Closed Population Capture-recapture Models

Chapter 14

Closed population capture-recapture modelsPaul Lukacs, Colorado Division of Wildlife

A fair argument could be made that marking individuals in a wild population was originallymotivated by the desire to estimate a fundamental parameter - abundance (i.e., population size). Bycomparing the relative proportions of marked and unmarked animals in successive samples, variousestimators of animal abundance could be derived. In this chapter, we consider the estimation ofabundance from closed population capture-recapture data using program MARK. The populationof interest is assumed to be closed geographically – no movement on or off the study area – anddemographically – no birth or death.

There is more than a century of literature on estimating abundance from capture-recapture data.Here we focus on the likelihood-based models currently available in MARK. Despite the descriptionin the ‘About’ window of MARK, the program does much more than estimate survival. All of thelikelihood-based models from Program CAPTURE can be built in MARK plus numerous modelsthat have been developed since then.

14.1. The basic idea

How many are there in the sampled population? Well, if you assume (or, if in fact) the populationis closed, then the number of individuals in the population being sampled is a constant over time.Meaning, the population size does not change at each sampling event. With a little thought, youquickly realize that the canonical estimate of population size is a function of (i) how many uniqueindividuals are encountered over all sampling events, and (ii) what the probability is of encounteringa individual at least once. For a single sampling event, we can express this more formally as

N =n

p

where the numerator (n) is the number of unique individuals encountered, and the denominator (p)is the probability that any individual will be encountered.

This expression makes good intuitive sense - for example, suppose that you capture 50 individuals(n = 50), and the encounter probability is p = 0.5, then clearly, since there is a 50:50 chance that youwill miss an individual instead of encountering it, then

c© Cooch & White (2010) c. 15/5/10

14.1.1. The Lincoln-Petersen estimator - a quick review 14 - 2

N =n

p=

500.5

= 100

14.1.1. The Lincoln-Petersen estimator - a quick review

In the preceding, we assume that we have an estimate of the encounter probability p. The mostgeneral approach to estimating abundance, and p, in closed populations is based on what is knownas the Lincoln-Petersen estimator (hereafter, the ‘LP estimator’). The LP estimator is appropriate whenthere are just two sampling occasions, and that the population is closed between the two occasions.Imagine you go out on the first occasion, capture a sample of individuals from the population youare interested in, mark and release them back into the population. Then, on the second occasion, youre-sample from (what you hope is) the same population. In this second sample, there will (potentially)be two types of individuals: those that are unmarked (not previously capture) and those with marks(individuals captured and marked on the first occasion).

For such a study, there are only 4 possible encounter histories: 11 (captured, marked and releasedon occasion 1, recaptured on occasion 2), 10 (captured, marked and released on occasion 1, notrecaptured on occasion 2), 01 (not captured on occasion 1, captured, marked and released for the firsttime on occasion 2), and 00 (not captured at either occasion 1 or occasion 2). Clearly, the number ofindividuals with encounter history 00 is not known directly, but must be estimated. So, the estimationof abundance proceeds by using the number of individuals observed who were encountered at leastonce.

Following convention, let x11 be the number of individuals caught on both occasions, x10 be thenumber of individuals caught on only the first occasion, and x01 be the number of inviduals caughtonly on the second occasion. Let n1 = x11 + x10 be the total number of individuals caught on thefirst occasion, and n2 = x11 + x01 be the total number of individuals caught on the second occasion.Finally, let m2 = x11 be the number of individuals caught on both occasions. Thus, the number ofdistinct individuals captured during the study is r = n1 + n2 − m2 (make sure you understand thealgebra).

After making some assumptions (i.e., (i) the population is closed, (ii) marks are not lost oroverlooked, and (iii) all animals are equally likely to be captured, regardless of whether or not they’vebeen previously captured), we can proceed with the estimation of abundance. The most intuitive (andeasily most familiar) approach to estimating abundance using an LP estimator is to note that theproportion of marked animals in the population after the first sample is simple n1/N, where N is thesize of the population (which, of course, is what we’re trying to estimate). Note that the numeratorof this expression (n1) is known. If our assumption that all individuals (marked or not) are equallycatchable, then this proportion should be equivalent to the proportion of marked individuals in thesecond sample. In other words,

n1

N=

m2

n2

A little algebra, and we come up with the familiar LP estimator for abundance, as

N =n1n2

m2

We might also use the canonical form noted earlier, estimating p as m2/n2. Thus

Chapter 14. Closed population capture-recapture models

14.2. Model Types 14 - 3

N =n1

p1=

n1n2

m2

which is clearly identical to the ‘intuitive’ LP estimator we derived earlier.

More formally, we can appeal to probability theory, and express the probability distribution for thetwo sample study as

P (n1, n2, m2|N, p1, p2) =N!

m2! (n1 − m1)! (n2 − m2)!(N − r)!

× (p1 p2)m2 (p1 (1 − p2))

n1−m2 ((1 − p1) p2)n2−m2 ((1 − p1) (1 − p2))

N−r

Note that the probability expression is written including a term for each encounter history, andwith the exponent representing the number of individuals with a given encounter history (expressedin the standard notation introduced earlier). For example, the probability of encounter history ‘11’ isp1 p2, the probability of encounter history ‘10’ is p1(1 − p2), and so on. The ML estimates under thismodel can be derived fairly easily.

It is sometimes convenient to use a conditional likelihood approach to estimating abundance, whereN is not actually considered as a parameter. This is possible if you condition the analysis only on thoseindividuals which are encountered (i.e., r). The probability that any individual in the population isencountered at least once during a two-sample study is

p′ = 1.0 − (1 − p1)(1 − p2)

Thus, we can re-write the conditional probability expression for the capture histories as

P({

xij

}|r, p1, p2

)=

r!x11!x10!x01!

×(

p1 p2

p′

)x11(

p1(1 − p2)

p′

)x10((1 − p1)p2

p′

)x01

The ML estimates for this model are again fairly easy to derive (see Williams, Nichols & Conroy2001 for the details).

Regardless of whether or not you include N in the likelihood, the key to understanding the fitting ofclosed capture models is in realizing that the event histories are governed by the encounter probability. In fact, the process of estimating abundance for closed models is in effect the process of estimatingdetection probabilities - the probability that an animal will be caught for the first time (if at all),and the probability that if caught at least once, that it will be caught again. The different closedpopulation models differ conceptually on how variation in the encounter probability (e.g., over time,among individuals) is handled. The mechanics of fitting these models in MARK is the subject of therest of this chapter.

14.2. Model Types

MARK currently supports 12 different closed population capture-recapture data types. These differ-ent data types can be classified within a hierarchy of dichotomous divisions - as shown in the diagramat the top of the next page. The first and most important split is between the models with abundancein the likelihood (Otis et al. 1978) and those with abundance conditioned out of the likelihood (Huggins1989). This is a major division that results in the two types of models not being comparable with



standard AIC-based model selection techniques. The remainder of the splits reflect one or moreconstraints on different parameters. As a matter of convention in this chapter, I will use bold p’s andc’s to indicate a set (vector) of parameters that are (potentially) time varying, italic, unsubscripted p’sand c’s to indicate constant parameters, and italic, subscripted p’s and c’s refer to specific samplingoccasions. At various points, I’ll indicate where my terminology differs from the original or traditionalterminology used to describe various models.

When you select ’closed captures’ in the data type specification window, MARK presents you witha popup window allowing you to select among these 12 different data types:

The first data type is labeled “Closed Captures". These are the models of Otis et al. (1978). They



are based on the full likelihood parameterization with three types of parameters; pi is the probabilityof first capture (i.e., the probability that an animal in the population will be captured - and marked -for the very first time), ci is the probability of recapture (conditional on having been captured at leastonce before), and N is abundance. Both pi and ci can be time specific, so long as a constraint is placedon pt (the final capture probability; this will be detailed later).

The second data type is labeled “Huggins Closed Capture’". These are the models of Huggins(1989). In this model, the likelihood is conditioned on the number of animals detected and N thereforedrops out of the likelihood. These models contain only pi and ci; the abundance N is estimated asa derived parameter. The primary advantage of the Huggins data type is that individual covariatescan be used to model p and c. Individual covariates cannot be used with the full likelihood approachbecause the term (1 − p1)(1 − p2)...(1 − pt) is included in the likelihood, and no covariate valueis available for animals that were never captured. In contrast, the Huggins parameterization hasconditioned this multinomial term out of the likelihood, and so an individual covariate can bemeasured for each of the animals included in the likelihood. When individual covariates are used, aHorvitz-Thompson estimator is used to estimate N:

N =Mt+1

∑i=1

11 − [1 − p1(xi)][1 − p2(xi)]...[1 − pt(xi)]

begin sidebar

Huggins models and ‘conditioned out of the likelihood’?

