Journal of Ornithology

, Volume 152, Supplement 2, pp 521–537 | Cite as

Parameter-expanded data augmentation for Bayesian analysis of capture–recapture models

EURING Proceedings


Data augmentation (DA) is a flexible tool for analyzing closed and open population models of capture–recapture data, especially models which include sources of hetereogeneity among individuals. The essential concept underlying DA, as we use the term, is based on adding “observations” to create a dataset composed of a known number of individuals. This new (augmented) dataset, which includes the unknown number of individuals N in the population, is then analyzed using a new model that includes a reformulation of the parameter N in the conventional model of the observed (unaugmented) data. In the context of capture–recapture models, we add a set of “all zero” encounter histories which are not, in practice, observable. The model of the augmented dataset is a zero-inflated version of either a binomial or a multinomial base model. Thus, our use of DA provides a general approach for analyzing both closed and open population models of all types. In doing so, this approach provides a unified framework for the analysis of a huge range of models that are treated as unrelated “black boxes” and named procedures in the classical literature. As a practical matter, analysis of the augmented dataset by MCMC is greatly simplified compared to other methods that require specialized algorithms. For example, complex capture–recapture models of an augmented dataset can be fitted with popular MCMC software packages (WinBUGS or JAGS) by providing a concise statement of the model’s assumptions that usually involves only a few lines of pseudocode. In this paper, we review the basic technical concepts of data augmentation, and we provide examples of analyses of closed-population models (M 0, M h , distance sampling, and spatial capture–recapture models) and open-population models (Jolly–Seber) with individual effects.


Hierarchical models Individual covariates Individual heterogeneity Markov chain Monte Carlo Occupancy models 



We thank Beth Gardner and Elise Zipkin for reviewing drafts of this manuscript. We thank Ullas Karanth (camera-trapping data) and Jim Nichols (Microtus data) for making data from their research available for our use. Use of trade, product, or firm names does not imply endorsement by the U.S. Government.


