundance is generally biased high in the presence ofindividual heterogeneity, and the resulting estimate ofabundance is biased low. This will be especially true insituations where the overall mean encounter probability islow; when the overall mean encounter probability is high,then even individuals with relatively low encounterprobability are likely to be encountered at least once duringthe sampling period.We illustrate the relationship between the mean and

variance in individual encounter probabilities and thenegative bias in estimated abundance, by a simple numericalexample. Consider a population consisting of N¼ 100individuals. Assume the population has no heterogeneity inp, such that all individuals have the same probability ofencounter, say, p¼ 0.3. If we assume p is constant over tsampling occasions, then we would expect p� ¼ 1� (1� p)t

to be the probability than an individual is encountered at leastonce, and thus we would expect to encounter N� p� uniqueindividuals over t occasions. If t¼ 5, then the expected valuefor p� ¼ 1� (1� 0.3)5¼ 0.83193, and therefore we wouldexpect to encounter 83.193 individuals out of N¼ 100 atleast once over t¼ 5 samples (the non-integer expectation isbased on a conceptual set of replicates).Now, consider the same population ofN¼ 100 individuals,

but in this case, we assume there is discrete heterogeneityamong individuals in encounter probability. Assume that 50individuals have a true encounter probability of plow¼ 0.1and the remaining 50 individuals have a higher encounterprobability, phigh¼ 0.5. The population mean encounterprobability is �p ¼ 0:3 (so, the same overall mean as in the firstscenario, but now with heterogeneity among individuals). Inthis case, p�low ¼ 0:40951, and p�high ¼ 0:96875. Thus, overt¼ 5 samples, we would expect to encounter 20.4755 of the50 low p individuals at least once, and 48.4375 of the highp individuals. Or, 68.913 total unique individuals, which (byJensen’s inequality; McShane 1937) is slightly negativelybiased with respect to the expected number encountered inthe previous example without heterogeneity (83.193),despite the same population mean encounter probabilityin both cases.

The expected encounter probability averaged over the trueprobabilities for those individuals encountered in the samplewill be

psample ¼20:4755� 0:1ð Þ þ 48:4375� 0:5ð Þ

20:4755þ 48:4375¼ 0:38115

which as anticipated is biased high, relative to the populationmean �p ¼ 0:3: Thus, the expected p�sample ¼ 1�1� 0:38115ð Þ5 ¼ 0:90923 for the encountered sample fort¼ 5 occasions. Our abundance estimate (based on thesampled individuals) is N¼ (68.913/0.90923)¼ 75.792,which is strongly negatively biased with respect to trueN¼ 100. This negative bias in estimated abundance is ageneral result, and is not an artifact of the structure of theexample: if one uses a model that assumes equal encounterprobabilities among individuals, the estimates of populationabundance will generally be too low (negatively biased) if, infact, there is heterogeneity in encounter probabilities.The magnitude of the bias can be substantial, and is a

function of the population mean detection probability, E(p),and variance, Var(p), in individual encounter probability(Fig. 1). Following Royle and Dorazio (2008), the expectedvalue of the encounter probability for encountered individu-als, assuming the variation is continuous Gaussian aroundthe mean, is given as

E½ pjcaptured� ¼ VarðpÞEðpÞ þ EðpÞ

Under conditions of individual heterogeneity (i.e., whenVar(p)> 0) the ratio Var(p)/E(p)> 0 and thus the expectedvalue of the encounter probability for encountered individu-als will be biased high, relative to the mean encounterprobability of the population. For a given mean encounterprobability, E(p), as the variance in individual heterogeneity,Var(p), increases, the denominator, and thus the propor-tional magnitude of the negative bias between estimated and

Figure 1. The magnitude of the negative bias in expected abundance withrespect to true N¼ 100, for a scenario where we assume 2 discrete classes ofindividuals with different encounter probabilities (i.e., low, high), where theoverall mean (E(p)) and variance differed between the 2 classes (e.g.,plow¼ 0.2, phigh¼ 0.4: mean¼ 0.3, variance¼ 0.02). Bias was calculatedassuming that both classes of low and high encounter probability individualswere in equal abundance at the start of t¼ 5 sampling occasions, and that theindividual probabilities of encounter were constant over time.

White and Cooch � Individual Heterogeneity and Abundance Estimation 323

true abundance, increases (Fig. 1). Conversely, for a givenvariance, Var(p), the proportional magnitude of the biastends to decrease with increasing mean encounter probabil-ity, which, of course, makes perfect sense—as the encounterprobability increases, the probability of all individuals in thepopulation being encountered at least once approaches 1.0(at a rate proportional to the number of sampling occasions),with bias in estimated abundance approaching 0. The worst-case scenario clearly involves high variance in individualencounter probabilities, and low mean encounter probabilityoverall. In such cases, the magnitude of the negative bias maybe very high.Largely because of concerns of the effect of individual