The closed captures data type consist of 2 versions of 3 different parameterizations of p and c: thefull likelihood version and Huggins version. For the Huggins (1989, 1991) version, the populationsize (N) is conditioned out of the likelihood. This basic idea was introduced earlier (p. 3).

An example is the best way to illustrate this concept. Consider the 8 possible encounter historiesfor 3 occasions with the p, c data type:

Encounter history probability

111 pcc

110 pc(1 − c)

101 p(1 − c)c

011 (1 − p)pc

100 p(1 − c)(1 − c)

010 (1 − p)p(1 − c)

001 (1 − p)(1 − p)p

000 (1 − p)(1 − p)(1 − p)

For each of the encounter histories except the last, the number of animals with the specificencounter history is known. For the last encounter history, the number of animals is (N − Mt+1), i.e.,the population size (N) minus the number of animals known to have been in the population (Mt+1).The approach described by Huggins (1989, 1991) was to condition this last encounter history out ofthe likelihood by dividing the quantity 1 minus this last history into each of the others. The resultis a new multinomial distribution that still sums to one.

The derived parameter N is then estimated as Mt+1/[1 − (1 − p)(1 − p)(1 − p)] for data withno individual covariates. A more complex estimator is required for models that include individualcovariates to model the p parameters.

end sidebar


14.2.1. What does closure really mean? 14 - 6

The third data type is the “Closed Captures with Heterogeneity’" These models incorporate a finite

mixture as an approximation to individual heterogeneity in the pi and ci parameters. In this model,

pi =

{pi,A with Pr(π)

pi,B with Pr(1 − π)

}

for the case with two mixtures A and B, although the model can be extended to >2 mixtures. Aswritten (above), the parameter π is the probability that the individual occurs in mixture A. For >2mixtures, additional π parameters must be defined (i.e., πA, πB,...), but constrained to sum to 1.

Note that the ‘heterogeneity models’ for both closed captures and the Huggins’ models comein one of two forms, differentiated by the presence of the word ‘full’ (e.g., ‘Closed Captures withHeterogeneity’ versus ’Full Closed Captures with Heterogeneity’). The former parameterizationsrepresent simple individual heterogeneity models, with parameters π, pi,A ≡ pA, and pi,B ≡ pB,and assumes no temporal or behavioral variation. Thus, estimates under model Mh are simple toobtain with this parameterization. In contrast, the full parameterization provides for all three effectsof time, behavior, and heterogeneity, so that estimates under models Mbh, Mth, and Mtbh are possible.Of course, any of the reduced models can be run from the full parameterization if the appropriateconstraints are applied.

The final six data types generalize the previous six data types to handle uncertainty in identificationof individuals, typically from genotyping error (Lukacs and Burnham 2005). These models include anadditional parameter, α, that is the probability that the individual was correctly identified on its firstobservation. In these models, N is estimated as a derived parameter. It is possible to construct modelsfor every data type with only the ‘Closed Capture with Heterogeneity’ and misidentification and‘Huggins Closed Captures’ with heterogeneity and misidentification data types. The other data typesare included to allow the user a less cumbersome set of parameters for building more constrainedmodels.

14.2.1. What does closure really mean?

The closed captures data types, as the name implies, all assume the population of interest is closedduring the sampling period. Strictly speaking, the models assume that no births or deaths occur andno immigration or emigration occurs. Typically, we refer to a closed population as one that is free ofunknown changes in abundance, as we can usually account for known changes. White et al. (1982:3–4)provide a good overview of the closure assumption.

There are a few modifications to closed populations that can be easily handled. For example,removal models explicitly remove all individuals that are caught from the population (at leasttemporarily). This is handled by setting c = 0. N then refers to the abundance at the beginningof the study. We’ll discuss removal models later in this chapter.

A few methods have been developed to test for closure violations. Program CloseTest exists totest the assumption of closure (Stanley and Burnham 1999). The Pradel model with recruitmentparameterization has also been used to explore closure violations (Boulanger et al. 2002; see chapter12 for details of the Pradel model). By analyzing closed population capture-recapture data withthe Pradel recruitment parameterization, one could test for emigration and immigration. To test foremigration, compare a model with φ = 1 to a model with φ unconstrained. To test for immigration,compare a model with f = 0 to an model with f unconstrained. A likelihood ratio test could be usedfor the comparison.


14.3. Likelihood 14 - 7

Heterogeneity in capture probability can cloud our ability to detect closure violations. In situationswhere the population is truly closed, heterogeneity in capture probability can cause both the tests ofimmigration and emigration to reject the null hypothesis of closure.

14.3. Likelihood

The closed population capture-recapture models are quite different than the open models we’vediscussed in previous chapters. Here, attention is focused on estimating abundance, N. So, now N

appears in the multinomial coefficient of the likelihood function. In addition, the encounter historyrepresenting individuals that were never caught (i.e., 000 for a three occasion case) also appears in thelikelihood, but not in the data (i.e., the encounter histories file) - since (obviously) there are no datafor individuals that were never captured!

The likelihood for the Closed Captures data type is

L(N, p, c|data) ∝N!

(N − Mt+1)!∏

h

Pr[h]nh · Pr[not encountered]N−Mt+1

where Mt+1 is the number of unique animals marked and nh is the number of individuals withencounter history h.

It is possible to rewrite the likelihood in terms of the number of individuals never caught, f0, suchthat f0 = N − Mt+1 (the notation f0 originates from the frequency (count) of animals observed 0times). The likelihood now becomes

L( f0, p, c|data) ∝( f0 + Mt+1)!

f0! ∏h

Pr[h]nh · Pr[not encountered] f0 .

The f0 parameterization is useful for computation because f0 is bounded on the interval [0, ∞],thus forcing the logical constraint that N ≥ Mt+1. MARK uses the f0 parameterization for ease ofcomputation by using the log link function to constrain f0 ≥ 0, but presents the results in terms of N

as both a real and derived parameter. Therefore, N = f0 + Mt+1 and var[N] = var[ f0].

In order to move on to the heterogeneity and misidentification extensions to the closed populationmodels, let’s first consider the encounter histories and their probabilities for a 4-occasion case for theClosed Captures data type.

History Cell Probability

1000 p1(1 − c2)(1 − c3)(1 − c4)

0100 (1 − p1)p2(1 − c3)(1 − c4)

0010 (1 − p1)(1 − p2)p3(1 − c4)

0001 (1 − p1)(1 − p2)(1 − p3)p4

1100 p1c2(1 − c3)(1 − c4)

1010 p1(1 − c2)c3(1 − c4)

1001 p1(1 − c2)(1 − c3)c4

1110 p1c2c3(1 − c4)

History Cell Probability

1101 p1c2(1 − c3)c4

1011 p1(1 − c2)c3c4

0110 (1 − p1)p2c3(1 − c4)

0101 (1 − p1)p2(1 − c3)c4

0011 (1 − p1)(1 − p2)p3c4

0111 (1 − p1)p2c3c4

1111 p1c2c3c4

0000 (1 − p1)(1 − p2)(1 − p3)(1 − p4)


14.3.1. constraining the final p 14 - 8

There are 2k possible encounter histories for a k-occasion study.

Now if we want to add a finite mixture to the cell probability, we now have the following probabili-ties (here, again, encounter histories and cell probabilities for a 4-occasion closed population capture-recapture model of the ‘Full Closed Captures with Heterogeneity’ data type, with two mixtures):

history cell probability

1000 ∑2a=1 (πa pa1(1 − ca2)(1 − ca3)(1 − ca4))

0100 ∑2a=1 (πa(1 − pa1)pa2(1 − ca3)(1 − ca4))

0010 ∑2a=1 (πa(1 − pa1)(1 − pa2)pa3(1 − ca4))

0001 ∑2a=1 (πa(1 − pa1)(1 − pa2)(1 − pa3)pa4)

1100 ∑2a=1 (πa pa1ca2(1 − ca3)(1 − ca4))

1010 ∑2a=1 (πa pa1(1 − ca2)ca3(1 − ca4))

1001 ∑2a=1 (πa pa1(1 − ca2)(1 − ca3)ca4)

1110 ∑2a=1 (πa pa1ca2ca3(1 − ca4))

1101 ∑2a=1 (πa pa1ca2(1 − ca3)ca4)

1011 ∑2a=1 (πa pa1(1 − ca2)ca3ca4)

0110 ∑2a=1 (πa(1 − pa1)pa2ca3(1 − ca4))

0101 ∑2a=1 (πa(1 − pa1)pa2(1 − ca3)ca4)

0011 ∑2a=1 (πa(1 − pa1)(1 − pa2)pa3ca4)

0111 ∑2a=1 (πa(1 − pa1)pa2ca3ca4)

1111 ∑2a=1 (πa pa1ca2ca3ca4)

0000 ∑2a=1 (πa(1 − pa1)(1 − pa2)(1 − pa3)(1 − pa4))

Note: The finite mixture models have a separate set of p’s and c’s for each mixture.

