The Annals of Applied Statistics

Bayesian binomial mixture models for estimating abundance in ecological monitoring studies

Guohui Wu, Scott H. Holan, Charles H. Nilon, and Christopher K. Wikle

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Abstract

Investigation of species abundance has become a vital component of many ecological monitoring studies. The primary objective of these studies is to understand how specific species are distributed across the study domain, as well as quantification of the sampling efficiency for detecting these species. To achieve these goals, preselected locations are sampled during scheduled visits, in which the number of species observed at each location is recorded. This results in spatially referenced replicated count data that are often unbalanced in structure and exhibit overdispersion. Motivated by the Baltimore Ecosystem Study, we propose Bayesian hierarchical binomial mixture models, including Binomial Conway–Maxwell Poisson (Bin-CMP) mixture models, that formally account for varying levels of spatial dispersion. Our proposed models also allow for variable selection of model covariates and grouping of dispersion parameters through the implementation of reversible jump Markov chain Monte Carlo methodology. Finally, using demographic covariates from the American Community Survey, we demonstrate the effectiveness of our approach through estimation of abundance for the American Robin (Turdus migratorius) in the Baltimore Ecosystem Study.

Article information

Source
Ann. Appl. Stat., Volume 9, Number 1 (2015), 1-26.

Dates
First available in Project Euclid: 28 April 2015

Permanent link to this document
https://projecteuclid.org/euclid.aoas/1430226082

Digital Object Identifier
doi:10.1214/14-AOAS801

Mathematical Reviews number (MathSciNet)
MR3341105

Zentralblatt MATH identifier
06446558

Keywords
American Community Survey American Robin Conway–Maxwell Poisson negative binomial overdispersion parallel computing unbalanced data underdispersion

Citation

Wu, Guohui; Holan, Scott H.; Nilon, Charles H.; Wikle, Christopher K. Bayesian binomial mixture models for estimating abundance in ecological monitoring studies. Ann. Appl. Stat. 9 (2015), no. 1, 1--26. doi:10.1214/14-AOAS801. https://projecteuclid.org/euclid.aoas/1430226082


