Capture–Recapture

In a typical capture–recapture experiment in biolog-ical and ecologic sciences, we place traps or netsin the study area and sample the population severaltimes. At the first trapping sample a number of ani-mals are captured; the animals are uniquely taggedor marked and released into the population. Thenat each subsequent trapping sample we record andattach a unique tag to every unmarked animal, recordthe capture of any animal that has been previouslytagged, and return all animals to the population. Atthe end of the experiment the complete capture his-tory for each animal is known. Such experiments arealso called mark-recapture, tag-recapture, and multi-ple record systems in the literature. The simplest typeonly includes two samples; one is the capture sampleand the other the recapture sample. This special two-sample case is often referred to as a “dual system”or a “dual-record system” in the context of censusundercount estimation.

The capture–recapture technique has been usedto estimate population sizes and related parameterssuch as survival rates, birth rates, and migration rates.Biologists and ecologists have long recognized that itwould be unnecessary and almost impossible to countevery animal in order to obtain an accurate estimateof population size. The recapture information (orthe proportion of repeated captures) by marking ortagging plays an important role because it can be usedto estimate the number missing in the samples underproper assumptions. Intuitively, when recaptures insubsequent samples are few, we know that the sizeis much higher than the number of distinct captures.However, if the recapture rate is quite high, then weare likely to have caught most of the animals.

According to Seber [15], the first use of thecapture–recapture technique can be traced back toLaplace, who used it to estimate the population size ofFrance in 1786. The earliest applications to ecologyinclude Petersen’s and Dahl’s work on fish popula-tions in 1896 and 1917, respectively, and Lincoln’suse of band returns to estimate waterfowl in 1930.More sophisticated statistical theory and inferenceprocedures have been proposed since the paper byDarroch [5], who founded the mathematical frame-work of this topic. See [15]–[17] and referencestherein for the historical developments, methodolo-gies, and applications.

The models are generally classified as eitherclosed or open. In a closed model the size of a pop-ulation, which is the main interest, is assumed to beconstant over the trapping times. The closure assump-tion is usually valid for data collected in a relativelyshort time during a nonbreeding season. In an openmodel, recruitment (birth or immigration) and losses(death or emigration) are allowed. It is usually usedto model the data from long-term investigations ofanimals or migrating birds. In addition to the popu-lation size at each sampling time, the parameters ofinterest also include the survival rates and numberof births between sampling times. Here we concen-trate on closed models because of their applicationsto epidemiology and health science.

Applications to Epidemiology

The capture–recapture model originally developedfor animal populations has been applied to humanpopulations under the term “multiple-record sys-tems”. A pioneering paper is that of Sekar & Deming[18], who used two samples to estimate the birth anddeath rates in India. Wittes & Sidel [19] were the firstto use three-sample records to estimate the numberof hospital patients. Related subsequent applicationswere given in an earlier overview by El-Khorazatyet al. [7].

Epidemiologists recently have shown renewed andgrowing interest in the use of the capture–recapturetechnique. As LaPorte et al. [13] indicated, the tradi-tional public-health approaches to counting the num-ber of occurrences of diseases are too inaccurate(surveillance), too costly (population-based registries;see Disease Registers), or too late (death certifi-cates) for broad monitoring. They felt that it wastime to start counting the incidences of diseases inthe same way as biologists count animals. Two recentreview articles [11, 12] by the International Societyfor Disease Monitoring and Forecasting proposed thatthe capture–recapture method would provide a tech-nique for enhancing our ability to monitor disease.Reference [12] also reviewed its applications to thefollowing categories: birth defects, cancers, drug use,infectious diseases, injuries, and diabetes as well asother areas of epidemiology.

The purpose of most applications to epidemiologyis to estimate the size of a certain target popula-tion by merging several existing but incomplete lists

2 Capture–Recapture

of names of the target population. If each list isregarded as a trapping sample and identification num-bers and/or names are used as “tags”, then it is simi-lar to a closed capture–recapture setup for wildlifeestimation. Now the “capture in a sample” corre-sponds to “being recorded or identified in a list”, and“capture probability” becomes “ascertainment proba-bility”. Two major differences between wildlife andhuman applications are (i) there are more trappingsamples in wildlife studies, whereas in human stud-ies only two to four lists are available; and (ii) inanimal studies there is a natural temporal or sequen-tial time order in the trapping samples, whereas forepidemiologic data such order does not exist in thelists, or the order may be different for some individ-uals. Researchers in wildlife and human applicationshave respectively developed models and methodolo-gies along separate lines. Three of these approachesare discussed after the data structure and assumptionsare explained.

