Estimators in capture–recapture studies with two sources

This paper investigates the applications of capture–recapture methods to human populations. Capture–recapture methods are commonly used in estimating the size of wildlife populations but can also be used in epidemiology and social sciences, for estimating prevalence of a particular disease or the size of the homeless population in a certain area. Here we focus on estimating the prevalence of infectious diseases. Several estimators of population size are considered: the Lincoln–Petersen estimator and its modified version, the Chapman estimator, Chao’s lower bound estimator, the Zelterman’s estimator, McKendrick’s moment estimator and the maximum likelihood estimator. In order to evaluate these estimators, they are applied to real, three-source, capture-recapture data. By conditioning on each of the sources of three source data, we have been able to compare the estimators with the true value that they are estimating. The Chapman and Chao estimators were compared in terms of their relative bias. A variance formula derived through conditioning is suggested for Chao’s estimator, and normal 95% confidence intervals are calculated for this and the Chapman estimator. We then compare the coverage of the respective confidence intervals. Furthermore, a simulation study is included to compare Chao’s and Chapman’s estimator. Results indicate that Chao’s estimator is less biased than Chapman’s estimator unless both sources are independent. Chao’s estimator has also the smaller mean squared error. Finally, the implications and limitations of the above methods are discussed, with suggestions for further development.

A capture–recapture approach for screening using two diagnostic tests with availability of disease status for the test positives only

The article considers screening human populations with two screening tests. If any of the two tests is positive, then full evaluation of the disease status is undertaken; however, if both diagnostic tests are negative, then disease status remains unknown. This procedure leads to a data constellation in which, for each disease status, the 2 × 2 table associated with the two diagnostic tests used in screening has exactly one empty, unknown cell. To estimate the unobserved cell counts, previous approaches assume independence of the two diagnostic tests and use specific models, including the special mixture model of Walter or unconstrained capture–recapture estimates. Often, as is also demonstrated in this article by means of a simple test, the independence of the two screening tests is not supported by the data. Two new estimators are suggested that allow associations of the screening test, although the form of association must be assumed to be homogeneous over disease status. These estimators are modifications of the simple capture–recapture estimator and easy to construct. The estimators are investigated for several screening studies with fully evaluated disease status in which the superior behavior of the new estimators compared to the previous conventional ones can be shown. Finally, the performance of the new estimators is compared with maximum likelihood estimators, which are more difficult to obtain in these models. The results indicate the loss of efficiency as minor.

CAMCR: Computer-Assisted Mixture model analysis for Capture–Recapture count data

Population size estimation with discrete or nonparametric mixture models is considered, and reliable ways of construction of the nonparametric mixture model estimator are reviewed and set into perspective. Construction of the maximum likelihood estimator of the mixing distribution is done for any number of components up to the global nonparametric maximum likelihood bound using the EM algorithm. In addition, the estimators of Chao and Zelterman are considered with some generalisations of Zelterman’s estimator. All computations are done with CAMCR, a special software developed for population size estimation with mixture models. Several examples and data sets are discussed and the estimators illustrated. Problems using the mixture model-based estimators are highlighted.

Completeness and timeliness: Cancer registries could/should improve their performance

Cancer registries must provide complete and reliable incidence information with the shortest possible delay for use in studies such as comparability, clustering, cancer in the elderly and adequacy of cancer surveillance. Methods of varying complexity are available to registries for monitoring completeness and timeliness. We wished to know which methods are currently in use among cancer registries, and to compare the results of our findings to those of a survey carried out in 2006.

Methods
In the framework of the EUROCOURSE project, and to prepare cancer registries for participation in the ERA-net scheme, we launched a survey on the methods used to assess completeness, and also on the timeliness and methods of dissemination of results by registries. We sent the questionnaire to all general registries (GCRs) and specialised registries (SCRs) active in Europe and within the European Network of Cancer Registries (ENCR).

