## The Annals of Applied Statistics

### Capture–recapture abundance estimation using a semi-complete data likelihood approach

#### Abstract

Capture–recapture data are often collected when abundance estimation is of interest. In this manuscript we focus on abundance estimation of closed populations. In the presence of unobserved individual heterogeneity, specified on a continuous scale for the capture probabilities, the likelihood is not generally available in closed form, but expressible only as an analytically intractable integral. Model-fitting algorithms to estimate abundance most notably include a numerical approximation for the likelihood or use of a Bayesian data augmentation technique considering the complete data likelihood. We consider a Bayesian hybrid approach, defining a “semi-complete” data likelihood, composed of the product of a complete data likelihood component for individuals seen at least once within the study and a marginal data likelihood component for the individuals not seen within the study, approximated using numerical integration. This approach combines the advantages of the two different approaches, with the semi-complete likelihood component specified as a single integral (over the dimension of the individual heterogeneity component). In addition, the models can be fitted within BUGS/JAGS (commonly used for the Bayesian complete data likelihood approach) but with significantly improved computational efficiency compared to the commonly used superpopulation data augmentation approaches (between about 10 and 77 times more efficient in the two examples we consider). The semi-complete likelihood approach is flexible and applicable to a range of models, including spatially explicit capture–recapture models. The model-fitting approach is applied to two different data sets: the first relates to snowshoe hares where model $M_{h}$ is applied and the second to gibbons where a spatially explicit capture–recapture model is applied.

#### Article information

Source
Ann. Appl. Stat. Volume 10, Number 1 (2016), 264-285.

Dates
First available in Project Euclid: 25 March 2016

https://projecteuclid.org/euclid.aoas/1458909916

Digital Object Identifier
doi:10.1214/15-AOAS890

Mathematical Reviews number (MathSciNet)
MR3480496

Zentralblatt MATH identifier
06586145

#### Citation

King, Ruth; McClintock, Brett T.; Kidney, Darren; Borchers, David. Capture–recapture abundance estimation using a semi-complete data likelihood approach. Ann. Appl. Stat. 10 (2016), no. 1, 264--285. doi:10.1214/15-AOAS890. https://projecteuclid.org/euclid.aoas/1458909916