Data Structure and Assumptions

Ascertainment data for all identified individuals areusually aggregated into a categorical data form. Wegive in Table 1 a three-list hepatitis A virus exam-ple for illustration. The purpose of this study was toestimate the number of people who were infected byhepatitis in an outbreak that occurred in and arounda college in northern Taiwan from April to July1995. Our data are restricted to those records fromstudents of that college. A total of 271 cases werereported from the following three sources: (i) P-list(135 cases): records based on a serum test con-ducted by the Institute of Preventive Medicine of Tai-wan. (ii) Q-list (122 cases): records reported by theNational Quarantine Service based on cases reportedby the doctors of local hospitals. (iii) E-list (126cases): records based on questionnaires collected byepidemiologists.

In Table 1, for simplicity, the presence or absencein any list is denoted by 1 and 0, respectively.There are seven observed cells Z100, Z010, Z001,Z110, Z011, Z101, and Z111. Here Z111 = 28 meansthat there were 28 people recorded on all three lists;Z100 = 69 means that 69 people were recorded onlist P only. A similar interpretation pertains to otherrecords. Let n1, n2, and n3 be the number of cases inP, Q and E, respectively. Then n1 = Z111 + Z110 +

Table 1 Data on hepatitis A virus

Hepatitis A virus list

P Q E Data

1 1 1 Z111 = 281 1 0 Z110 = 211 0 1 Z101 = 171 0 0 Z100 = 690 1 1 Z011 = 180 1 0 Z010 = 550 0 1 Z001 = 630 0 0 Z000 = ??

Z101 + Z100 = 135. Similar expressions hold for n2

and n3. There is one missing cell, Z000, the numberof uncounted. The purpose is to predict Z000 or toestimate the total population size.

A crucial assumption in the traditional approachis that the samples are independent. Since indi-viduals can be cross classified according to theirpresence or absence in each list, the independencefor two samples is usually interpreted from a 2 ×2 categorical data analysis in human applications.This assumption in animal studies is expressed interms of the “equal-catchability assumption”: allanimals have the same probability of capture ineach sample. However, this assumption is rarelyvalid in most applications. Lack of independenceamong samples leads to a bias for the usual esti-mators derived under the independence assumption.The bias may be caused by the following twosources:

1. List dependence within each individual (or sub-stratum): that is, inclusion in one sample has adirect causal effect on any individual’s inclusionin other samples. For example, an individual witha positive for the serum test of hepatitis is morelikely to go to the hospital for treatment andthus the probability of being identified in localhospital records is larger than that of the sameindividual given as negative by the serum test.Therefore, the “capture” of the serum test and the“capture” of hospital records become positivelydependent. This type of dependence is usuallyreferred to as “list dependence” in the literature.

2. Heterogeneity between individuals (or substrata):even if the two lists are independent within indi-viduals, the ascertainment of the two lists maybecome dependent if the capture probabilities

Capture–Recapture 3

are heterogeneous among individuals. This phe-nomenon is similar to Simpson’s paradox incategorical data analysis. That is to say, aggregat-ing two independent 2 × 2 tables might result ina dependent table. Hook & Regal [10] providedan example.

The above two types of dependences are usuallyconfounded and cannot be easily disentangled in adata analysis without further assumptions. We discussthree approaches (the ecologic model, the loglinearmodel, and the sample coverage approach) that allowfor the above two types of dependences.

Ecologic Models

Pollock, in his 1974 Ph.D. thesis and subsequentpapers (e.g. Pollock [14]), proposed a sequence ofmodels mainly for wildlife studies to relax the equal-catchability assumption. This approach aims to modelthe dependences by specifying various forms of “cap-ture” probability. The basic models include (i) modelMt, which allows capture probabilities to vary withtime; (ii) model Mb, which allows the capture ofbehavioral responses; and (iii) model Mh, whichallows heterogeneous animal capture probabilities.Various combinations of these three types of unequalcapture probabilities (i.e. models Mtb, Mth, Mbh, andMtbh) are also proposed.