14.3.1. constraining the final p

A subtlety of the closed population models is that only the initial capture probabilities are used toestimate N, as demonstrated in the equation above for how N is estimated for the Huggins model.

However, an even deeper subtlety occurs in that the last p parameter is not identifiable unless aconstraint is imposed. So, for example, in model Mt, the constraint of pi = ci is imposed, providingan estimate of the last p from the last c. Likewise, under model Mb, the constraint of pi ≡ p. isimposed, so that the last p is assumed equal to all the other p values. A similar constraint is usedfor model Mbh, i.e., pi,A ≡ pA, pi,B ≡ pB, and so on. Under model Mtb, the pi and ci are modeledas a constant offset (Obeh) of one another, i.e., ci = (pi + Obeh). This relationship will depend onthe link function used, but the last pi is still obtained as ci minus the offset (where the offset isestimated from the data on the other pi and ci ). Under model Mtbh, the offset between the pi and ci

is applied, with an additional offset(s) included to model the relationship among the mixtures, i.e.,pi,B = (pi,A + OB), pi,C = (pi,A + OC), with a different offset applied to each succeeding mixture.Similarly, ci,B = (pi,B + Obeh) = (pi,A + OB +Obeh), with the resulting relationship depending on thelink function applied. With this model, the relationship between the mixtures of the pi is maintained,i.e., the ordering of the mixtures is maintained across occasions.Model Mth can also be modeled asan additive offset between the mixtures, although other relationships are possible because the last


14.3.2. Including misidentification 14 - 9

pi for each mixture is estimated from the corresponding last ci. Although other relationships thanthose of the preceding paragraph can be proposed to provide identifiability, the proposed modelsmust provide identifiability of all the initial capture probabilities. When no constraint is imposed onthe last pi , the likelihood is maximized with the last p = 1, giving the estimate N = Mt+1. Thus, adiagnostic to determine whether the estimated model is of value is to check to see whether N = Mt+1,and if so, to see if the last pi estimate equals 1.

14.3.2. Including misidentification

The likelihoods and cell probabilities get more complicated when we want to include the possibilityof misidentification into the cell probabilities. In order to do this we must assume that (i) an individualencountered more than once is correctly identified (i.e., individuals captured on multiple occasionsare correctly identified - presumably owing to the greater amount of information gathered on whichto base the identification), and (ii) individuals encountered only once may or may not be correctlyidentified.

First, we consider the closed capture cell probabilities without finite mixtures. We will add the pos-sibility of misidentification to the probabilities (here, again, encounter histories and cell probabilitiesfor a 4-occasion closed population capture-recapture model of the Closed Captures, but now withmisidentification data type):

history cell probability

1000 p1α(1 − c2)(1 − c3)(1 − c4) + p1(1 − α)

0100 (1 − p1) [p2α(1 − c3)(1 − c4) + p2(1 − α)]

0010 (1 − p1)(1 − p2) [p3α(1 − c4) + p3(1 − α)]

0001 (1 − p1)(1 − p2)(1 − p3) [p4α + p4(1 − α)]

1100 p1αc2(1 − c3)(1 − c4)

1010 p1α(1 − c2)c3(1 − c4)

1001 p1α(1 − c2)(1 − c3)c4

1110 p1αc2c3(1 − c4)

1101 p1αc2(1 − c3)c4

1011 p1α(1 − c2)c3c4

0110 (1 − p1)p2αc3(1 − c4)

0101 (1 − p1)p2α(1 − c3)c4

0011 (1 − p1)(1 − p2)p3αc4

0111 (1 − p1)p2αc3c4

1111 p1αc2c3c4

0000 (1 − p1)(1 − p2)(1 − p3)(1 − p4)

In the encounter histories for individuals encountered only once their probability expression isa summation across the two possible ways the history could have occurred; for example, considerhistory 0100; captured for the first time, marked and released alive at occasion 2. Conditional onbeing alive and in the sample (i.e., available for capture) over the entire sampling period, then theprobability of observing encounter history 0100 is (1 − p1) (the probability of not being captured at


14.4. Encounter Histories Format 14 - 10

the first occasion), times the sum of (1) the probability the individual was correctly identified and notseen again (p2α (−c3) (1 − c4), or (2) the individual was misidentified and therefore unable to be seenagain p2 (1 − α).

When misidentification occurs, the constraint that N ≥ Mt+1 no longer holds. It is possible thatenough animals are misidentified such that the number detected is greater than the number thatactually exist in the population. Now abundance is estimated as

N = α(

f0 + Mt+1

).

Therefore, in these models where misidentification is possible MARK presents f0 in the realparameter output and N in the derived parameter output as it is a function of more than oneparameter.

The Huggins model data types condition on the number of individuals detected to estimate p

and c. Therefore, all of the cell probabilities are divided by the probability that that an individualis encountered at least once. The 0000 history is no longer in the likelihood. Because we no longerconsider the history of animals not encountered, we can use individual covariates to model captureprobability because each animal included in the likelihood was actually captured, and hence itsindividual covariates could be measured. When the 0000 history is included in the likelihood, wecannot have measured individual covariates because these animals were never captured.

begin sidebar

Individual heterogeneity – the bane of abundance estimation

Individual heterogeneity refers the the variation among individual animals in their probabilityof detection. Most capture-recapture models assume that capture probability is constant acrossindividuals within a group. When individuals vary in the their capture probabilities, the mostcatchable animals are likely to be caught first and more often. This leads to capture probabilitybeing over estimated and abundance being underestimated.

Many attempts have been made to deal with heterogeneity in capture probability over the past30+ years. No single method has really solved the problem, although several methods are useful.MARK allows individual heterogeneity to be approximated with finite mixtures or with individualcovariates. It also allows the jackknife and frequency of capture methods to be fit with CAPTURE.

The single best way to minimize the bias caused by individual heterogeneity is to get p as high aspossible – the ‘big law’ of capture-recapture design. When p is high there is little room for variationand little chance that an individual is not detected. Link (2003) demonstrated that different modelsof the form of individual heterogeneity can lead to very different estimates of abundance and fit thedata equally well. The magnitude of the differences in abundance estimates is related to p; when p

is small the differences can be large. Therefore, to have much hope of estimating abundance withlittle bias, capture probability must be relatively high.

end sidebar

14.4. Encounter Histories Format

All of the closed capture-recapture models use the LLLL encounter histories format (see chapter 2for more detail). By the definition of a closed population, animals are not dying, therefore a deadencounter is not possible. On the same line of reasoning, time between sampling occasions is notrelevant because there is no survival or movement process to consider. Encounter histories arefollowed by group frequencies. For the Huggins models, group frequencies can be followed with


14.5. Building Models 14 - 11

individual covariates. All encounter histories end with the standard semicolon.

/* Closed capturerecapture data for a Huggins model.

tag #, encounter history, males, females, length */

/* 001 */ 1001 1 0 22.3;

/* 002 */ 0111 1 0 18.9;

/* 003 */ 0100 0 1 20.6;

If you wish to analyze a data set that contains covariates in the input with both full and conditionallikelihoods, you must initially import that data set by selecting a Huggins data type. The ClosedCaptures data type will not allow individual covariates to be specified. In this case, it is likely best tocreate two separate MARK files for the analysis because the AICc values are not comparable betweenthe Closed Captures and Huggins data types.

14.5. Building Models

Now it is time to move on to the actual operation of MARK. I will base this on simulated datacontained in (simple_closed1.inp). In this simulated data set (which consists of 6 encounter occasions),true N = 350. The total number of individuals encountered was Mt+1 = 339 (so, 11 individuals werenever seen). Open MARK and create a new database using the ‘File’ menu and ‘New’ option. Selectthe ‘Closed Captures’ radio-button. When you click on the ‘Closed Captures’ radio-button, a windowwill open that allows you to select a model type, shown earlier in this chapter. To start, select ‘ClosedCaptures.’ Then enter a title for the analysis, select the input file, and set the number of encounteroccasions to 6.