Results
With a response rate of 66% among GCRs and 59% among SCRs, we obtained data for analysis from 116 registries with a population coverage of ∼280 million. The most common methods used were comparison of trends (79%) and mortality/incidence ratios (more than 60%). More complex methods were used less commonly: capture–recapture by 30%, flow method by 18% and death certificate notification (DCN) methods with the Ajiki formula by 9%.

The median latency for completion of ascertainment of incidence was 18 months. Additional time required for dissemination was of the order of 3–6 months, depending on the method: print or electronic. One fifth (21%) did not publish results for their own registry but only as a contribution to larger national or international data repositories and publications; this introduced a further delay in the availability of data.

Conclusions
Cancer registries should improve the practice of measuring their completeness regularly and should move from traditional to more quantitative methods. This could also have implications in the timeliness of data publication.

Estimating the size of the population of persons who inject drugs in the island of Montréal, Canada, using a six-source capture-recapture model

BACKGROUND: To plan and implement services to persons who inject drugs (PWID), knowing their number is essential. For the island of Montréal, Canada, the only estimate, of 11,700 PWID, was obtained in 1996 through a capture-recapture method. Thirteen years later, this study was undertaken to produce a new estimate. METHODS: PWID were defined as individuals aged 14-65 years, having injected recently and living on the island of Montréal. The study period was 07/01/2009 to 06/30/2010. An estimate was produced using a six-source capture-recapture log-linear regression method. The data sources were two epidemiological studies and four drug dependence treatment centres. Model selection was conducted in two steps, the first focusing on interactions between sources and the second, on age group and gender as covariates and as modulators of interactions. RESULTS: A total of 1480 PWID were identified in the six capture sources. They corresponded to 1132 different individuals. Based on the best-fitting model, which included age group and sex as covariates and six two-source interactions (some modulated by age), the estimated population was 3910 PWID (95% confidence intervals (CI): 3180-4900) which represents a prevalence of 2.8 (95% CI: 2.3-3.5) PWID per 1000 persons aged 14-65 years. CONCLUSIONS: The 2009-2010 estimate represents a two-third reduction compared to the one for 1996. The multisource capture-recapture method is useful to produce estimates of the size of the PWID population. It is of particular interest when conducted at regular intervals thus allowing for close monitoring of the injection phenomenon.

An extension of an over-dispersion test for count data

While over-dispersion in capture–recapture studies is well known to lead to poor estimation of population size, current diagnostic tools to detect the presence of heterogeneity have not been specifically developed for capture–recapture studies. To address this, a simple and efficient method of testing for over-dispersion in zero-truncated count data is developed and evaluated. The proposed method generalizes an over-dispersion test previously suggested for un-truncated count data and may also be used for testing residual over-dispersion in zero-inflation data. Simulations suggest that the asymptotic distribution of the test statistic is standard normal and that this approximation is also reasonable for small sample sizes. The method is also shown to be more efficient than an existing test for over-dispersion adapted for the capture–recapture setting. Studies with zero-truncated and zero-inflated count data are used to illustrate the test procedures.

Some general comparative points on Chao’s and Zelterman’s estimators of population size

Two simple and frequently used capture–recapture estimates of the population size are compared: Chao's lower-bound estimate and Zelterman's estimate allowing for contaminated distributions. In the Poisson case it is shown that if there are only counts of ones and twos, the estimator of Zelterman is always bounded above by Chao's estimator. If counts larger than two exist, the estimator of Zelterman is becoming larger than that of Chao's, if only the ratio of the frequencies of counts of twos and ones is small enough. A similar analysis is provided for the binomial case. For a two-component mixture of Poisson distributions the asymptotic bias of both estimators is derived and it is shown that the Zelterman estimator can experience large overestimation bias. A modified Zelterman estimator is suggested and also the bias-corrected version of Chao's estimator is considered. All four estimators are compared in a simulation study.