#### References

• Bonner, S. and Schofield, M. R. (2014). MC(MC)MC: Exploring Monte Carlo integration within MCMC for mark-recapture models with individual covariates. Methods in Ecology and Evolution. 5. 1305–1315.
• Borchers, D. L., Buckland, S. T. and Zucchini, W. (2002). Estimating Animal Abundance, Closed Populations. Springer, London.
• Borchers, D. L. and Efford, M. G. (2008). Spatially explicit maximum likelihood methods for capture–recapture studies. Biometrics 64 377–385, 664.
• Buckland, S. T., Anderson, D. R., Burnham, K. P., Laake, J. L., Borchers, D. L. and Thomas, L. (2001). Introduction to Distance Sampling. Oxford Univ. Press, Oxford.
• Coull, B. A. and Agresti, A. (1999). The use of mixed logit models to reflect heterogeneity in capture–recapture studies. Biometrics 55 294–301.
• Dorazio, R. M. and Royle, J. A. (2003). Mixture models for estimating the size of a closed population when capture rates vary among individuals. Biometrics 59 351–364.
• Durban, J. W. and Elston, D. A. (2005). Mark-recapture with occasion and individual effects: Abundance estimation through Bayesian model selection in a fixed dimensional parameter space. J. Agric. Biol. Environ. Stat. 10 291–305.
• Fienberg, S. E., Johnson, M. S. and Junker, B. W. (1999). Classical multilevel and Bayesian approaches to population size estimation using multiple lists. J. Roy. Statist. Soc. Ser. A 162 383–405.
• Gimenez, O. and Choquet, R. (2010). Individual heterogeneity in studies on marked animals using numerical integration: Capture–recapture mixed models. Ecology 91 951–957.
• King, R. and Brooks, S. P. (2001). On the Bayesian analysis of population size. Biometrika 88 317–336.
• King, R. and Brooks, S. P. (2008). On the Bayesian estimation of a closed population size in the presence of heterogeneity and model uncertainty. Biometrics 64 816–824.
• King, R., Morgan, B. J. T., Gimenez, O. and Brooks, S. P. (2009). Bayesian Analysis for Population Ecology Coba. CRC Press, Boca Raton, FL.
• King, R., Bird, S. M., Overstall, A. M., Hay, G. and Hutchinson, S. J. (2014). Estimating prevalence of injecting drug users and associated heroin-related death rates in England by using regional data and incorporating prior information. J. Roy. Statist. Soc. Ser. A 177 209–236.
• King, R., McClintock, B. T., Kidney, D. and Borchers, D. (2016). Supplement to “Capture–recapture abundance estimation using a semi-complete data likelihood approach.” DOI:10.1214/15-AOAS890SUPP.
• Langrock, R. and King, R. (2013). Maximum likelihood estimation of mark-recapture-recovery models in the presence of continuous covariates. Ann. Appl. Stat. 7 1709–1732.
• Link, W. A. (2013). A cautionary note on the discrete uniform prior for the binomial $N$. Ecology 94 2173–2179.
• Lunn, D., Jackson, C., Best, N., Thomas, A. and Spiegelhalter, D. (2013). The BUGS Book: A Practical Introduction to Bayesian Analysis. CRC Press, Boca Raton, FL.
• Madigan, D. and York, J. C. (1997). Bayesian methods for estimation of the size of a closed population. Biometrika 84 19–31.
• Manrique-Vallier, D. and Fienberg, S. E. (2008). Population size estimation using individual level mixture models. Biom. J. 50 1051–1063.
• McClintock, B. T., White, G. C., Burnham, K. P. and Pryde, M. A. (2009). A generalized mixed effects model of abundance for mark-resight data when sampling is without replacement. In Modeling Demographic Processes in Marked Populations (D. L. Thomson, E. G. Cooch and M. J. Conroy, eds.) 271–289. Springer, New York.
• McCrea, R. S. and Morgan, B. J. T. (2015). Analysis of Capture–Recapture Data. Chapman & Hall/CRC, Boca Raton, FL.
• Morgan, B. J. T. and Ridout, M. S. (2008). A new mixture model for capture heterogeneity. J. Roy. Statist. Soc. Ser. C 57 433–446.
• Otis, D. L., Burnham, K. P., White, G. C. and Anderson, D. R. (1978). Statistical inference from capture data on closed animal populations. Wildl. Monogr. 62 1–135.
• Overstall, A. M., King, R., Bird, S. M., Hutchinson, S. J. and Hay, G. (2014). Incomplete contingency tables with censored cells with application to estimating the number of people who inject drugs in Scotland. Stat. Med. 33 1564–1579.
• Pledger, S. (2000). Unified maximum likelihood estimates for closed capture–recapture models using mixtures. Biometrics 56 434–442.
• Pledger, S. (2005). The performance of mixture models in heterogeneous closed population capture–recapture. Biometrics 61 868–876.
• Pledger, S., Efford, M., Pollock, K. H., Collazo, J. A. and Lyons, J. E. (2009). Stopover duration analysis with departure probability dependent on unknown time since arrival. In Modeling Demographic Processes in Marked Populations (D. L. Thomson, E. G. Cooch and M. J. Conroy, eds.) 349–363. Springer, New York.
• Plummer, M. (2013). rjags: Bayesian graphical models using MCMC. R package version 3-11.
• Plummer, M., Best, N., Cowles, K. and Vines, K. (2015). CODA: Convergence diagnosis and output analysis for MCMC. R News 6 7–11.
• R Core Team (2014). R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. Available at http://www.R-project.org/.
• Royle, J. A. and Dorazio, R. M. (2012). Parameter-expanded data augmentation for Bayesian analysis of capture–recapture models. J. Ornithol. 152 S521–S537.
• Royle, J. A., Dorazio, R. M. and Link, W. A. (2007). Analysis of multinomial models with unknown index using data augmentation. J. Comput. Graph. Statist. 16 67–85.
• Royle, J. A., Karanth, K. U., Gopalaswamy, A. M. and Kumar, N. S. (2009). Bayesian inference in camera trapping studies for a class of spatial capture–recapture models. Ecology 90 3233–3244.
• Schofield, M. R. and Barker, R. J. (2014). Hierarchical modeling of abundance in closed population capture–recapture models under heterogeneity. Environ. Ecol. Stat. 21 435–451.
• Williams, B. K., Nichols, J. D. and Conroy, M. J. (2002). Analysis and Management of Animals Populations. Academic Press, San Diego, CA.

#### Supplemental materials

• Supplement to “Capture–recapture abundance estimation using a semi-complete data likelihood approach”. The supplement consists of Appendices A and B that provide sample JAGS codes for the examples provided in the text using the different model-fitting algorithms (referenced in Sections 3.2, 4.1 and 4.2).