To start, we’ll construct some of the ‘standard’ closed capture models, as originally described inOtis et al. (1978). Model notation for the closed capture-recapture models in the literature often stillfollows that of Otis et al. (1978). Now that more complex models can be built, it seems appropriateto use a notation that is similar to the notation used for other models in MARK. Thus, my notationin this chapter will be based on a description of the parameters in the models - here is a tablecontrasting model notation based on Otis et al. (1978) and expanded notation based on a descriptionof the parameters. Combinations of the models described below are possible.

Otis notation Expanded notation Description

M0 {N, p(.) = c(.)} Constant p

Mt {N, p(t) ≡ c(t)} Time varying p

Mb {N, p(.), c(.)} Behavioral response

Mh or Mh2 {N, pa(.) = ca(.), pb(.) = cb(.), π} Heterogeneous p

If you look closely at the ‘expanded notation’, you’ll see that models are differentiated based onrelationships between the p and c parameters. This is important - the closed capture-recapture modelsare one of the model types in MARK where different types of parameters are modeled as functionsof each other. In this case p and c are commonly modeled as functions of one another. This makesintuitive sense because both p and c relate to catching animals.

With that said, let’s begin building a few models to learn some of the tricks of using MARK toestimate abundance. We’ll start with models {N, p(.) = c(.)}, {N, p(t) ≡ c(t)}, and {N, p(.), c(.)} (i.e.,models M0, Mt and Mb). Let’s first examine the default PIM chart for the ‘Closed Capture’ models.



MARK defaults to a general time-varying parameter structure where there is a different p andc for each occasion. You may recall (from 14.3.1) that abundance is not estimable with this modelstructure because no constraint is imposed to estimate pt. If this model is fit to the data, N = Mt+1and p10 = 1.0 regardless of the data. Therefore, in every model we build we must put some constrainton pi for the last encounter occasion so that this parameter is estimated.

If we open the PIM windows (say, for p), we’ll notice that the p’s and c’s have only a single row oftext boxes.

In the closed capture models, every individual is assumed to be in the population and at risk ofcapture on every occasion. Therefore, there is no need for cohorts (expressed as multiple rows in thePIM window) as there is in some of the open-population models.

We’ll start with {N, p(.) = c(.)} - for this model, there is no temporal variation in either p and c,and the two parameters are set equal to each other. This model can be easily and quickly constructedusing the PIM chart:



Go ahead and run this model, and add the results to the browser. Couple of important things tonote. First, it is common for AICc values to be negative for the full likelihood closed captures models.Negative AICc values are legitimate and interpreted in the same way as positive AICc values. Thenegative number merely arises due to the portion of the multinomial coefficient that is computed.Keep in mind that minimum AICc remains the target and the model with the ‘most negative’ AICc,i.e., the one furthest from zero, is the most parsimonious model. AICc values from the conditionallikelihood models are typically positive.

In addition, note that MARK defaults to a sin link just as it does with all other data types when anidentity design matrix is specified. In the case of the closed models, the sin link is used for the p’s andc’s, but a log link is used for N (more precisely f0). The log link is used because f0 must be allowedto be in the range of [0 → ∞]. Therefore, no matter what link function you select a log link will beused on f0. If you choose the ‘Parm-Specific’ option to set different link functions for each parameter,be sure you choose a link that does not constrain f0 to the [0 → 1] interval. Choose either a log oridentity link (log is preferable).

Now, we’ll build model {N, p(t) ≡ c(t)} (i.e., model Mt). It is important to note that there is no c

for the first occasion because it is impossible for an animal to be recaptured until it has been capturedonce. Therefore, MARK offers an easy way to assure that the correct p’s line up with the correct c’s:under the ‘Initial’ menu select ‘make c=p’ and renumber with overlap. The constraint on p5 in thismodel is that p5 = c5. Here is the PIM chart:

Finally, we’ll build model {N, p(.), c(.)} (i.e., model Mb). Here, we’re accounting for possibledifferences in ‘behavior’ (i.e., encounter probability) between the first encounter, and subsequentencounters. Such a ‘behavioral’ effect might indicate some permanent ‘trap effect’ (trap ‘happiness’or trap aversion). For model {N, p(.), c(.)}, there is a ‘behavior’ effect, but no temporal variation. ThePIM chart for this model is shown at the top of the next page. Note that there is no ‘overlap’ (i.e.,no function relating p and c) for this model - this is analogous to the default model {N, p(t), c(t)},


14.5.1. Closed population models and the design matrix 14 - 15

shown at the bottom of p. 14-12. However, in this instance, all parameters are estimable because ofthe constraint that p and c are constant over time - the lack of estimability for the final p occurs onlywhen both p and c are time-dependent.

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simple extension - removal models

Now let’s consider a removal model. These are commonly used in fisheries work where the re-searcher does not want to subject a fish to multiple passes of electricity. Therefore, the fish thatare encountered are held aside until all sampling has occurred. To do this, build an {N, p(t), c(.)} or{N, p(.), c(.)} model (in this case I use the PIMs). Then click ‘Run’ to open the run window. Click thefix parameters button. A window will open listing all of the real parameters in the model. Simplyfix c = 0, and run the model. Note - a removal model requires that the number of captures decreasewith occasion.

end sidebar

14.5.1. Closed population models and the design matrix

In the preceding, we constructed 3 simple models using the PIM chart. While using the PIM chartwas very straightforward for those models, through the design matrix MARK allows models to be fitthat were not possible with the PIM chart. For example, it is possible to build an {N, p(t) = c(t) + b}model where capture probability and recapture probability are allowed to vary through time, butconstrained to be different by an additive constant on the logit scale. It is also worth noting that theseextended models are also not available in program CAPTURE - and represent one of several reasonsthat CAPTURE is not longer preferred for fitting closed population abundance models.



As introduced in Chapter 6, one approach to doing this is to first build a general model usingPIMs, and then construct the design matrix corresponding to this general model. Then, once youhave the general model constructed using the design matrix, all other models of interest can beconstructed simply by modifying the design matrix. In this case, the most general model we canbuild is {N, p(t), c(t)}. As noted in above, we know before the fact that this is particular model is nota useful model, but it is convenient to build the design matrix for this model as a starting point. Todo this we need the PIMs in the full time varying setup (again, as shown at the bottom of p. 4-12).Go ahead and run this model, and add the results to the browser. Look at the parameter estimates- note that (i) p5 = 1.0, and (ii) N = Mt+1 = 339. Note as well that the reported SEs for both p5and N are impossibly small - a general diagnostic that there is ‘something wrong’ with this model(as expected for this model). In fact, as discussed earlier, this is not a useful model without imposingsome constraints since the estimate of N = Mt+1.

Now, the design matrix. Recall that there are 12 parameters specifying this model: 1 → 6 for p,7 → 11 for c, and 12 for N. Thus, our design matrix will have 12 columns. Now, if you select ‘Design| Full’, MARK will respond with the following default DM:

Here, we see a DM which is strictly analogous to what we might have expected for 3 parameters -each parameter (in this case, p, c and N) has a separate ‘block’ within the matrix: p in the upper-left,c in the middle, and N in the lower-right. If you go ahead and run this model, you’ll see that it givesyou exactly the same model deviance as the general model built with PIMs.

You’ll also note, however, that the AICc reported for this DM-built general model is not the sameas the AICc reported for the general model built with PIMs (-530.1030 versus -528.0812). If the modeldeviances are the same, but the reported AICc values are different, then this implies that the numberof estimated parameters is different. In fact, we see that the number estimated for the ‘full default DM’model is 10, whereas for the model built with PIMs, the number reported is 11. In fact, for this model,the difference in the number reported isn’t particularly important, since this is not a ‘reasonable’model in the first instance (as mentioned several times earlier in this chapter). The fact that the modeldeviances ‘match’ indicates that the DM is correct.



However, while this is ‘technically’ true, the default DM assumes that there is no interest increating a functional relationship between any of the parameters. While normally this is a reasonableassumption (e.g., in a CJS live encounter study, there is no plausible reason to create a functionalrelationship between φ and p), this is clearly not the case for closed population abundance models,where many of the models of interest are specified by imposing a particular relationship between p

and c. For example, model {N, p(t) ≡ c(t)} imposes a relationship between p and c at each samplingoccasion t.