encounter heterogeneity on bias in abundance estimate,deriving models for estimating abundance from closedpopulations that account for individual heterogeneity hasbeen of historical interest, and has generally proven difficult.The underlying conceptual challenge is straightforward.Consider the situation where each individual i has a uniqueprobability of encounter, p (p1, p2,. . ., pN¼ {pi} over Nindividuals in the population). Imagine an inference modelwhere we attempted to estimate each individual pi. Such amodel would completely account for heterogeneity amongindividuals. However, estimation of a time-specific encoun-ter probability for individual i over time step t to tþ 1requires inference about a binomial parameter based on asingle Bernoulli trial. Based on whether the individual isencountered or not at time tþ 1, we must somehow estimatethe underlying probability of encounter, and there is simplynot enough information in this single observation (adoptinga frequentist paradigm) to allow us to do this in a robust way(i.e., we are attempting to estimate an individual parameterbased on an effective sample size of 1).As such, we are generally left constructing a model that

represents a compromise between one where we assume noheterogeneity in encounter probabilities, and a fully saturatedmodel, where pi is inestimable for each individual. Therehave been 2 fairly distinct, but not mutually exclusive,approaches to development of such intermediate models.The first considers models where the source of theheterogeneity among individuals is assumed observable,and can potentially be accounted for by �1 individual,measurable covariates—some individual trait that influencesthe detection probability of that individual. The individualcovariate approaches have the advantage of potentiallyinforming as to the underlying causes of the heterogeneity inencounter probability, in addition to reducing bias inabundance estimators. For some taxa, identifying plausiblecovariates that might induce encounter heterogeneity is oftenstraightforward. For example, fish of a certain size are morelikely to be encountered if their size is at least as large as, say,the mesh size of the net used to encounter samples thanwould smaller fish. In other cases, heterogeneity induced bythe spatial interaction of a trapping grid with individualdifferences in distribution and movement over the landscapemight be accommodated by explicitly modeling the spatialattributes of the encounter process (Efford 2004, Efford andFewster 2013, Royle et al. 2013).

Alternatively, there is a class of models where individualheterogeneity in encounter probability is unobservable (i.e.,not modeled as a function of �1 factors or covariates), and ismodeled as an individual random effect. Such models arevery general because they do not require specification of thepossible source(s) of the heterogeneity. Instead, they posit aparametric probability distribution for {pi} (i.e., the set {pi} isa random sample of size N drawn from some probabilitydistribution), and use appropriate methods of parametricinference. Such models can be broadly classified in terms ofwhether the distribution of individuals is modeled as either adiscrete- or continuous-mixture, where the population isimplicitly a mixture of individuals with different probabilitiesof encounter. Norris and Pollock (1996) and Pledger (2000,2005) proposed discrete-mixture models where eachindividual pi may belong to one of a discrete set of classes(recently reviewed by Pledger and Phillpot 2008); becausethe discrete set of classes is finite, these models are oftenreferred to synonymously as finite-mixture models. Such adiscrete, finite-mixture model was the basis for the examplespresented earlier. Models based on finite-mixtures generallyconsist of a discrete number of binomial components,although mixtures of binomial and beta-binomial distribu-tions have recently been proposed by Morgan and Ridout(2008). Alternatively, the mixture distribution can becontinuous infinite (Burnham and Overton 1978, 1979,Dorazio and Royle 2003). A commonly used distributionis the logit-normal (Coull and Agresti 1999), whereindividuals pi are drawn from a normal distribution (onthe logit scale) with specified mean m and variance s2, that is,logit(pi)N(m, s2). In this paper, we consider continuousmixture models, based on the logit-normal distribution.In principle, the construction of an unobservable hetero-

geneity model is straightforward. We seek to model thedistribution of the data (number of unique individualsencountered, n, and how many individuals were encounteredyf times, where f¼ 1,. . .,t, where

Ptf ¼1 yf ¼ n), typically as

functions of both N and p. Following Seber (1982), andassuming for simplicity that the encounter probability p isconstant over time and identical for all individuals, thelikelihood for N and p can be written as a productmultinomial of the binomial probability of each encounterfrequency, y, such that

L N ; pjy1; . . . ; yt� � ¼ N !

N � mð Þ!Ptj¼1p

yjj 1� St

j¼1pj

� �N�m

where m ¼Ptj¼1 yj is the number of individuals observed at

least once (i.e., traditionally denoted as Mtþ1), and pj is thebinomial probability mass function for a given encounterfrequency,

pj ¼y

j

!pj 1� p� �y�j

Heterogeneity is modeled by conceiving the encounterprobability for each individual i as drawn as a randomsample from some continuous probability density function,g, with moments specified by the parameter vector u (the


same idea applies in principle for discrete probability massfunctions):

pi g pju� �To accommodate this heterogeneity, the cell probabilities p

of each encounter frequency yf are marginalized (i.e.,averaged) over all realizations of the random distribution,g, by integrating over the joint probability of the binomialencounter process given p, and the probability density forp (frequently referred to as integrating out the random effect)using the random encounter probability:

pn ¼ Pr½y ¼ n� ¼Z 1

0

njt; p� �g pju� �

dp

Analytical solutions for these marginalized cell probabilitiesare available in only a few instances, including finite-mixturemodels. In general practice, the integration must be carried outnumerically. In a seminal paper onmodeling individual randomeffects in CMR models, Royle (2008) used Markov chainMonte Carlo (MCMC) algorithms in a Bayesian frameworkthat directly generate (by numerical simulation) random valuesfrom a Markov chain whose stationary distribution is theposterior distribution of the parameters under interest.Although Monte Carlo resampling is increasingly popular