A covariate adjustment for zero-truncated approaches to estimating the size of hidden and elusive populations

In this paper we consider the estimation of population size from onesource capture–recapture data, that is, a list in which individuals can potentially be found repeatedly and where the question is how many individuals are missed by the list. As a typical example, we provide data from a drug user study in Bangkok from 2001 where the list consists of drug users who repeatedly contact treatment institutions. Drug users with 1, 2, 3, . . . contacts occur, but drug users with zero contacts are not present, requiring the size of this group to be estimated. Statistically, these data can be considered as stemming from a zero-truncated count distribution.We revisit an estimator for the population size suggested by Zelterman that is known to be robust under potential unobserved heterogeneity. We demonstrate that the Zelterman estimator can be viewed as a maximum likelihood estimator for a locally truncated Poisson likelihood which is equivalent to a binomial likelihood. This result allows the extension of the Zelterman estimator by means of logistic regression to include observed heterogeneity in the form of covariates. We also review an estimator proposed by Chao and explain why we are not able to obtain similar results for this estimator. The Zelterman estimator is applied in two case studies, the first a drug user study from Bangkok, the second an illegal immigrant study in the Netherlands. Our results suggest the new estimator should be used, in particular, if substantial unobserved heterogeneity is present.

Estimating the hidden number of scrapie affected holdings in great Britain using a simple, truncated count model allowing for heterogeneity.

None of the current surveillance streams monitoring the presence of scrapie in Great Britain provide a comprehensive and unbiased estimate of the prevalence of the disease at the holding level. Previous work to estimate the under-ascertainment adjusted prevalence of scrapie in Great Britain applied multiple-list capture–recapture methods. The enforcement of new control measures on scrapie-affected holdings in 2004 has stopped the overlapping between surveillance sources and, hence, the application of multiple-list capture–recapture models. Alternative methods, still under the capture–recapture methodology, relying on repeated entries in one single list have been suggested in these situations. In this article, we apply one-list capture–recapture approaches to data held on the Scrapie Notifications Database to estimate the undetected population of scrapie-affected holdings with clinical disease in Great Britain for the years 2002, 2003, and 2004. For doing so, we develop a new diagnostic tool for indication of heterogeneity as well as a new understanding of the Zelterman and Chao’s lower bound estimators to account for potential unobserved heterogeneity. We demonstrate that the Zelterman estimator can be viewed as a maximum likelihood estimator for a special, locally truncated Poisson likelihood equivalent to a binomial likelihood. This understanding allows the extension of the Zelterman approach by means of logistic regression to include observed heterogeneity in the form of covariates—in case studied here, the holding size and country of origin. Our results confirm the presence of substantial unobserved heterogeneity supporting the application of our two estimators. The total scrapie-affected holding population in Great Britain is around 300 holdings per year. None of the covariates appear to inform the model significantly.

A simple variance formula for population size estimators by conditioning

This note considers the variance estimation for population size estimators based on capture–recapture experiments. Whereas a diversity of estimators of the population size has been suggested, the question of estimating the associated variances is less frequently addressed. This note points out that the technique of conditioning can be applied here successfully which also allows us to identify sources of variation: the variance due to estimation of the model parameters and the binomial variance due to sampling n units from a population of size N. It is applied to estimators typically used in capture–recapture experiments in continuous time including the estimators of Zelterman and Chao and improves upon previously used variance estimators. In addition, knowledge of the variances associated with the estimators by Zelterman and Chao allows the suggestion of a new estimator as the weighted sum of the two. The decomposition of the variance into the two sources allows also a new understanding of how resampling techniques like the Bootstrap could be used appropriately. Finally, the sample size question for capture–recapture experiments is addressed. Since the variance of population size estimators increases with the sample size, it is suggested to use relative measures such as the observed-to-hidden ratio or the completeness of identification proportion for approaching the question of sample size choice.