How do we accommodate our interest in specifying these relationships between p and c in theDM? In fact, it is very easy, with a simple conceptual ‘trick’ - we’re going to treat the two parametersp and c as if they were levels of some putative ‘treatment’ - in precisely the same way we handledage (TSM) effects for individuals marked as young in age (TSM) models (Chapter 7 - section 7.2). Asa reminder, recall how we would construct the design matrix to correspond to the PIM for survivalfor a simple age model, with 2 age classes, and time-dependence in each age class. Assume that wehave 7 occasions. Recall that the PIM for this model looks like:

1 7 8 9 10 112 8 9 10 11

3 9 10 114 10 11

5 116

So, based on the number of indexed parameters in the PIM, we know already that our designmatrix for survival would need to have 11 rows and 11 columns. What does the linear model looklike? Again, writing out the linear model is often the easiest place to start. In this case we see thatover a given time interval, we have, in effect, 2 kinds of individuals: juveniles (individuals in theirfirst year after marking), and adults (individuals at least 2 years after marking). Thus, for a given TIME

interval, there are 2 groups: juvenile and adult. If we call this group effect AGE, then we can write outour linear model as

‘survival’ = AGE+TIME+AGE.TIME

= β0 + β1(AGE) + β2(T1) + β3(T2) + β4(T3) + β5(T4)

+ β6(T5) + β7(AGE.T2) + β8(AGE.T3) + β9(AGE.T4) + β10(AGE.T5)

Again, recall from Chapter 7 that there is no (AGE.TIME1) interaction term. Also remember, we’retreating the two age classes as different groups - this will be the key ‘conceptual step’ in seeing howwe apply the same idea to closed population abundance models.



The design matrix corresponding to this linear model is:

So, column B2 in this design matrix indicates a putative ‘age group’ - for a given cohort, and agiven time step, is the individual young (indicated with the dummy ‘1’) or adult (indicated with thedummy ‘0’). If you don’t recall this connect, go back an re-read section 7.2.

Now, what does this have to do with build design matrices for closed abundance estimationmodels? The connection relates to the idea of creating a ‘logical group’. For age models, we used theage of an individual for a given cohort and time step as a grouping variable. For closed populationabundance models, we do the same thing - except that instead of age, we’re going to ‘group’ as afunction of whether or not the individual has been captured at least once or not. In other words,we’re going to treat the parameters p (caught for the first time) and c (caught subsequently) as levelsof a putative ‘encounter’ group (analogous to young and adult, respectively).

This will make more sense when you see how we set up the DM. Here it is - note that it is identical

to the age (TSM) model (above):



Column B1 is the common intercept - this is necessary step (and a key difference from the defaultDM) in order to allow us to specify a functional relationship between p and c. Column B2 is thecolumn which specifics the putative ‘encounter group’ - first encounter (corresponding to parameterp) or subsequent encounter (corresponding to parameter c). Note that there are 5 ‘1’s; for p, but only4 ‘0’s’ for c (since there is no c parameter for occasion 1). This is entirely analogous to having no adultsin the first occasion for individuals marked as young. Columns B3 → B7 correspond to the time steps- again, note that for parameter c, there is no time coding for occasion 1. These are followed by theinteraction columns B8 → B11. Again, there is no logical interaction of p and c for occasion 1, so theinteraction columns start with occasion 2. Finally, column B12 for the abundance parameter N.

Go ahead, run this model, and add the results to the browser:

We see that the model deviances for the general model constructed with (i) PIMs, (ii) the defaultDM (which used a separate intercept for each parameter), and (iii) the modified DM which used acommon intercept, are all identical.

Now, let’s see how to use the DM to build the 3 models we constructed previously using PIMs. First,model {N, p(.) = c(.)}. We see that (i) there is no temporal variation (meaning, we simply delete thecolumns corresponding to time and interactions with time from the DM - columns B3 → B11), and (ii)p = c (meaning, we delete the column specifying difference between the putative ‘encounter groups’- column B2):



Run this model and add the results to the browser:

We see the model results match those of the same model constructed using PIMs.

What about model {N, p(.), c(.)}? Here, we again delete all of the time and interaction columns,but retain the column coding for the ‘encounter group’ term in the model:

Again, we see that the results of fitting this model constructed using the DM approach exactlymatch those from the same model constructed using PIMs:



Finally, model {N, p(t) = c(t)}. Here, we have no ‘encounter group’ effect, but simple temporalvariation in p and c. We simple delete the interaction and ‘encounter group’ columns:

We see (below) that the model deviances are identical, regardless of whether or not the PIM or DMapproach was used.

Now, let’s consider a model which we couldn’t build using the PIM-only approach (or, as noted, ifwe’d relied on the default DM): a model with an additive ‘offset’ between p and c. As we introducedin Chapter 6, to build such additive models, all you need to do is delete the interaction columnsfrom the DM - this ‘additive’ model is shown at the top of the next page. Remember that this modelconstrains time-specific estimates of p and c to parallel each other by a constant offset. Whether ornot this is a ‘meaningful’ model is up to you.


14.6. Heterogeneity models 14 - 22

14.6. Heterogeneity models

As noted earlier, MARK allows you to fit a class of models which are parameterized based on whatare known as ‘finite mixtures’. These models have proven to be very useful for modeling unspecifiedheterogeneity among individuals in the pi and ci parameters. In these models,

pi =

{pi,A with Pr(π)

pi,B with Pr(1 − π)

}

for the case with two mixtures A and B, although the model can be extended to >2 mixtures. Aswritten (above), the parameter π is the probability that the individual occurs in mixture A. For >2mixtures, additional π parameters must be defined (i.e., πA, πB,...), but constrained to sum to 1. Inpractice, most data sets generally support no more than 2 mixtures. Note that the π parameter isassumed to be constant over time (i.e., an individual in a given mixture is always in that particularmixture over the sampling period). This has important implications for constructing the DM, whichwe discuss later.

We will demonstrate the fitting of finite mixture (‘heterogeneity’) models to a new sample data set(mixed_closed1.inp). These data were simulated assuming a finite mixture (i.e., heterogeneity) usingthe true model {N, π, p(.) ≡ c(.) = constant} - 9 occasions, 2 mixtures, N = 2000, π = 0.40, andpπA = 0.25, pπB = 0.75. In other words, two mixtures, one with an encounter probability of p = 0.25,the other with an encounter probability of p = 0.75, with the probability of being in the first mixtureπ = 0.40.

Start a new project, select the input data file, and specify the Full Closed Captures with Heterogeneity

data type. Remember the distinction between ‘Full Closed Capture...’ models and ‘Closed Capture...’models - the later is just a simple subset of the former. Once we’ve selected a closed data typewith heterogeneity, we’re asked how many mixtures we want to model. We’ll use 2 mixtures forthis example.



Once we’ve specified the number of mixtures, open the PIM chart for this data type (when youswitch data types, the underlying model will default to a general time-specific model):

You’ll notice immediately that there are now twice as many p’s and c’s as you might have expectedgiven there are 9 occasions represented in the data. This increase represents the parameters for eachof the two mixture groups. The PIM for the p’s now has two rows defaulting to parameters 2 → 10and 11 → 19.

Parameters 2 → 10 represent the p’s for the first mixture and 11 → 19 represent the p’s forthe second mixture. It becomes more important with the mixture models to keep track of whichoccasion each c corresponds to because now both parameter 2 and 11 relate to occasion 1 which hasno corresponding c parameter.

We’ll follow the approach used in the preceding section, by first fitting a general model basedon PIMs to the data. You might consider model {N, π, p(t), c(t)} as a reasonable starting model.However, there are two problems with using this as a general, starting model. First, you’ll recall thatthere are estimation problems (in general) for a closed abundance model where both p and c arefully time-dependent. Normally, we need to impose some sort of constraint to achieve identifiability.



However, even if we do so, there is an additional, more subtle problem here - recall we are fittinga heterogeneity ‘mixture’ model, where the parameter π is assumed to be constant over time. Assuch, there is no interaction among mixture groups possible over time. Such an interaction wouldimply time-varying π. Thus, the most general model we could fit would be an additive model, withadditivity between the mixture groups. We’ll allow for possible interaction of p and c within a givenmixture group. Recall that we can’t construct this model using PIMs - building an additive modelrequires use of the design matrix.