for problems involving analytically intractable integrals,MCMC in general, and not specifically in a Bayesiancontext, is typically many order of magnitudes computation-ally slower than other numerical methods, which makesit difficult, and in many cases, entirely intractable, tothoroughly assess estimator performance in terms of bias andprecision. An alternative to MCMC for the numericalintegration of random effects, using Gaussian–Hermitequadrature (Liu and Pierce 1994, Givens andHoeting 2005),has been proposed byMcClintock et al. (2009) and Gimenezand Choquet (2010). The Gaussian–Hermite quadratureapproach approximates the integral of functions that can bewritten as Hermite polynomials by constructing a grid ofpoints at which the likelihood function is evaluated. Asdiscussed by Gimenez and Choquet (2010), integration byGaussian–Hermite quadrature is very robust under theassumption that the random effect is Gaussian (we revisitthis assumption in the discussion), and computationally ismuch more efficient than approaches based on Monte Carlosampling. Further, because Gaussian–Hermite integrationcan be embedded in a classical likelihood-based modelingframework, we can use established methods for goodness-of-fit testing and model selection to evaluate the relativeperformance of different heterogeneity models in estimatingabundance from closed population encounter data.We conducted a series of numerical simulation experi-

ments, assessing the performance of models for theestimation of abundance from closed population encounterdata, using Gaussian–Hermite quadrature to numericallyintegrate out the effects of random heterogeneity inindividual encounter probability that we simulated in ourencounter data. We evaluated several measures of the biasand precision of estimates of abundance, for differentamounts of heterogeneity in the simulated population.

METHODS

Our approach was to 1) simulate closed populationencounter data, where individual encounter probabilitieswere randomly drawn from a logit-normal distributionspecified by a known mp and sp; 2) fit each simulated dataset with a set of approximating models, based on theconditional likelihood for abundance estimation describedby Huggins (1989, 1991) and Alho (1990), where individualheterogeneity in the encounter probability was numericallyintegrated out of the marginal likelihood using Gaussian–Hermite quadrature; and 3) compare and contrast variousstatistics derived from the distribution of the parameterestimates collected over the set of simulations, with thetrue values of the parameters used in generating thesimulated data.

Huggins Random Effects EstimatorThe Huggins random effects estimator is based on the CMRestimator conditioned on individuals encountered �1 time(Huggins 1989, 1991; Alho 1990). We consider only theHuggins estimator in this study because it has the potentialadvantage of also allowing the inclusion of individualcovariates in the analysis.The Huggins estimator is extended by including an

individual random effect for the encounter probability (pik) ofeach individual i constant across occasions k¼ 1,. . ., t on thelogit scale following McClintock et al. (2009) and Gimenezand Choquet (2010):

logitðpikÞ ¼ bk þ ei

with bk a fixed effect modeling time, and ei a normallydistributed random effect with mean zero and unknownvariance s2

p :Hence

pik ¼1

1þ exp �ðbk þ spZi

� �Þwhere Zi N(0, 1). Therefore, individual i on occasion k hasthe probability of being encountered

pik ¼Z þ1

�1

1

1þ exp � bk þ spZi

� �� w Zið ÞdZi

where wðZiÞ is the probability density function of thestandard normal distribution. The estimate of populationabundance, N , is obtained following Huggins (1989) as thesummation across animals encountered �1 time,

N ¼XMtþ1

i¼1

1

p�i

� �

where

p�i ¼ 1�Z þ1

�1Pk

i¼1

1

1þ exp � bk þ spZi

� �� !

w Zið ÞdZi

(i.e., the probability of being detected �1 time during the toccasions).


Because this integral does not have a closed form, thelikelihood must be integrated numerically using Gaussian–Hermite quadrature (Givens andHoeting 2005,McClintocket al. 2009, Gimenez and Choquet 2010). This estimator isimplemented in Program MARK (White and Burnham1999, White 2007) with a default of 101 quadrature points.

Evidence of OverdispersionAn important issue with individual heterogeneity in closedencounterspopulationestimation isdecidingwhether there is asignificant overdispersion in the data, justifying fitting of anindividual random effects model. Typically, estimating over-dispersion cð Þ from the observed number of individualsassociated with each possible encounter history is complicatedby the large number of encounter histories with very lowexpected frequencies, especially when the average encounterprobability is low.Commonly, an estimate of overdispersion isbased on Pearson’s x2 lack-of-fit statistic. Fletcher (2012)proposed a new estimator with smaller variance,

c ¼ c x�r;where�r ¼ 1

H

XH

i¼1

yihi

Here, c x is theestimatorofoverdispersionbasedonthePearsonx2 statistic (i.e., the Pearson x2 statistic divided by the degreesof freedom,wheresp is included in the parameter count for therandom effects models because it is an estimated parameter(D. J. Fletcher, University ofOtago, personal communication)following Sandland and Cormack 1984, Cormack 1989), yiand hi are the observed and expected number of individualswith encounter history i, and H ¼ 2t � 1 is the number ofobservable histories over t occasions.One of the problems with using Pearson’s statistic for

sparse data is that the ith term involves dividing by hi, whichwill often be very small. The new estimator makes anallowance for this because the ith term in the denominatoralso involves dividing by hi. Simulations suggest that thisnew estimator also performs better than those based on thedeviance.