Only for model Mt are the samples independent.List dependence is present for models Mb and Mtb;heterogeneity arises for model Mh; and both typesof dependences exist for models Mbh and Mtbh. Forany model involving behavioral response, the captureprobability of any animal depends on its “previous”capture history. However, there is usually no sequen-tial order in the lists, so those models have limiteduse in epidemiology. Models Mh and Mth might beuseful for epidemiological studies. Various estima-tion procedures have been proposed. See [15]–[17]for reviews.

Loglinear Models

The loglinear model approach is a commonly usedtechnique for epidemiological data. Loglinear modelsthat incorporate list dependence were first proposedby Fienberg [8] for dealing with human populations.

Cormack [4] proposed the use of this technique forseveral ecologic models.

In this approach the data are regarded as a formof an incomplete 2t contingency table (t is the num-ber of lists) for which the cell corresponding to thoseindividuals uncounted by all lists is missing. A basicassumption is that there is no t-sample interaction.This assumption implies an extrapolation formula forthe number of uncounted. For three lists, the mostgeneral model is a model with main effects andthree two-sample interaction terms. Various loglin-ear models are fitted to the observed cells and aproper model is selected using deviance statisticsand the Akaike information criterion. The cho-sen model is then projected onto the unobservedcell.

List dependences correspond to some specificinteraction terms in the model. As for heterogeneity,quasi-symmetric and partial quasi-symmetric mod-els of loglinear models can be used to model sometypes of heterogeneity, i.e. Rasch models and theirgeneralizations; see [1] and [6]. Since the quasi-symmetric or partial quasi-symmetric models areequivalent to assuming that some two-factor interac-tion terms are identical, the heterogeneity correspondsto some common interaction effects in loglinear mod-els. Details of the theory and development are fullydiscussed in [11] and [12].

Sample Coverage Approach

The idea of sample coverage, originally fromI.J. Good and A.M. Turing (Good [9]), has been usedin species and animal population size estimation; seeChao & Lee [2]. The same approach was also appliedto epidemiologic data in [3].

This approach aims to model dependences bysome parameters, which are called “coefficients ofvariation”, defined for two or more samples. Themagnitude of the parameters measures the degree ofdependence of samples. The two types of depen-dences are confounded in these measures. In theindependent case, all dependence measures are zero.This general model encompasses the Rasch modeland the ecologic models as special cases.

A common definition for the sample coverage of agiven sample is the probability-weighted fraction ofthe population that is discovered in that sample. Formultiple-sample type of data the sample coverage is

4 Capture–Recapture

modified to be the probability-weighted fraction ofthe population that is jointly covered by the avail-able samples. See [3] for a formal definition. Thebasic motivation here is that sample coverage can bewell estimated even in the presence of two sourcesof dependences. Thus an estimate of population sizecan be obtained via the relation between the popula-tion size and sample coverage. Chao et al. [3] haveshown that an estimator of C is C = 1 − (Z100/n1 +Z010/n2 + Z001/n3)/3, which is one minus the aver-age of the proportion of individuals listed in onlyone sample (i.e. singletons). Let D be the averageof the distinct cases for three pairs of samples. Inthis approach, when all three samples are indepen-dent, a valid estimator is N0 = D/C. If any type ofdependence arises, then Chao et al. [3] attempt toaccount for the dependences by adjusting D/C basedon a function of the estimates of the coefficients ofvariation. In the same reference, estimators of pop-ulation size are proposed separately for high samplecoverages (e.g. if C is over 55%) and low samplecoverages.

Analysis of the Hepatitis Example (LowSample Coverage)

Several loglinear models were fitted to the hepatitisdata given in Table 1. Except for the saturated model,the loglinear models that do not take heterogeneityinto account (e.g. models with one or two interactionterms) do not fit the data well, whereas all othermodels that take heterogeneity into account (quasi-symmetric and partial quasi-symmetric models) fitwell. All those adequate models yielded very similarestimates – 1300 with an approximate estimatedstandard error of 520.

