A Generalized Estimating Equations Approach to Model Heterogeneity and Time Dependence in Capture-Recapture Studies

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Abdus S Akanda
Russell Alpizar-Jara


Individual heterogeneity in capture probabilities and time dependence are fundamentally important for estimating the closed animal population parameters in capture-recapture studies. A generalized estimating equations (GEE) approach accounts for linear correlation among capture-recapture occasions, and individual heterogeneity in capture probabilities in a closed population capture-recapture individual heterogeneity and time variation model. The estimated capture probabilities are used to estimate animal population parameters. Two real data sets are used for illustrative purposes. A simulation study is carried out to assess the performance of the GEE estimator. A Quasi-Likelihood Information Criterion (QIC) is applied for the selection of the best fitting model. This approach performs well when the estimated population parameters depend on the individual heterogeneity and the nature of linear correlation among capture-recapture occasions.

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Akanda, A. S., & Alpizar-Jara, R. (2017). A Generalized Estimating Equations Approach to Model Heterogeneity and Time Dependence in Capture-Recapture Studies. European Journal of Ecology, 3(1), 9–17. https://doi.org/10.1515/eje-2017-0002


Agresti, A. (1994) Simple capture-recapture models permitting unequal
catchability and variable sampling effort. Biometrics, 50(2), 494–
Akanda, M.A.S. & Alpizar-Jara, R. (2014a) A generalized estimating
equations approach for capture-recapture closed population
models. Environmental and Ecological Statistics, 21(4), 667–688.
Akanda, M.A.S. & Alpizar-Jara, R. (2014b) Estimation of capture probabilities
using generalized estimating equations and mixed effects
approaches. Ecology and Evolution, 4(7), 1158–1165.
Briand, L.C., Emam, K.E., Freimut, B. & Oliver (1997) Quantitative evaluation
of capture-recapture models to control software inspections,
Proceedings of the Eighth International Conference on
Software Reliability Engineering, Albuquerque, NM, 234–244.
Chao, A., Lee, S.M. & Jeng, S.L. (1992) Estimating population size for
capture-recapture data when capture probabilities vary by time
and individual animal. Biometrics, 48(1), 201–216.
Chao, A. & Lee, S.M. (1993) Estimating population size for continuous
time capture-recapture models via sample coverage. Biometrical
Journal, 35(1), 29–45.
Chao, A., Tsay, P.K., Lin, S.H., Shau, W.Y. & Chao, D.Y. (2001) The applications
of capture-recapture models to epidemiological data. Stat.
Med., 20(20), 3123–57.
Chao, A. & Huggins, R.M. (2005) Modern closed-population capture-recapture
models. In: C. Amstrup, T.L. McDonald and B.F.J. Manly
(eds), Handbook of capture-recapture analysis. Princeton University
Press, Princeton, NJ, 58–87.
Diggle, P., Heagerty, P., Liang, K.Y. & Zeger, S. (2013) Analysis of longitudinal
data. 2nd Edition, Oxford University Press, New York.
Hardin, J. W. & Hilbe, J.M. (2013) Generalized estimating equations, 2nd
Edition. Chapman and Hall/CRC Press, Boca Ratan, FL.
Horvitz, D.G. & Thompson, D.J. (1952) A generalization of sampling
without replacement from a finite universe. J. Amer. Statist. Assoc.,
47(260), 663–685.
Huggins, R.M. (1989) On the statistical analysis of capture experiments.
Biometrika, 76(1), 133–140.
Huggins, R.M. (1991) Some practical aspects of a conditional likelihood
approach to capture experiments. Biometrics, 47(2), 725–732.
Huggins, R.M. & Yip, P.S.F. (1997) Statistical analysis of removal experiments
with the use of auxillary variables. Stat. Sin., 7(3), 705–
Hwang, W.H. & Huggins, R.M. (2005) An examination of the effect of
heterogeneity on the estimation of population size using capture-
recapture data. Biometrika, 92(1), 229–233.
King, R. & Brooks, S.P. (2008) On the Bayesian estimation of a closed
population size in the presence of heterogeneity and model uncertainty.
Biometrics, 64(3), 816–824.
Liang, K.Y. & Zeger, S.L. (1986) Longitudinal data analysis using generalized
linear models. Biometrika, 73(1), 13–22.
Lloyd, C. & Yip, P. (1991) A unification of inference from capture-recapture
studies through martingale estimating functions. In: Godambe,
V.P. (eds) Estimating functions. Oxford: Clarendon Press,
Otis, D.L., Burnham, K.P., White, G.C. & Anderson, D.R. (1978) Statistical
inference from capture data on closed animal populations. Wildlife
Monographs, 62, 1–135.
Pan, W. (2001) Akaike’s information criterion in generalized estimating
equations. Biometrics, 57(1), 120–125.
Pledger, S. (2000) Unified maximum likelihood estimates for closed
capture-recapture models using mixtures. Biometrics, 56(2),
Pollock, K., Hines, J. & Nichols, J. (1984) The use of auxiliary variables in
capture-recapture and removal experiments. Biometrics, 40(2),
Pradel, R. & Sanz-Aguilar, A. (2012) Modeling trap-awareness and related
phenomena in capture-recapture studies. PLoS ONE, 7(3),
Qaqish, B.F. (2003) A family of multivariate binary distributions for simulating
correlated binary variables with specified marginal means
and correlations. Biometrika, 90(2), 455–463.
R Development Core Team (2016) R: A language and environment for
statistical computing. R Foundation for Statistical Computing,
Vienna, Austria. ISBN 3-900051-07-0. Available from: http://
Rexstad, E. & Burnham, K. (1991) User’s guide for interactive program
CAPTURE. Colorado Cooperative Fish and Wildlife Research Unit,
Seber, G.A.F. (2002) The estimation of animal abundance and related
parameters. 2nd Edition, The Blackburn Press, London, Edward
Stanley, T.R. & Richards, J.D. (2005) Software review: a program for testing
capture-recapture data for closure. Wildlife Society Bulletin,
33(2), 782–785.
Stoklosa, J. & Huggins, R.M. (2012) A robust p-spline approach to closed
population capture-recapture models with time dependence
and heterogeneity. Comput. Stat. Data Anal., 56(2), 408–417.
Williams, B.K., Nichols, J.D. & Conroy, M.J. (2002) Analysis and management
of animal populations. Academic Press, San Diego, California.
Yang, H.C. & Chao, A. (2005) Modeling animals behavioral response by
Markov chain models for capture-recapture experiments. Biometrics,
61(4), 1010–1017.
Zeger, S.L. & Liang, K.Y. (1986) Longitudinal data analysis for discrete and
continuous outcomes. Biometrics, 42(1), 121–130.
Zhang, S. (2012) A GEE approach for estimating size of hard-to-reach
population by using capture-recapture data. Statistics, 46(2),