SimulationsWe simulated individual encounter data using a generatingmodel with no temporal variation in the encounterprobability, following a factorial design with 2 levels ofsampling occasions (t¼ 5, 10), 3 levels of N (100, 500,1,000), 4 levels of median p (0.1, 0.2, 0.4, 0.6 on the realprobability scale), and 4 levels of sp (0, 0.5, 1, 2 on the logitnormal scale), resulting in a design space consisting of 96(2�3�4�4) simulation scenarios. We assumed individual piwas constant over t sampling occasions. Because theindividual random effect was simulated on the logit-normalscale, the distribution of the resulting pi on the back-transformed (0,1) probability scale is asymmetrical (exceptfor median p¼ 0.5; Fig. S1, available online in SupportingInformation), and median p is the value for which half theanimals have a lower detection probability, and half have alarger detection probability (Table 1).At each design point, we simulated 1,000 replicates using

ProgramMARK,with 4Hugginsmodels estimated followingthe naming conventions of Otis et al. (1978): Mt, M0, MtRE,

andM0RE, where the t subscript indicates time dependence inthe encounter probability, and theREsubscript corresponds tothe random effects model described above. We useMtRE andM0RE, rather than the more generic Mh notation (Otis et al.1978), to make it explicit that we are using an individualrandom effects (RE) approach, in conjunction with classicalM0 or Mt model ultrastructure to account for encounterheterogeneity.For each of the estimating models, we computed the

percent relative bias (PRB) of a parameter as:

PRBðuÞ ¼ 100� EðuÞ � u� �u

with EðuÞ computed as the mean of the estimates N ; sp

� �for

1,000 replicates (because of division by zero when sp¼ 0 wereport bias in estimated sp for sp> 0 only). The distributionof N was frequently strongly skewed, especially for highervalues of sp, with a long tail of high values, such that themean overestimated the most likely outcome of oursimulations. The distribution of sp was similarly skewed,although the skew was generally less than observed for N :Thus, we also report percent relative difference of the medianvalue of both N and sp, as PRMD N

� �and PRMD sp

� �;

respectively. In addition, we summarized the Fletcher (2012)estimator of over-dispersion, c , as a mean, and percentiles90% and 95% to assess the effectiveness of this estimator.We computed 95% confidence intervals for N assuming a

log-normal distribution as lower confidence bound¼ðN �Mtþ1Þ=C þMtþ1, upper confidence bound¼ðN �Mtþ1Þ � C þMtþ1; and

C ¼ exp 1:96�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiln 1þ SEðN Þ

N �Mtþ1

!224

35

vuuut8><>:

9>=>;

Wecomputed confidence interval coverage as the percentageof confidence intervals that covered true N, although thisprocedure does not perform well for the simulation scenarioswith p¼ 0.6 because the probability of detecting allN animalsis 0.99 for 5 occasions and effectively 1 for 10 occasions.We constructed likelihood ratio tests for 1)M0RE (the true

model under which the simulated data were generated)

Table 1. Expected values of mean detection probability of the population,E(p), with standard deviation in parenthesis for the scenarios simulatedover the range of median detection probability (p) and standard deviation ofthe distribution of individual heterogeneity (sp). Note the mirror images ofthe values around 0.5 for median p¼ 0.4 and 0.6. As sp increases for allmedian p> 0, the expected value approaches 0.5; at the limit (sp verylarge), half of the individuals will be caught with probability 1.0 (effectivep¼ 1.0), and half will never be caught (effective p¼ 0.0), with an averageE(p)¼ 0.5.

sp

Median p 0 0.5 1 2

0.1 0.1 (0) 0.109 (0.049) 0.134 (0.113) 0.203 (0.237)0.2 0.2 (0) 0.211 (0.081) 0.239 (0.162) 0.300 (0.281)0.4 0.4 (0) 0.405 (0.114) 0.417 (0.204) 0.439 (0.311)0.6 0.6 (0) 0.595 (0.114) 0.583 (0.204) 0.561 (0.311)


versus the null model of M0, and 2) MtRE versus the nullmodel of Mt to test for statistical significance of individualheterogeneity. We computed the rejection rate of these tests(test size for sp¼ 0 test power for sp> 0). However, becausethe null hypothesis of both of these tests is on the boundaryof the parameter space, classical inference no longer holds(Self and Liang 1987). The asymptotic null distribution ofthe likelihood ratio test is a 50:50 mixture of x2 distributionswith 0 and 1 degrees of freedom (Stram and Lee 1994).Therefore, following Gimenez and Choquet (2010), wecomputed a corrected version of the test, although we alsocomputed a na€ıve version for comparison to evaluate thenecessity of the correction.