We see from the PIM chart (shown on the preceding page) that the default reduced DM has 24columns. Note: if you select ’Design | Full’, MARK will respond with an error message, telling youit can’t build a default fully time-dependent DM. Basically, for heterogeneity models, you’ll need tobuild the DM by hand - meaning, starting with a reduced DM. So, we select ‘Design | Reduced’, andkeep the default 24 columns (we can change the number of columns manually later, if need be).

Now, how do we build the DM? We might start by first writing out the linear model. To do so,we need to first consider the ‘groups’ in our model. Here, we have in fact 2 groups: (i) the putative‘encounter group’ (ENCGRP) representing the p and c parameters (as we saw in the preceding section),and (ii) a ‘heterogeneity’ group (HETGRP) representing what we might call the ‘π’ and ‘1 − π’ groups,for convenience. So, two ‘ encounter groups’, 2 ‘heterogeneity groups’, 9 occasions (TIME), and thevarious plausible interactions among them. Here is our linear model (which we write only in terms ofparameters p and c - parameters π and N are simple scalar constants):

f = ENCGRP+HETGRP+TIME+(ENCGRP.TIME)+(HETGRP.TIME)+(ENCGRP.HETGRP.TIME)

= β0

+ β1(ENCGRP)

+ β2(HETGRP)

+ β3(T1) + β4(T2) + β5(T3) + β6(T4) + β7(T5) + β8(T6) + β9(T7) + β10(T8)

+ β11(HETGRP.T1) + β12(HETGRP.T2) + β13(HETGRP.T3) + · · ·+ β18(HETGRP.T8)

+ β19(ENCGRP.T2) + β20(ENCGRP.T3) + β21(ENCGRP.T4) + · · ·+ β25(ENCGRP.T8)

+ β26(ENCGRP.HETGRP.T1) + β27(ENCGRP.HETGRP.T2) + · · ·+ β33(ENCGRP.HETGRP.T8)

So, 34 parameters in this linear model. If we add 2 (for π and N, respectively), we get 36 total. Thedesign matrix corresponding to this model is shown at the top of the next page (you might need toput on some ‘special reading glasses’ to see it all).

Now, some important things to note from the linear model and corresponding DM. First, the two‘groups’ (encounter and heterogeneity; ENCGRP and HETGRP, respectively) are each coded by a singlecolumn (single β) - columns B3 and B4. 9 sampling occasions, so 8 columns for time (B5 → B12). Theremaining columns code for the two-way interactions between ENCGRP (E), HETGRP (H) and time (T),and the three-way interaction (H.E.Tx)



Now, if you run this model, and look at the parameter estimates, you’ll quickly notice that some ofthe parameters aren’t identifiable. In particular, the final p estimates for the two mixture groups haveproblems, and the estimate of N is simply Mt+1 (the SE of the estimate for abundance is also clearlywrong).

Why the problems? Simple - despite the fact we have 2 mixture groups, this is still model{p(t), c(t)}, which we know is not identifiable without constraints. One possible constraint is tomodel p and c as additive functions of each other. How can we modify the DM to apply thisconstraint? Simply by eliminating the interactions between ENCGRP and TIME. In other words, deletingcolumns B21 → B27 (coding for the interaction of ENCGRP and TIME), and columns B28 → B35(coding for the 3-way interaction of HETGRP, ENCGRP, and TIME) from the DM shown at the top ofthis page. This model allows time variation, behavioral variation and individual heterogeneity incapture probability, yet does so in an efficient and parsimonious (and estimable) manner. We canuse this DM to create additional, reduced parameter models. For example, we could build model{N, pa(t) = ca(t) = pb(t) + z = cb(t) + z } representing capture probability varying through time andadditive difference between mixture groups, but with no interaction between p and c over time (nobehavior effect). We do this simply by deleting the ENCGRP column from the DM.

Final test - how do we modify the DM to generate the true generating model, which for these datawas model {N, π, pA = cA, pB = cB, N}? To build this model from our DM, we simply delete (i) allthe time columns, (ii) any interactions with time, and (iii) the encounter group column (ENCGRP). Wedelete the encounter group column because we’re setting p = c. We retain the heterogeneity (mixture)group column (HETGRP) since we want to allow for the possibility that encounter probability differsbetween mixtures (which of course is logically necessary for a mixture model!).



Here is the modified DM:

Looking at the real parameter estimates generated by fitting this reduced DM to the data, wesee that the estimates (π = 0.607, pπA = 0.250, pπB = 0.754, N = 1995.494). Clearly most of theseestimates are very close to the true parameter values used in the generating model. But, what aboutπ? Well, the true value we used in the simulation was π = 0.40 - the estimated value π = 0.607 issimply the complement.

We can confirm that this corresponds to model {N, π, pA = cA, pB = cB} by comparing the modelfit with that from the PIM-based equivalent. We can do this in one of two ways - we can either (i)stay within the ‘full closed captures with heterogeneity’ data type, and build the appropriate PIMs,or (ii) change data type to the simpler ’closed captures with heterogeneity’, which in fact defaults toour desired model.


14.6.1. Interpreting π 14 - 27

If we take the first approach, all we need to do is modify the two encounter probability PIMs asfollows, for p and c, respectively:

So, constant over time and no behavior effect (i.e., p = c) within mixture group. If you run thismodel, you’ll see that it yields an identical model deviance (555.1792) as the model built using themodified DM (above).

What about changing data types to ‘closed captures with heterogeneity’? Well, you might thinkthat you need to restart MARK, and begin a new project after first specifying the new data type. Infact, you don’t need to - you can simply ‘tell’ MARK that you want to switch data types (somethingMARK lets you do within certain types of models - in this instance, closed population abundanceestimators). All you need to do is select ‘PIM | change data type’ on the main menu bar, and thenselect ‘closed captures with heterogeneity’ from the resulting popup window. As noted earlier, thedefault model for this data type is the model we’re after - it is simply a reduced parameter version ofthe full model.

14.6.1. Interpreting π

So, you do an analysis using a closed population heterogeneity abundance model, and derive anestimate of π. Perhaps you’ve built several such models, and have a model averaged estimate of π.So, what do you ‘say’ about this estimate of π?

Easy answer - essentially nothing. The estimate of π is based on fitting a finite mixture model,with a number (typically small) of discrete states. When we simulated such data (above), we useda discrete simulation approach - we simply imagined a population where 40% or the individualshad one particular detection probability, and 60% had a different encounter probability . In that case,because the distribution of individuals in the simulated population was in fact discrete, then the realestimate of π reflected the true generating parameter. However, if in fact the variation in detection


14.7. Goodness-of-fit 14 - 28

probability was (say) continuous, then in fact the estimate of π reflects a ‘best estimate’ as to where adiscrete ‘breakpoint’ might be (breaking the data into a set of discrete, finite mixtures). And, as such,such an estimate π is not, generally, interpretable. Our general advice is to avoid post hoc story-tellingwith respect to π, no matter how tempting (or satisfying) the story might seem.

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Running CAPTURE from MARK

Program CAPTURE fits models outside of the maximum likelihood framework of MARK. Someof these models remain useful, especially the jackknife. Therefore, it is possible to call CAPTURE

directly from MARK. To run CAPTURE, have the closed capture-recapture data set you wish toanalyze open in MARK. Select ‘Tests’ and ‘Program CAPTURE’ from the menu. Click the checkboxes for the model(s) you wish to run. After you click ‘OK’, CAPTURE will perform the analysisand the results will be displayed in a Notepad window. The results from CAPTURE will not beappended to the MARK results browser.

end sidebar

14.7. Goodness-of-fit

Testing model fit in the closed-population capture-recapture models remains an unresolved issue,even more so than in other capture-recapture model types. A central component of the problemstems from the fact that there is no unique way to compute a saturated model. If one was onlyconcerned about time variation in capture probability, then goodness-of-fit is fairly straightforward.When individual heterogeneity is added into the problem there is an infinite set of possible modelsfor heterogeneity. Thus, no unique goodness-of-fit exists.

Several tests of model assumptions have been developed for the closed-population capture-recapturemodels (Otis et al. 1978:50–67, White et al. 1982:77–79). The seven tests examine the fit of specific modelforms relative to other specific models or vague alternatives (i.e., the model fails to fit for unspecifiedreasons). These tests are available in MARK through CAPTURE by selecting the ‘Appropriate’ checkbox in the CAPTURE window. The tests were developed largely as a means of model selection in theabsence of another method. Now that MARK employs AICc as a selection criterion and that it hasbeen shown the model averaged estimates of N have better properties than single-model estimates(Stanley and Burnham 1998), the tests of Otis et al. (1978) have fallen out of use.