The coverage estimate is C = 51.27%, which isconsidered to be low. The average of the distinctcases for three pairs of samples is D = 208.667.If the incorrect independence is assumed, then anestimate would be N0 = D/C = 407. (The loglin-ear independent model yields a similar estimate of388.) It follows from Chao et al. [3] that the esti-mates for dependence measures are relatively large,which indicates that the three samples are pairwisepositively dependent and N0 would generally under-estimate. However, one cannot distinguish which typeof dependence is the main cause of the bias. Incorpo-rating the bias due to dependences along the sample

coverage approach results in an estimate of 508.An estimated standard error of 40 is calculated byusing a bootstrap method based on 1000 replica-tions. The resulting 95% confidence interval is (407,591) based on the same bootstrap replications.

This example shows that the loglinear and samplecoverage approaches may give widely differentestimates. Moreover, the example in [11] furthershows that several loglinear models that fit the dataequally well might also result in quite differentestimates. Simulation comparisons of the twoapproaches and other examples with high samplecoverages are provided in Chao et al. [3].

References

[1] Agresti, A. (1994). Simple capture-recapture modelspermitting unequal catchability and variable samplingeffort, Biometrics 50, 494–500.

[2] Chao, A. & Lee S.-M. (1992). Estimating the numberof classes via sample coverage, Journal of the AmericanStatistical Association 87, 210–217.

[3] Chao, A., Tsay, P.K., Shau, W.-Y. & Chao, D.-Y.(1996). Population size estimation for capture–recapturemodels with applications to epidemiological data,Proceedings of Biometrics Section, American StatisticalAssociation, pp. 108–117.

[4] Cormack, R.M. (1989). Loglinear models for capture-recapture, Biometrics 45, 395–413.

[5] Darroch, J.M. (1958). The multiple recapture censusI. Estimation of a closed population, Biometrika 45,343–359.

[6] Darroch, J.N., Fienberg, S.E., Glonek, G.F.V. &Junker, B.W. (1993). A three-sample multiple-recaptureapproach to census population estimation with hetero-geneous catchability, Journal of the American StatisticalAssociation 88, 1137–1148.

[7] El-Khorazaty, M.N., Imery, P.B., Koch, G.G. & Wells,H.B. (1977). A review of methodological strategies forestimating the total number of events with data frommultiple-record systems, International Statistical Review45, 129–157.

[8] Fienberg, S.E. (1972). The multiple recapture censusfor closed populations and incomplete 2k contingencytables, Biometrika 59, 591–603.

[9] Good, I.J. (1953). On the population frequencies ofspecies and the estimation of population parameters,Biometrika 40, 237–264.

[10] Hook, E.B. & Regal, R.R. (1993). Effects of varia-tion in probability of ascertainment by sources (“vari-able catchability”) upon capture–recapture estimatesof prevalence, American Journal of Epidemiology 137,1148–1166.

[11] International Society for Disease Monitoring andForecasting (1995). Capture–recapture and multiple-record systems estimation I: history and theoretical

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development, American Journal of Epidemiology 142,1047–1058.

[12] International Society for Disease Monitoring and Fore-casting (1995). Capture–recapture and multiple-recordsystems estimation II: Applications in human diseases,American Journal of Epidemiology 142, 1059–1068.

[13] LaPorte, R.E., McCarty, D.J., Tull, E.S. & Tajima, N.(1992). Counting birds, bees, and NCDs, Lancet 339,494–495.

[14] Pollock, K.H. (1991). Modelling capture, recapture,and removal statistics for estimation of demographicparameters for fish and wildlife populations: past,present, and future, Journal of the American StatisticalAssociation 86, 225–238.

[15] Seber, G.A.F. (1982). The Estimation of Animal Abun-dance, 2nd Ed. Griffin, London.

[16] Seber, G.A.F. (1986). A review of estimating animalabundance, Biometrics 42, 267–292.

[17] Seber, G.A.F. (1992). A review of estimating ani-mal abundance II, International Statistical Review 60,129–166.

[18] Sekar, C. & Deming W.E. (1949). On a method ofestimating birth and death rates and the extent of regis-tration, Journal of the American Statistical Association44, 101–115.

[19] Wittes, J.T. & Sidel, V.W. (1968). A generalization ofthe simple capture–recapture model with applications toepidemiological research, Journal of Chronic Diseases21, 287–301.

(See also Structural and Sampling Zeros)

ANNE CHAO