RESULTS

The random effects estimatorsM0RE andMtRE showed littlebiasof N except at the lower levels of theencounterprobability,p. The bias was much more pronounced for 5 occasionscompared to 10 occasions, and for N¼ 100 compared toN¼ 500 or N¼ 1,000 (Fig. 2a, left-hand panels). Thesampling distribution from these estimators was highlyskewed, especially at the design point with 5 occasions andp¼ 0.1, producing some very large estimates of N , whichinflated the estimate of the mean PRB. This reflected themarked non-linearity of the back-transformation of thedistribution of the encounter probabilities from the logit-normal scale, which were symmetrical around mp, to the realprobability scale, where the distribution was often stronglyasymmetrical with respect toE(p), especially for low p (Fig. S1,available online in Supporting Information). However, withthe percent relative median difference (Fig. 2b, left-handpanels), the performance of these estimators was stillreasonable even for the worst cases of bias.In contrast, as was expected, the bias of the M0 and Mt

models was consistently negative, and increased rapidly withincreasing levels of heterogeneity with sp (Fig. 2, right-handpanels). In contrast, the heterogeneity parameter sp wasestimated with little bias; the mean (Fig. 3) and medianestimates both were generally quite close, reflecting markedlyless skew in the distribution of sp as compared to thedistribution of N : The small bias observed for sp¼ 0 (Fig. 3)is due to the use of a log link to estimate sp because the loglink forces sp> 0.In either case, the magnitude and direction of bias (Fig. 2a)

and the percentage difference in median N (Fig. 2b) werevirtually identical between models based onM0 (i.e., no timevariation in p) or Mt (i.e., time variation in p). This result isconsistent with Seber (1982:164) who originally noted thateven when the generating model is M0, there is no gain inreduced bias or, in fact, increased parsimony, by using modelM0 instead of model Mt (see also Borchers et al. 2002).Generally, confidence interval coverage ofN appeared close

to the nominal 95% expected. Coverage of the 95%confidence interval for N appeared to be imperfect forp¼ 0.6, and even somewhat for p¼ 0.4, particularly for 10occasions (Fig. 4). The reason for this observed poor coverageis that the detection probabilities are so high that effectivelyall of the animals are encountered. When this occurs, the

log-normal confidence intervals always fail to cover thetrue N. As an example, for 10 occasions, sp¼ 0, and p¼ 0.6,the probability of detecting an individual is 1–(1–0.6)10¼ 0.9989. As sp increases, this high probabilitydeclines because some animals have considerably lowerdetection probabilities and are not encountered, and so thelog-normal confidence interval procedure demonstratesbetter performance. Thus, performance of the log-normalconfidence interval procedure seems adequate with therandom effects models. As was expected, confidence intervalcoverage for M0 and Mt models declined with increasing spbecause of the increasing bias of these estimators (Fig. 2).The Fletcher (2012) estimator c showed negligible bias for

the random effects models M0RE and MtRE (Fig. 5). As spand the number of occasions increased, the value of cincreased for modelsM0 andMt, illustrating the usefulness ofthis estimator in detecting individual heterogeneity. The90th and 95th percentiles (Fig. 6) for models M0RE andMtRE provide guidelines on how large c might be just bychance.The size of the likelihood ratio tests for individual

heterogeneity was close to the nominal a¼ 0.05 (Fig. 7),and power increased with increasing sp, number of occasions,population abundance, and median p. All the tests nearlyalways rejected the null hypothesis for sp¼ 2, whichrepresents fairly extreme individual heterogeneity. Howevereven for sp¼ 1, the test power was still nearly 100% for 10occasions, and close to 100% for 5 occasions andN� 500. Asexpected, the na€ıve test without the correction showed justslightly less power to detect an effect (Fig. 7).

Direction of Bias: M0 Versus M0RE

Estimates from a model that assumes equal encounterprobabilities at a particular sampling occasion (i.e., modelsM0,Mt) when in fact there is individual heterogeneity will benegatively biased because individuals with a high encounterprobability are on average over-represented in the encoun-tered sample, leading to a positive bias in average p�, andthus, an overall negative bias in estimated abundance. This isprecisely what we observed when fitting models M0 and Mt

to the data simulated with heterogeneity (Fig. 2, right-handpanels). In contrast, the bias in mean PRB or PRMD fromfitting M0RE and MtRE to the simulated data was generallypositive (Fig. 2, left-hand panels), suggesting that estimatesof average p� are negatively biased, at least for some of thesimulated samples.To understand why this might be the case, we consider

one particular design point from our simulations in moredetail: trueN¼ 100, sp¼ 0.5, median p¼ 0.2, correspondingto E(p)¼ 0.211 (Table 1). Mean estimated N from modelM0RE was 112.30 (i.e., positive bias), and mean estimatedN for M0 was 91.337 (i.e., negative bias). Average Mtþ 1

was the same in either case because the same simulateddata were being analyzed under both models. If the meanMtþ 1 is the same under both models, then differences inmean estimated abundance implies that average p� differsbetween the 2 models. This is precisely what is observed:mean p ¼ 0:233 for M0 (i.e., positive bias with respect to


Figure 2. Percent relative bias of N , where relative bias is calculated as mean N � true N, as percentage of true N (a), and percent relative difference ofmedian of N relative to true N (b) for 96 simulation scenarios with 1,000 replicates for each scenario. Each panel shows results for N¼ 100, 500, and 1,000,and occasions t¼ 5 and 10 for each of the 4 models M0RE (no time variation in p, individual random effects), MtRE (time variation in p, individual randomeffects), M0 (no time variation in p, no individual random effects), and Mt (time variation in p, no individual random effects). True median detectionprobability (p) is plotted on the x-axis. The 4 lines on each panel show results for true standard deviation of distribution of individual encounter probabilitiessp¼ 0, 0.5, 1, and 2.


E(p)¼ 0.211), whereas mean p ¼ 0:192 for M0RE (i.e.,negative bias with respect to E(p)¼ 0.211). This negativebias is the result of a strong left skew in the distribution ofp from model M0RE, especially for designs with relativelylow number of sampling occasions, and low encounterprobabilities (Fig. 8 ).In contrast, the frequency distribution from model M0,

although positively biased with respect to E(p), is notablysymmetrical around the mean, with little evidence for skew.