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References

  • Aronson, M. F., La Sorte, F. A., Nilon, C. H., Katti, M., Goddard, M. A., Lepczyk, C. A., Warren, P. S., Williams, N. S., Cilliers, S., Clarkson, B. et alet al. (2014). A global analysis of the impacts of urbanization on bird and plant diversity reveals key anthropogenic drivers. Proc. R. Soc. B Biol. Sci. 281 20133330. 1780.
  • Brooks, S. P. and Gelman, A. (1998). General methods for monitoring convergence of iterative simulations. J. Comput. Graph. Statist. 7 434–455.
  • Cameron, A. C. and Trivedi, P. K. (1998). Regression Analysis of Count Data. Econometric Society Monographs 30. Cambridge Univ. Press, Cambridge.
  • Carroll, R. J. and Lombard, F. (1985). A note on $N$ estimators for the binomial distribution. J. Amer. Statist. Assoc. 80 423–426.
  • Chandler, R. B., Royle, J. A. and King, D. I. (2011). Inference about density and temporary emigration in unmarked populations. Ecology 92 1429–1435.
  • Conway, R. and Maxwell, W. (1962). A queuing model with state dependent service rates. Int. J. Ind. Eng. 12 132–136.
  • Cressie, N. and Johannesson, G. (2008). Fixed rank kriging for very large spatial data sets. J. R. Stat. Soc. Ser. B. Stat. Methodol. 70 209–226.
  • Cressie, N. and Wikle, C. K. (2011). Statistics for Spatio-Temporal Data. Wiley, Hoboken, NJ.
  • Dail, D. and Madsen, L. (2011). Models for estimating abundance from repeated counts of an open metapopulation. Biometrics 67 577–587.
  • Denison, C. (2010). Effects of socioeconomics on European Starling (Sturnus Vulgaris) abundance in Baltimore, Maryland. Master’s thesis, Univ. Missouri, Columbia, MO.
  • Furrer, R., Nychka, D. and Sain, S. (2012). fields: Tools for spatial data. R package version 6.7.
  • George, E. and McCulloch, R. (1993). Variable selection via Gibbs sampling. J. Amer. Statist. Assoc. 88 881–889.
  • George, E. and McCulloch, R. (1997). Approaches for Bayesian variable selection. Statist. Sinica 7 339–374.
  • Graves, T., Kendall, K., Royle, J., Stetz, J. and Macleod, A. (2011). Linking landscape characteristics to local grizzly bear abundance using multiple detection methods in a hierarchical model. Animal Conservation 14 652–664.
  • Green, P. J. (1995). Reversible jump Markov chain Monte Carlo computation and Bayesian model determination. Biometrika 82 711–732.
  • Herbers, J. M. (1989). Community structure in North temperate ants: Temporal and spatial variation. Oecologia 81 201–211.
  • Hodges, J. S. and Reich, B. J. (2010). Adding spatially-correlated errors can mess up the fixed effect you love. Amer. Statist. 64 325–334.
  • Holan, S., Wang, S., Arab, A., Sadler, E. J. and Stone, K. (2008). Semiparametric geographically weighted response curves with application to site-specific agriculture. J. Agric. Biol. Environ. Stat. 13 424–439.
  • Hooten, M. B. and Hobbs, N. T. (2015). A guide to Bayesian model selection for ecologists. Ecol. Mono. 85 3–28.
  • Kéry, M. (2008). Estimating abundance from bird counts: Binomial mixture models uncover complex covariate relationships. The Auk 125 336–345.
  • Kéry, M. and Royle, J. (2010). Hierarchical modelling and estimation of abundance and population trends in metapopulation designs. J. Anim. Ecol. 79 453–461.
  • Kéry, M., Royle, J. and Schmid, H. (2005). Modeling avian abundance from replicated counts using binomial mixture models. Ecol. Appl. 15 1450–1461.
  • King, R. and Brooks, S. P. (2002). Bayesian model discrimination for multiple strata capture–recapture data. Biometrika 89 785–806.
  • Little, R. J. A. and Rubin, D. B. (2002). Statistical Analysis with Missing Data, 2nd ed. Wiley, Hoboken, NJ.
  • Loss, S. R., Ruiz, M. O. and Brawn, J. D. (2009). Relationships between avian diversity, neighborhood age, income, and environmental characteristics of an urban landscape. Biol. Conserv. 142 2578–2585.
  • Maloney, M. and Auffrey, C. (2013). The Social Areas of Cincinnati, 5th ed. School of Planning, Univ. Cincinnati, Cincinnati, OH. Available at http://www.socialareasofcincinnati.org/files/FifthEdition/SASBook.pdf.
  • Minka, T., Shmueli, G., Kadane, J., Borle, S. and Boatwright, P. (2003). Computing with the COM-Poisson distribution. Technical report, Statistics Dept., Carnegie Mellon Univ., Pittsburgh, PA.
  • Müller, N., Ignatieva, M., Nilon, C. H., Werner, P. and Zipperer, W. C. (2013). Patterns and trends in urban biodiversity and landscape design. In Urbanization, Biodiversity and Ecosystem Services: Challenges and Opportunities (T. Elmqvist, S. Parnell, M. Fragkias, M. Schewenius, J. Goodness, M. Sendstad, B. Güneralp, K. C. Seto, P. J. Marcotullio, C. Wilkinson and R. I. McDonald, eds.) 123–174. Springer, Berlin.
  • O’Hara, R. B. and Sillanpää, M. J. (2009). A review of Bayesian variable selection methods: What, how and which. Bayesian Anal. 4 85–117.