Some general comparative points on Chao's and Zelterman's estimators of the population size

Two simple and frequently used capture–recapture estimates of the population size are compared: Chao's lower-bound estimate and Zelterman's estimate allowing for contaminated distributions. In the Poisson case it is shown that if there are only counts of ones and twos, the estimator of Zelterman is always bounded above by Chao's estimator. If counts larger than two exist, the estimator of Zelterman is becoming larger than that of Chao's, if only the ratio of the frequencies of counts of twos and ones is small enough. A similar analysis is provided for the binomial case. For a two-component mixture of Poisson distributions the asymptotic bias of both estimators is derived and it is shown that the Zelterman estimator can experience large overestimation bias. A modified Zelterman estimator is suggested and also the bias-corrected version of Chao's estimator is considered. All four estimators are compared in a simulation study.

A simple variance formula for population size estimators by conditioning

This note considers the variance estimation for population size estimators based on capture–recapture experiments. Whereas a diversity of estimators of the population size has been suggested, the question of estimating the associated variances is less frequently addressed. This note points out that the technique of conditioning can be applied here successfully which also allows us to identify sources of variation: the variance due to estimation of the model parameters and the binomial variance due to sampling n units from a population of size N. It is applied to estimators typically used in capture–recapture experiments in continuous time including the estimators of Zelterman and Chao and improves upon previously used variance estimators. In addition, knowledge of the variances associated with the estimators by Zelterman and Chao allows the suggestion of a new estimator as the weighted sum of the two. The decomposition of the variance into the two sources allows also a new understanding of how resampling techniques like the Bootstrap could be used appropriately. Finally, the sample size question for capture–recapture experiments is addressed. Since the variance of population size estimators increases with the sample size, it is suggested to use relative measures such as the observed-to-hidden ratio or the completeness of identification proportion for approaching the question of sample size choice.

Use of the ratio plot in capture-recapture estimation

Statistical graphics are a fundamental, yet often overlooked, set of components in the repertoire of data analytic tools. Graphs are quick and efficient, yet simple instruments of preliminary exploration of a dataset to understand its structure and to provide insight into influential aspects of inference such as departures from assumptions and latent patterns. In this paper, we present and assess a graphical device for choosing a method for estimating population size in capture-recapture studies of closed populations. The basic concept is derived from a homogeneous Poisson distribution where the ratios of neighboring Poisson probabilities multiplied by the value of the larger neighbor count are constant. This property extends to the zero-truncated Poisson distribution which is of fundamental importance in capture–recapture studies. In practice however, this distributional property is often violated. The graphical device developed here, the ratio plot, can be used for assessing specific departures from a Poisson distribution. For example, simple contaminations of an otherwise homogeneous Poisson model can be easily detected and a robust estimator for the population size can be suggested. Several robust estimators are developed and a simulation study is provided to give some guidance on which should be used in practice. More systematic departures can also easily be detected using the ratio plot. In this paper, the focus is on Gamma mixtures of the Poisson distribution which leads to a linear pattern (called structured heterogeneity) in the ratio plot. More generally, the paper shows that the ratio plot is monotone for arbitrary mixtures of power series densities.

Extending the Lincoln–Petersen estimator for multiple identifications in one source

The Lincoln–Petersen estimator is one of the most popular estimators used in capture–recapture studies. It was developed for a sampling situation in which two sources independently identify members of a target population. For each of the two sources, it is determined if a unit of the target population is identified or not. This leads to a 2 × 2 table with frequencies f11, f10, f01, f00 indicating the number of units identified by both sources, by the first but not the second source, by the second but not the first source and not identified by any of the two sources, respectively. However, f00 is unobserved so that the 2 × 2 table is incomplete and the Lincoln–Petersen estimator provides an estimate for f00. In this paper, we consider a generalization of this situation for which one source provides not only a binary identification outcome but also a count outcome of how many times a unit has been identified. Using a truncated Poisson count model, truncating multiple identifications larger than two, we propose a maximum likelihood estimator of the Poisson parameter and, ultimately, of the population size. This estimator shows benefits, in comparison with Lincoln–Petersen’s, in terms of bias and efficiency. It is possible to test the homogeneity assumption that is not testable in the Lincoln–Petersen framework. The approach is applied to surveillance data on syphilis from Izmir, Turkey.