When both the number of sampling occasions, and theencounter probability, are higher (t¼ 10, median p¼ 0.4),then the frequency distributions from models M0 and M0RE

are effectively identical (Fig. 8), with no bias inmean estimatesof p, and as a result, no bias in estimated abundance fromeithermodel.Note that the frequencies are based onfitting the2 different models to exactly the same set of simulated data.It is clear (Fig. 8) that for some of the design points in our

simulations, modelM0RE generates much lower estimates of

Figure 3. Mean of estimates of standard deviation of distribution of individual encounter probabilities, sp, from modelsM0RE (no time variation in encounterprobability, individual random effects) and MtRE (time variation in encounter probability, individual random effects) for 96 simulation scenarios with 1,000replicates for each scenario. Each panel shows results forN¼ 100, 500, and 1,000, and occasions t¼ 5 and 10. True median detection probability p is plotted onthe x-axis, and mean sp on the y-axis. The 4 lines on each panel show results for sp¼ 0, 0.5, 1, and 2.


p, and thus positively biased estimated of N , than does modelM0 for the same data sets. Superficially, this negative bias in �pfor model M0RE seems inconsistent with the overallexpectation that heterogeneity will tend to positively biasthe mean encounter probability from the sample (Fig. 1). Allother things being equal, if sp goes up, we might anticipatethat E(p) should also increase (Table 1).However, if we plot estimates of themean value of sp for each

of the discrete frequency bins (0.02) used for p (Fig. 8), for adesign point with a low number of sampling occasions (t¼ 5),and lowencounter probability (median p¼ 0.2),where the skewin the distribution of p wasmost pronounced (Fig. 8), we find astrong inverse relationship between the estimate of p and meansp (Fig. 9a). For simulations where estimated p > 0:211; sp <0:5 (true sp ¼ 0:5), whereas for simulations where estimatedp < 0:211; sp > 0:5, with sp increasing rapidly with decreas-ing p. In contrast, for t¼ 10 occasions, median p¼ 0.4, wherethe likelihoodofencounteringevery simulated individual at leastonce is very high p� ¼ 1� 1� 0:4ð Þ10 ffi 0:99

� �, sp did not

vary systematically with p, and was essentially constant at truesp ¼ 0:5 (Fig. 9b).Clearly, for some simulated data sets, Gaussian–Hermite

integration in model M0RE results in either 1) very low

estimates of p, relative to E(p), and high estimates of sp,relative to true sp under which the data were simulated, or 2)estimates of p that are somewhat higher than E(p), withcorresponding estimates of which are lower than the truevalue. Both situations tend to occur only under circumstanceswith low number of sampling occasions, particularly whencombined with low probability of encounter. In that case, theformer situation (i.e., low estimates of p and high estimatesof sp) occur more often that the opposite situation, asrepresented by the strong left skew in the distribution ofp (Fig. 8). As noted by Morgan and Ridout (2008), when asignificant amount of the mass of the encounter probabilityis nearer 0, occasionally extremely large estimates of Ncan result, which will strongly positively bias mean PRB.Further, Borchers (1996) showed that conditional (Horvitz–Thompson) estimators can exhibit severe positive bias whentrue encounter probabilities are low and sample size is not verybig.Norris (2012) has considered approaches to penalizing thelikelihood to mitigate this bias.Although it is difficult to precisely characterize when

Gaussian–Hermite integration will result in low estimates ofp, and high estimates of sp (or, for that matter, when it willyield higher than expected estimates of p, with lower

Figure 4. Confidence interval coverage ofN for 96 simulation scenarios with 1,000 replicates for each scenario. Each panel shows results forN¼ 100, 500, and1,000, and occasions t¼ 5 and 10 for each of the 4modelsM0RE (no time variation in p, individual random effects),MtRE (time variation in p, individual randomeffects),M0 (no time variation in p, no individual random effects), andMt (time variation in p, no individual random effects). True median detection probability(p) is plotted on the x-axis, and coverage on the y-axis. The 4 lines on each panel show results for true standard deviation of distribution of individual encounterprobabilities sp¼ 0, 0.5, 1, and 2.


estimates of sp), it is reasonable to predict that the encounterhistories in either case are over-sparse (i.e., too manyindividuals in the sample that are observed only once) orover-saturated (i.e., too many individuals in the sample thatare observed more than once, or, too few observed only once),respectively. A superficial examination of the simulatedencounter histories appears to support this conclusion. Amore rigorous but somewhat ad hoc approach to addressingthis hypothesis is to evaluate the relationship of the mean ofthe calculated Fletcher c values as a function of mean sp.Recall that the Fletcher c is based on departures from theexpected frequency of encounter histories in the sample,under multinomial expectations, and as such, we expect it toreflect certain systematic differences in the structure of oursimulated encounter data. In fact, consistent with ourhypothesis, there is a strong positive correlation between the2 (Fig. 10); for samples where the encounter histories hadsignificant lack of fit (i.e., Fletcher c was either above orbelow 1), there was a corresponding deviation of sp relativeto the true value. We believe these results are importantbecause they suggest that for situations where trueabundance, number of sampling occasions and encounterprobabilities are all low, such that there may be too many or

too few individuals encountered only once in the encounterhistory data, there will be a tendency for estimates ofabundance from model M0RE to be biased high, whereas forthe same data, estimates frommodelM0 (i.e., assuming thereis no heterogeneity in the encounter data) will tend to bebiased low.