14.8. Model Output

After models are run, the model selection statistics are presented in the results browser. By default,models are sorted by AICc. You might note large differences in the AICc between the full likelihoodmodels and the Huggins conditional likelihood models in the browser. This occurs because the twotypes of models are based on different likelihoods, therefore AICc is in fact not comparable betweenthe two types. If you wish to use both full and conditional likelihood models, it is often easier (andarguably more appropriate) to create a separate MARK file for each type of likelihood.

In the browser itself, parameter estimates are available by clicking on the ‘beta’ and ‘real’ parameterbuttons on the results browser toolbar. The β parameter estimates are the parameter estimates priorto any transformation through the link functions. The real parameter estimates are the abundanceand capture probability estimates after being back-transformed with the link functions. The real


14.9. Model averaging and closed models 14 - 29

parameters may be functions of one or more beta parameters. Covariance matrices for the beta andreal parameters can be output by selecting ‘Output | Specific Model Output | Variance-CovarianceMatrices’ and then selecting the type of parameter and mode of output. Parameter estimates andcovariance matrices can be output to the Clipboard, an Excel Spreadsheet, or Notepad.

For the closed capture-recapture data types, the derived parameter button will be available forall data types regardless of whether the data type has a derived parameter. This occurs becausethe N is a derived parameter in the misidentification models, yet can be model averaged with N

from the models without misidentification. In the real parameter output from some results from amisidentification model, the abundance parameter is labeled f0:

Parameter Estimate SE LCI UCI

p 0.1195709 0.0189652 0.0870975 0.1620023c 0.1477397 0.0141956 0.1220228 0.1777792

alpha 0.9410335 0.0727814 0.5496410 0.9952308f0 109.48029 35.142385 59.261430 202.25524

N is available only in the derived parameter output. Therefore, N is available from both the realparameters and derived parameters for some data types.

14.9. Model averaging and closed models

Model averaging is particularly important in the closed models because selecting a single modeltends to be especially problematic when a parameter, in this case N, is in the multinomial coefficient.Typically, abundance would be the only parameter for which we’re interested in a model averagedestimate. The basic concepts and mechanics of model averaging were introduced in earlier chapters.

To compute a model averaged estimate, select ‘Output | Model Averaging’ then either ‘Real’ or‘Derived’ from the menu (shown at the top of the next page). Select the appropriate parameter bychecking the box from the PIM window that opens.


14.9.1. estimating CI for model averaged abundance estimates 14 - 30

Note the check box in the lower lefthand corner of the model averaging window (highlighted in thered oval). The check box selects whether model averaging is performed across multiple data types. Itis legitimate to model average across data types that are based on the same likelihood, but not acrossthose based on different likelihoods.

14.9.1. estimating CI for model averaged abundance estimates

The usual (simplest) approach to estimating the confidence interval for a given parameter makes useof asymptotic variances, covariances - typically, these can be generated from the information matrixfor models with maximum likelihood estimates (this is discussed elsewhere).

However, there is a basic problem with applying this ‘classical’ approach to estimates of abundance- specifically, the classical approach requires asymptotic normality of point estimates N, and thisassumption is frequently not met for any number of reasons.

An alternative approach is to focus on the number of animals that are not caught ( f0), where f0 =N − Mt+1 (this relation was introduced earlier in this chapter). On the assumption that this quantityfollows a log-normal distribution (which has been generally confirmed by various authors), thenlower and upper CI interval bounds for N are given by∗

[Mt+1 +

(f0/C

), Mt+1 +

(f0 × C

)]

where

f0 = N − Mt+1

C = exp

1.96

ln

1 +

var

(N)

f 20

1/2

Note that since N = Mt+1 + f0, then var(N) is exactly the same as the variance of f0, because Mt+1is a known constant. As such,

var

(N)

f 20

=

var

(f0

)

f 20

= CV ( f0)2

Commonly in these kinds of calculations, the square of the CV (coefficient of variation) of f0 isembedded in the formula.

It is important to note that the lower bound of this confidence interval cannot be smaller than Mt+1,but the upper bound frequently is larger than the upper bounds computed with the informationmatrix under the assumption of normality. This is the approach used by MARK to derive the CI forN (regardless of whether N is a derived or real parameter).

∗ there is a typographical error in the equation for C in the Williams, Nichols & Conroy book (p. 304, section 14.2.4). Theversion presented here is correct.



Now, how do we handle the calculation of the CI for the model averaged estimate of abundance?From Buckland et al. (1997), the estimated unconditional (i.e., model averaged) variance var(θ),calculated over models {M1, M2, . . . , Mi} is given as

var

(θ)=

R

∑i=1

wi

(var(θi|Mi) + (θi − ˆθ)2

)where θ =

R

∑i=1

wiθi

and the wi are the Akaike weights (∆ i) scaled to sum to 1. The subscript i refers to the ith model. Thevalue θa is a weighted average of the estimated parameter θ over R models (i = 1, 2, . . . , R).

This estimator of the unconditional variance is clearly the sum of 2 components: (i) the conditional

sampling variance var(θi|Mi) (i.e., conditional on model Mi), and (ii) a term for the variation in theestimates across the R models (θi − ˆθ)2. The sum of these terms is then merely weighted by theAkaike weights wi. Thus, the unconditional standard error would be given as

SE(θ)=

√

var(

ˆθ)

OK - given all this, back to the original question - how do you estimate the confidence intervalfor model averaged abundance estimates? We’ll demonstrate the mechanics by means of a workedexample. Suppose you fit 3 different models to some close capture data, where Mt+1 = 47. We’ll callthese models M1, M2 and M3 - assume they’re all based on the same likelihood form, and differ onlyin parameter structure. Here is a tabulation of the relevant results of fitting these models to the data:

model QAICc wi N var(N)

M1 272.5064 0.63665 59.1161 63.8798M2 273.6392 0.36134 58.1909 56.0402M3 284.0209 0.00201 56.4051 35.0438

Now, we first need to calculate the unconditional variance of N. Since our model averaged estimateof θ is given as

θ =R

∑i=1

wiθi

then N is given as

N =R

∑i=1

wiNi

= (0.63665× 59.11611) + (0.36134× 58.1909) + (0.00201× 56.4051)

= 58.7814

then



var

(ˆN)=

R

∑i=1

wi

(var(θi|Mi) + (θi − ˆθ)2

)

=[(

0.63665(63.8798+ (59.1161− 58.7814)2)+(

0.36134(56.0402+ (58.1909− 58.7814)2)

+(

0.00201(35.0438+ (56.4051− 58.7814)2)]

= [40.7404+ 20.3756+ 0.0818]

= 61.1978

Next, we calculate

C = exp

1.96

ln

1 +

var

( N)

f 20

1/2

Since Mt+1 = 47 for this data set, and since N = 58.7814, then

f 0 = Na − Mt+1

= 58.7814− 47

= 11.7814

and thus

C = exp

1.96

ln

1 +

var

( N)

f 20

1/2

= exp

{1.96

[ln(

1 +61.1978

(11.7814)2

)]1/2}

= 3.2693

Last step. Now that we have a value for C, we can derive the 95% CI as

[47 + (11.7814/3.2693) , 47+ (11.7184× 3.2693)] = [50.604, 85.311]

OK, this seems like a lot of work, but in this particular example, it was necessary. If we had simplyused the ‘automatic’ model averaging in MARK, the CI reported by MARK for N was [43.443, 74.116].Major problem with this CI, since the lower bound is less than Mt+1 (43.443 < 47). Clearly, this makesno sense whatsoever. In contrast, the CI we derived ‘by hand’ does not bound Mt+1. Not only wasthe reported lower-limit of the CI too low, but the upper limit was as well.



Now, in this example, there was an obvious ‘problem’ with the simple model-averaged CI for N;however, even if the lower bound of the reported CI is ≥ Mt+1, don’t take this as evidence thatthe reported CI is correct. For example, consider fitting models {N, p(.) = c(.)} and {N, p(.), c(.)}(corresponding to model M0 and Mb in the older, ‘Otis’ notation) to the ‘Carothers A’ data set (foundin the examples subdirectory created when you installed MARK).