  • Oh, J., Washington, S. P. and Nam, D. (2006). Accident prediction model for railway-highway interfaces. Accident Anal. Prev. 38 346–356.
  • Pickett, S. T. A., Cadenasso, M. L., Grove, J. M., Boone, C. G., Groffman, P. M., Irwin, E., Kaushal, S. S., Marshall, V., McGrath, B. P., Nilon, C. H., Pouyat, R. V., Szlavecz, K., Troy, A. and Warren, P. (2011). Urban ecological systems: Scientific foundations and a decade of progress. J. Environ. Econ. Manage. 92 331–362.
  • Pickett, S., Brush, G., Felson, A., McGrath, B., Grove, J., Nilon, C., Szlavecz, K., Swan, C. and Warren, P. (2012). Understanding and working with urban biodiversity: The Baltimore Ecosystem Study. CityGreen 4 68–77.
  • Pollock, K. H. (1982). A capture–recapture design robust to unequal probability of capture. J. Wildl. Manag. 46 752–757.
  • Ridout, M. S. and Besbeas, P. (2004). An empirical model for underdispersed count data. Stat. Model. 4 77–89.
  • Royle, J. A. (2004). $N$-mixture models for estimating population size from spatially replicated counts. Biometrics 60 108–115.
  • Royle, J. and Dorazio, R. (2006). Hierarchical models of animal abundance and occurrence. J. Agric. Biol. Environ. Stat. 11 249–263.
  • Royle, J. A. and Dorazio, R. M. (2008). Hierarchical Modeling and Inference in Ecology: The Analysis of Data from Populations, Metapopulations and Communities. Academic Press, San Diego, CA.
  • Royle, J. and Link, W. (2005). A general class of multinomial mixture models for anuran calling survey data. Ecology 86 2505–2512.
  • Royle, J. A. and Wikle, C. K. (2005). Efficient statistical mapping of avian count data. Environ. Ecol. Stat. 12 225–243.
  • Ruppert, D., Wand, M. P. and Carroll, R. J. (2003). Semiparametric Regression. Cambridge Univ. Press, Cambridge.
  • Sallabanks, R. and James, F. C. (1999). American Robin (Turdus migratorius). In The Birds of North America 462 (A. Pool and F. Gill, eds.) 1–27. Academy of Natural Sciences, Philadelphia, PA.
  • Sellers, K. F., Borle, S. and Shmueli, G. (2012). The COM-Poisson model for count data: A survey of methods and applications. Appl. Stoch. Models Bus. Ind. 28 104–116.
  • Shmueli, G., Minka, T. P., Kadane, J. B., Borle, S. and Boatwright, P. (2005). A useful distribution for fitting discrete data: Revival of the Conway–Maxwell–Poisson distribution. J. Roy. Statist. Soc. Ser. C 54 127–142.
  • Shochat, E., Lerman, S. and Fernández-Juricic, E. (2010). Birds in urban ecosystems: Population dynamics, community structure, biodiversity, and conservation. In Urban Ecosystem Ecology (J. A. Peterson and A. Volder, eds.) Agronomy Monographs 55 75–86. ASA-CSSA-SSSA Madison, WI.
  • Smallbone, L. T., Luck, G. W. and Wassens, S. (2011). Anuran species in urban landscapes: Relationships with biophysical, built environment and socio-economic factors. Landsc. Urb. Plan. 101 43–51.
  • Spiegelhalter, D. J., Best, N. G., Carlin, B. P. and van der Linde, A. (2002). Bayesian measures of model complexity and fit. J. R. Stat. Soc. Ser. B. Stat. Methodol. 64 583–639.
  • R Development Core Team (2013). R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria. ISBN 3-900051-07-0.
  • Ver Hoef, J. M. V. and Boveng, P. L. (2007). Quasi-Poisson vs. negative binomial regression: How should we model overdispersed count data? Ecology 88 2766–2772.
  • Vignola, F., Michalsky, J. and Stoffel, T. (2012). Solar and Infrared Radiation Measurements. CRC Press, Boca Raton, FL.
  • Webster, R. A., Pollock, K. H. and Simons, T. R. (2008). Bayesian spatial modeling of data from avian point count surveys. J. Agric. Biol. Environ. Stat. 13 121–139.
  • Wenger, S. J. and Freeman, M. C. (2008). Estimating species occurrence, abundance, and detection probability using zero-inflated distributions. Ecology 89 2953–2959.
  • Wikle, C. K. (2010). Low-rank representations for spatial processes. In Handbook of Spatial Statistics (A. E. Gelfand, P. J. Diggle, M. Fuentes and P. Guttorp, eds.) 107–118. Chapman & Hall, Boca Raton, FL.
  • Williams, B., Nichols, J. and Conroy, M. (2002). Analysis and Management of Animal Populations. Academic Press, San Diego, CA.
  • Wu, G., Holan, S. H. and Wikle, C. K. (2013). Hierarchical Bayesian spatio-temporal Conway–Maxwell Poisson models with dynamic dispersion. J. Agric. Biol. Environ. Stat. 18 335–356.
  • Wu, G., Holan, S. H., Nilon, C. H. and Wikle, C. K. (2015). Supplement to “Bayesian binomial mixture models for estimating abundance in ecological monitoring studies.” DOI:10.1214/14-AOAS801SUPP.

Supplemental materials

  • Supplement to “Bayesian binomial mixture models for estimating abundance in ecological monitoring studies”.: The supplementary material contains the MCMC sampling algorithm, details regarding computation times for the models implemented, and additional figures.