DISCUSSION

Assessing performance of a statistical method requirescomparing estimates of �1 parameters against some datagenerated for known true value(s) of the parameter(s). Thedata for these evaluations can be either from empirical studieswhere the true value is known (Edwards and Eberhardt 1967,Greenwood et al. 1985, Conn et al. 2006, Grimm et al.2014), or based on numerical simulation (Norris and Pollock1996, Chao et al. 2001, Borchers et al. 2002). In this study,we have considered the performance of models forabundance estimation from closed populations in thepresence of individual heterogeneity in the probability ofencounter. Such heterogeneity is likely ubiquitous, to somedegree, and leads to significant bias in estimates of abundancein many cases. Our approach was to account for theheterogeneity by integrating out individual differences using

Figure 5. Mean of the Fletcher (2012) estimator (c) for 96 simulation scenarios with 1,000 replicates for each scenario. Each panel shows results for N¼ 100,500, and 1,000, and occasions t¼ 5 and 10 for each of the 4 models M0RE (no time variation in p, individual random effects), MtRE (time variation in p,individual random effects),M0 (no time variation in p, no individual random effects), andMt (time variation in p, no individual random effects). True mediandetection probability (p) is plotted on the x-axis, and mean c on the y-axis. The 4 lines on each panel show results for true standard deviation of distribution ofindividual encounter probabilities sp¼ 0, 0.5, 1, and 2.


Gaussian–Hermite quadrature, which has been implementedin several software applications (Program MARK, Whiteand Burnham 1999; Program E-SURGE, Choquet et al.2009; R package multimark, McClintock 2015).Overall, our results indicate that closed population

abundance estimates based on numerical integration usingGaussian–Hermite quadrature were generally unbiased,except for extreme cases of sparse data determined byrelatively low encounter rates (p� 0.1), low number ofsampling occasions (t� 5), or both. This was in markedcontrast to estimates from models that did not account forindividual heterogeneity, which were strongly negativelybiased, even for a fairly large number of sampling occasionsand moderate encounter probabilities. One reason theGaussian–Hermite quadrature was very good at accountingfor heterogeneity in our analysis is because our simulatedencounter data were based on a logit-normal generatingmodel; Gaussian–Hermite quadrature is optimal when theprobability density function for the data is normal Gaussian.Although it remains to be determined how well thisapproach would work if the distribution of encounter rateswas strongly asymmetric, the underlying model of normallydistributed individual random effects on the logit scale for p

provides a more realistic biological model of heterogeneitythan Pledger (2005) mixture models when individualheterogeneity is thought to occur over a continuous scalerather than a discrete set of mixtures. One extension toconsider would be to apply the logit-normal random effectseparately to discrete mixtures in a Pledger model, creating ahybrid of the Pledger model and the random effects modelapplied in this paper. Such an approach has been consideredfor analysis of species richness data (Daley and Smith 2016).The decline in PRBðN Þ from 5 to 10 occasions suggest that

the random effects estimators can provide unbiased estimatesfor low P values if enough occasions are available and theestimating model is the same as the model generating thedata. This is positive news for surveys such as camera surveysof individually identifiable animals where detection proba-bilities are often low (i.e.,<0.01/occasion). However, the flipside is that typical small-mammal trapping studies with�5 occasions will likely not detect important individualheterogeneity. As noted earlier, we have not simulatedscenarios where the model generating the data is not a logit-normal model, so we cannot speak to how well the logit-normal estimator might perform if other models are used togenerate data. Further, we also assumed that individual

Figure 6. The 90th and 95th percentiles of the Fletcher (2012) goodness-of-fit estimator (c) for 96 simulation scenarios with 1,000 replicates for each scenario.Each panel shows results for N¼ 100, 500, and 1,000, and occasions t¼ 5 and 10 for each of the percentile levels, and 2 modelsM0RE (no time variation in p,individual random effects), andMtRE (time variation in p, individual random effects). True median detection probability (p) is plotted on the x-axis, and mean con the y-axis. The 4 lines on each panel show results for true standard deviation of distribution of individual encounter probabilities sp¼ 0, 0.5, 1, and 2.


encounter probability is constant over occasions; in particu-lar, we assumed that individual encounter probabilities werenot influenced by the encounter process (i.e., no effects ofinitial encounter on subsequent encounter behavior). Chaoand Hsu (2000) reported that under heterogeneity andeffects of initial encounter (i.e., trap effects), there could besignificant differences in estimates between the conditional(Huggins) and unconditional likelihood estimators forabundances, whereas Fewster and Jupp (2009) reportedthe 2 approaches were near equivalent under heterogeneityalone.Several approaches are available to assess the significance of

individual heterogeneity in a single survey, and whether theindividual random effects estimators are preferred overestimators assuming no individual heterogeneity. First, theestimate of sp and its standard error provide inference onindividual heterogeneity. Second, the Fletcher (2012) estima-tor c shows good potential to detect individual heterogeneity(Figs. 5 and 6). Third, the likelihood ratio tests described byGimenez and Choquet (2010) show excellent power to detect