Here is a tabulation of the relevant results of fitting these models to the data:

model QAICc wi N var(N)

{N, p(.) = c(.)} -99.7370 0.63460 368.128 212.944

{N, p(.), c(.)} -98.6330 0.36540 392.480 1234.986

If we had used the model averaging option in MARK, the model averaged estimate for N = 377.027,and the reported 95% CI is [324.292, 429.761]. For this data set, Mt+1 = 283, so, in one sense at least,the reported CI for the model average abundance estimate seems reasonable, since the lower limit ofthe CI is greater than Mt+1 (i.e., 324.292 > 293). How does the reported CI compare with the onederived using the calculations presented above? Again, we start by first deriving an estimate of thevariance of the model averaged abundance:

var

(ˆN)=

R

∑i=1

wi

(var(Ni|Mi) + (Ni − ˆN)2

)

= 0.63460(

212.944+ (368.128− 377.027)2)

+ 0.36540(

1234.986+ (392.480− 377.027)2)

= 723.910

Next, we calculate

C = exp

1.96

ln

1 +

var

( N)

f 20

1/2

Since Mt+1 = 283 for this data set, and since N = 377.027, then f 0 = N − Mt+1 = (377.027− 283) =94.027.

Thus,

C = exp

1.96

ln

1 +

var

( N)

f 20

1/2

= exp

{1.96

[ln(

1 +723.910(94.027)2

)]1/2}

= 1.324



Final step. Now that we have a value for C, we can derive the 95% CI for N = 377.027 as

[283 + (94.027/1.324) , 283+ (94.027× 1.324)] = [354.02, 407.49]

Recall that if we had used the model averaging option in MARK, the reported model averaged 95%CI was [324.292, 429.758]. Again, the reported lower- and upper-limits of the CI are both different thanthe ones we just calculated ‘by hand’. The general recommendation, then, is to calculate the 95% CIfor the model averaged abundance ‘by hand’, using the procedure outlined above.

begin sidebar

Profile confidence intervals - careful!

In chapter 1, we introduced the concept of a profile likelihood as a means of constructing confidenceintervals that ‘made more sense’ (in some instances) than intervals constructed the usual way (as asimple function of the SE of the parameter estimate). As discussed in chapter 1, we use the critical χ2

value of 1.92 to derive the 95% CI - you simply take the value of the log likelihood at the maximum(-16.30 in the example shown below), add 1.92 to it (preserving the sign), and look to see where theline corresponding to this sum (-18.22=-(16.30+1.92)) intersects with the profile of the log likelihoodfunction.

The two intersection points of this line and the profile correspond to the upper- and lower-boundsof the CI.

In chapter 1, we also suggested that the biggest limit to using a profile likelihood approach togenerating confidence intervals (and why it wasn’t the default procedure in MARK) was computa-tional - it simply takes more work (computational time) to derive it. However, for closed abundanceestimators, there is another reason to be cautious in using profile likelihoods to generate CI, havingto do with the fact that abundance estimates are not simple [0, 1] bounded parameters, but areminimally bound at 0. The maximum bound (if in fact one exists) is determined by the likelihood.And, as such, there are situations for some closed models where the upper bound of the likelihoodprofile → ∞.



For example, take the likelihoods plotted for a set of simulated closed capture data for model{N, P(.) = c(.)} (i.e., model Mt)

N

50 100 150 200 250 300 350 400 450 500

Lik

elih

oo

d

6

8

10

12

14

16

18

20

22

Model M(t)

and model {N, p(.), c)(.)} (i.e., model Mb)

N

50 100 150 200 250 300 350 400 450 500

Lik

elih

oo

d

34

35

36

37

38

39

40

41

Model M(b)

We see that for model {N, P(.) = c(.)}, the likelihood profile rises to the MLE (vertical dottedline), and then falls, such that the horizontal dashed line corresponding to the MLE−1.92 intersectsthe likelihood at 2 points (which represent the two bounds of the 95% CI). However, for model{N, p(.), c)(.)}, the likelihood rises, but then never falls to < 2 units from the MLE - and, as such,there is no upper bound for the profile likelihood.

So, clearly, there are some potential ‘issues’ with respect to using a profile likelihood approach forgenerating 95% CI for closed abundance estimates.

end sidebar


14.10. Parameter estimability in closed models 14 - 36

14.10. Parameter estimability in closed models

It is important to examine the real parameter results to see if pt = 1.0 and N = Mt+1. This wouldindicate that the model you constructed was not estimable. Be careful – incorrectly built models mayappear very good in terms of AICc. If you don’t know what Mt+1 is for a particular data set, it canbe found in the full model output labeled as ‘M(t+1):’.

In addition, it has been noted several times that a constraint must be placed on pt in order toproperly estimate N. It is straightforward to demonstrate that an estimate of pt is necessary to get anestimate of N. Consider the following estimator of N a for t = 3 occasion capture-recapture study,

N =Mt+1

1 − (1 − p1)(1 − p2)(1 − p3).

Now if p3 = 1, then the denominator in the estimator above equals 1. Thus, the estimate of N = Mt+1.

Let’s consider the estimability of the p’s, now that we know we need pt to get N. The first p isestimable because we have information in the subsequent capture occasions about the proportion ofmarked and unmarked animals captured. This goes for each p until we get to pt. On the last occasion,there are no future occasions from which to pull information. Thus, we must place a constraint of pt.The constraint can be in the form of modeling pt as a function of previous p’s or as a function of therecaptures.

14.11. Other Applications

Closed population capture-recapture models have been used for other applications beyond estimatingthe number of individuals in a population. There is a natural extension to estimating the numberof species in an area. In this case, encounter histories represent detections of species rather thanindividuals. Heterogeneity in detection probability among species is virtually guaranteed.

Closed capture-recapture models and modifications thereof are widely used in human demogra-phy. There they are typically referred to as multiple list sampling. Several lists containing people froma population of interest, for example drug users in a city, act as sampling occasions. Individuals arematched across lists to estimate abundance.

The closed population capture-recapture models underpin the secondary sampling periods in arobust design (Kendall et al. 1997; see Chapter 15). It is therefore essential to understand the closedcaptures models in order to fully understand the robust design

14.12. Summary

Despite a seemingly simple goal, estimating abundance can be quite difficult. The closed capture-recapture models contain numerous, subtle complications. MARK offers a framework for a varietyof models addressing different assumptions, compares models and most importantly model averagesestimated abundance.

An additional advantage of MARK is the ability to combine data from multiple study sites withease. It is too often argued in the ecological literature that capture-recapture is not useful because thesample size at any one trapping grid is too small. Through the use of groups, MARK allows data frommultiple grids to be used to jointly estimate detection probability. While this may bias the estimate of


14.13. References 14 - 37

N somewhat for each individual grid, it remains a far better solution than using minimum numberknown alive as an index. Moreover, MARK handles all of the covariances among the N’s estimatedfrom common data.

14.13. References

Boulanger, J., G. C. White, B. N. McLellan, J. Woods, M. Proctor, and S. Himmer. 2002. A meta-analysisof grizzly bear DNA mark-recapture projects in British Columbia, Canada. Ursus 13: 137–152.

Carothers, A. D. 1973. Capture-recapture methods applied to a population with known parameters.Journal of Animal Ecology 42: 125–146.

Huggins, R. M. 1989. On the statistical analysis of capture experiments. Biometrika 76: 133–140.

Kendall, W. L., J. D. Nichols, and J. E. Hines. 1997. Estimating temporary emigration using capture-recapture data with Pollock’s robust design. Ecology 78: 563–578.

Link, W. A. 2004. Nonidentifiability of population size from capture-recapture data with heteroge-neous detection probabilities. Biometrics 59: 1123–1130.

Lukacs, P. M., and K. P. Burnham. 2005. Estimating population size from DNA-based closed capture-recapture data incorporating genotyping error. Journal of Wildlife Management 69: 396–403.

Otis, D. L., K. P. Burnham, G. C. White, and D. R. Anderson. 1978. Statistical inference from capturedata on closed animal populations. Wildlife Monographs 62.

Pledger, S. 2000. Unified maximum likelihood estimates for closed capture-recapture models usingmixtures. Biometrics 56: 434–442.

Stanley, T.R., and K.P. Burnham. 1999. A closure test for time-specific capture-recapture data. Envi-ronmental and Ecological Statistics 6: 197–209.

Stanley, T.R., and K.P. Burnham. 1998. Information-theoretic model selection and model averaging forclosed-population capture-recapture studies. Biometrical Journal 40: 475–494.

White, G. C., D. R. Anderson, K. P. Burnham, and D. L. Otis. 1982. Capture-recapture and removalmethods for sampling closed populations. Los Alamos National Laboratory Publication LA-8787-NERP. Los Alamos, NM.


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Closed Population Capture-recapture Models