individual heterogeneity (Fig. 7).Note that comparingmodelsM0 and Mt to M0RE and MtRE using Akaike’s InformationCriterion (AIC) is not appropriate because the parameter sp isat the boundary of zero. Once the structure of the randomeffects has been selected via likelihood ratio tests (Fig. 7),standard model selection procedures using AIC or itsderivatives can be used to determine a structure on fixedeffects, for example, comparing p(t) versus p(.).Link (2003, 2004) pointed out that various models of

individual heterogeneity can fit a set of data identically, ornearly so, and provide widely different estimates of N. Thus,the choice of a model for individual heterogeneity shouldbe based on biological explanations of what might behappening, and not just blindly follow automated modelselection procedures, which will not be useful when multiplemodels fit the data equally well. Estimation of abundancewhen individual heterogeneity is hypothesized requiresmodel-based inference, so careful consideration of modelsthat are appropriate for the biology is required for modelselection.

Figure 7. Estimated likelihood ratio test power or size for true standard deviation of distribution of individual encounter probabilities, sp¼ 0, of the nullhypothesis of no individual heterogeneity for 96 simulation scenarios with 1,000 replicates for each scenario. Each panel shows results for N¼ 100, 500, and1,000, and occasions t¼ 5 and 10 for each of the 2 tests of modelsM0RE (no time variation in p, individual random effects) versusM0 (no time variation in p, noindividual random effects) andMtRE (time variation in p, individual random effects) versusMt (time variation in p, no individual random effects), for both thena€ıve uncorrected test and for the corrected test following Gimenez and Choquet (2010). True median detection probability (p) is plotted on the x-axis, andmean likelihood ratio on the y-axis. The 4 lines on each panel show results for true standard deviation of distribution of individual encounter probabilities sp¼ 0,0.5, 1, and 2.


The Huggins individual random effects estimator has beenimplemented in Program MARK for robust designs, multi-state robust designs, Pradel robust designs, and the Ivan et al.(2013) density estimators. Detailed descriptions of theseadvanced models are provided in Cooch and White (2016).The data simulator in Program MARK provides users theability to evaluate bias and confidence interval coverage fortheir applications using these models.In addition, design of studies can be conducted using

simulations to assess performance of estimators givenexpected specifications for prospective studies. The primaryissue with using the individual random effects estimator inthese more complex models is obtaining an adequate numberof occasions to be able to estimate sp. A lower level of >5occasions is likely necessary to achieve reasonable perfor-mance of this estimator. To visualize why 5 occasions areusually required, consider that the effect of individualheterogeneity is to increase the proportion of animals caughtfew and many times, and to decrease the proportion ofanimals encountered an average number of times. In effect,

the histogram of number of encounters is lowered in themiddle, and raised in the tails (Fig. 8 and 9). To detect thiseffect requires a reasonable number of occasions, with themore the better.

MANAGEMENT IMPLICATIONS

Individual heterogeneity is a common source of bias causingcapture-mark-reencounter estimates of population abundanceto be biased low. We have demonstrated that Gaussian–Hermite quadrature to numerically integrate out individualrandom effects is an effective approach to eliminate this biaswhen an adequate number of capture occasions are availableand detection probabilities are relatively high. To have muchhope of estimating abundance with little bias, encounterprobabilitymust be relatively high. In sampling situationswithlowp, and lownumberof samples (theworst-case scenario), theinvestigator will need to be aware of the potential for biasedestimates, and evaluate whether not the direction of bias(which reflects the estimator) is important in the appliedcontext. For example, for management of small, threatened

Figure 8. Comparison of frequency distribution of estimated encounter probabilities over 1,000 simulations, from model M0 (no time variation in p, noindividual random effects) versus modelM0RE (no time variation in p, individual random effects), as a function of the number of encounter occasions (t¼ 5 or10) and median detection probability (p¼ 0.2 or 0.4). True standard deviation of distribution of individual encounter probabilities, sp¼ 0.5, andN¼ 100. Solidvertical lines indicatemean estimate of p overall all simulations, for bothmodels. Dashed vertical line indicates truemean detection probability of the population,E(p), given median p used in the simulation.


populations, the potential ramifications of under-estimating(negativebias)orover-estimating (positivebias) abundanceareclearly of some importance.

ACKNOWLEDGMENTS

We thank D. Borchers, K. Burnham, B. McClintock, and A.Royle for comment and discussion during the developmentof this manuscript. We especially thank D. Fletcher foruseful discussions on the application of his lack-of-fitstatistic to closed population abundance estimation. We alsoacknowledge the thoughtful and helpful comments from theEditor, J. Laake, S. McCorquodale, and B. Steidl in revisingthe manuscript.

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Figure 10. Plot of mean estimated Fletcher c versus mean estimatedheterogeneity s p (standard deviation of distribution of individual encounterprobabilities) from model M0RE (no time variation in p, individual randomeffects), t¼ 5 occasions, median detection probability p¼ 0.2, N¼ 100, andsp¼ 0.5. Mean for both parameters is based on binning of encounterprobability data at discrete intervals of 0.1. Dotted lines indicate true value ofsp (0.5), and value of Fletcher c assuming perfect fit of the model to the data(c ¼ 1.0).


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Associate Editor: Scott McCorquodale.

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