  1. Bled F, Royle JA, Cam E (2010) Hierarchical modeling of an invasive spread: case of the Eurasian collared-dove Streptopelia decaocto in the USA. Ecol Appl (in press)Google Scholar
  2. Bonner S, Schwarz C (2006) An extension of the Cormack Jolly Seber model for continuous covariates with application to Microtus pennsylvanicus. Biometrics 62:142–149PubMedCrossRefGoogle Scholar
  3. Borchers DL, Efford MG (2008) Spatially explicit maximum likelihood methods for capture–recapture studies. Biometrics 64:377–385PubMedCrossRefGoogle Scholar
  4. Burnham KP, Overton WS (1978) Estimation of the size of a closed population when capture probabilities vary among animals. Biometrika 65:625–633CrossRefGoogle Scholar
  5. Converse SJ, Royle JA (2010) Dealing with incomplete and variable detectability in multi-year, multi-site monitoring of ecological populations. In: Design and analysis of long-term ecological monitoring studies (in press)Google Scholar
  6. Cooch E, White G (2001) Using MARK: a gentle introduction. Cornell University, IthacaGoogle Scholar
  7. Coull BA, Agresti A (1999) The use of mixed logit models to reflect heterogeneity in capture–recapture studies. Biometrics 55:294–301PubMedCrossRefGoogle Scholar
  8. Crosbie SF, Manly BFJ (1985) Parsimonious modelling of capture–mark-recapture studies. Biometrics 41:385–398CrossRefGoogle Scholar
  9. Dorazio RM, Royle JA (2003) Mixture models for estimating the size of a closed population when capture rates vary among individuals. Biometrics 59:350–363CrossRefGoogle Scholar
  10. Dorazio RM, Royle JA (2005) Estimating size and composition of biological communities by modeling the occurrence of species. J Am Stat Assoc 100:389–398CrossRefGoogle Scholar
  11. Dorazio RM, Royle JA, Soderstrom B, Glimskar A (2006) Estimating species richness and accumulation by modeling species occurrence and detectability. Ecology 87:842–854PubMedCrossRefGoogle Scholar
  12. Dorazio RM, Kéry M, Royle JA, Plattner M (2010) Models for inference in dynamic metacommunity systems. Ecology 91:2466–2475PubMedCrossRefGoogle Scholar
  13. Dupuis JA, Schwarz CJ (2007) A Bayesian approach to the multistate Jolly-Seber capture–recapture model. Biometrics 63:1015–1022PubMedCrossRefGoogle Scholar
  14. Durban JW, Elston DA (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 10:291–305CrossRefGoogle Scholar
  15. Efford M (2004) Density estimation in live-trapping studies. Oikos 106:598–610CrossRefGoogle Scholar
  16. Gardner B, Royle JA, Wegan MT, Rainbolt RE, Curtis PD (2010) Estimating black bear density using DNA data from hair snares. J Wildl Manag 74:318–325CrossRefGoogle Scholar
  17. Gardner B, Reppucci J, Lucherini M, Royle JA (2010b) Spatially-explicit inference for open populations: estimating demographic parameters from camera-trap studies. Ecology 91:3376–3383Google Scholar
  18. Gimenez O, Rossi V, Choquet R, Dehais C, Doris B, Varella H, Vila JP, Pradel R (2007) State-space modelling of data on marked individuals. Ecol Model 206:431–438CrossRefGoogle Scholar
  19. Gimenez O, Choquet R (2010) Individual heterogeneity in studies on marked animals using numerical integration: capture–recapture mixed models. Ecology 91:148–154Google Scholar
  20. Green PJ (1995) Reversible jump Markov chain Monte Carlo computation and Bayesian model determination. Biometrika 82:711–732CrossRefGoogle Scholar
  21. Johnson DH (1999) The insignificance of statistical significance testing. J Wildl Manag 63:763–772CrossRefGoogle Scholar
  22. Jolly G (1965) Explicit estimates from capture–recapture data with both death and immigration—stochastic model. Biometrika 52:225–247PubMedGoogle Scholar
  23. Karanth KU (1995) Estimating tiger (Panthera tigris) populations from camera-trap data using capture–recapture models. Biol Conserv 71:333–338CrossRefGoogle Scholar
  24. Karanth KU, Nichols JD (1998) Estimation of tiger densities in India using photographic captures and recaptures. Ecology 79:2852–2862CrossRefGoogle Scholar
  25. Karanth K, Nichols JD, Kumar N, Hines JE (2006) Assessing tiger population dynamics using photographic capture–recapture sampling. Ecology 87:2925–2937PubMedCrossRefGoogle Scholar
  26. Kéry M, Royle JA (2009) Inference about species richness and community structure using species-specific occupancy models in the National Swiss Breeding Bird Survey MHB. In: Thomson DL, Cooch EG, Conroy MJ (eds) Modeling demographic processes in marked populations. Springer, New York, pp 639–656CrossRefGoogle Scholar
  27. Kéry M, Royle JA, Plattner M, Dorazio RM (2009) Species richness and occupancy estimation in communities subject to temporary emigration. Ecology 90:1279–1290PubMedCrossRefGoogle Scholar
  28. King R, Brooks SP (2001) On the Bayesian analysis of population size. Biometrika 88:317–336CrossRefGoogle Scholar
  29. King R, Brooks SP (2008) On the Bayesian estimation of a closed population size in the presence of heterogeneity and model uncertainty. Biometrics 64:816–824PubMedCrossRefGoogle Scholar
  30. King R, Brooks SP, Coulson T (2008) Analysing complex capture–recapture data in the presence of individual and temporal covariates and model uncertainty. Biometrics 64:1187–1195PubMedCrossRefGoogle Scholar
  31. Langtimm CA, Dorazio RM, Stith BM, Doyle TJ (2010) A new aerial survey design to monitor manatee abundance for Everglades restoration. J Wildl Manag (in press)Google Scholar
  32. Lebreton JD, Burnham K, Clobert J, Anderson DR (1992) Modeling survival and testing biological hypotheses using marked animals: a unified approach with case studies. Ecol Monogr 62:67–118CrossRefGoogle Scholar
  33. Link WA (2003) Nonidentifiability of population size from capture–recapture data with heterogeneous detection probabilities. Biometrics 59:1123–1130PubMedCrossRefGoogle Scholar
  34. Link WA, Barker RJ (2010) Bayesian inference: with ecological applications. Academic, New YorkGoogle Scholar
  35. Liu JS, Wu YN (1999) Parameter expansion for data augmentation. J Am Stat Assoc 94:1264–1274CrossRefGoogle Scholar
  36. Lunn D, Spiegelhalter D, Thomas A, Best N (2009) The BUGS project: evolution, critique and future directions (with discussion). Stat Med 28:3049–3082PubMedCrossRefGoogle Scholar
  37. MacKenzie DI, Nichols JD, Lachman GB, Droege S, Royle JA, Langtimm CA (2002) Estimating site occupancy rates when detection probabilities are less than one. Ecology 83:2248–2255CrossRefGoogle Scholar
  38. MacKenzie DI, Nichols JD, Hines JE, Knutson MG, Franklin AB (2003) Estimating site occupancy, colonization, and local extinction when a species is detected imperfectly. Ecology 84:2200–2207CrossRefGoogle Scholar
  39. Nichols JD, Karanth KU (2002) Statistical concepts: assessing spatial distributions. In: Monitoring tigers and their prey: a manual for researchers, managers, and conservationists in tropical Asia. Centre for Wildlife Studies, pp 29–38Google Scholar
  40. Patil A, Huard D, Fonnesbeck CJ (2010) PyMC 2.0: Bayesian stochastic modelling in python. J Stat Softw (in press)Google Scholar
  41. Pledger S (2005) The performance of mixture models in heterogeneous closed population capture–recapture. Biometrics 61:868–873Google Scholar
  42. Pledger S, Pollock KH, Norris JL (2003) Open capture–recapture models with heterogeneity: I. Cormack-Jolly-Seber model. Biometrics 59:786–794PubMedCrossRefGoogle Scholar
  43. Pollock K (1982) A capture–recapture design robust to unequal probability of capture. J Wildl Manag 46:757–760CrossRefGoogle Scholar
  44. Royle JA (2006) Site occupancy models with heterogeneous detection probabilities. Biometrics 62:97–102PubMedCrossRefGoogle Scholar
  45. Royle JA (2008) Modeling individual effects in the Cormack-Jolly-Seber model: a state-space formulation. Biometrics 64:364–370Google Scholar
  46. Royle JA (2009) Analysis of capture—recapture models with individual covariates using data augmentation. Biometrics 65:267–274PubMedCrossRefGoogle Scholar
  47. Royle JA, Kéry M (2007) A Bayesian state-space formulation of dynamic occupancy models. Ecology 88:1813–1823Google Scholar
  48. Royle JA, Dorazio RM, Link WA (2007) Analysis of multinomial models with unknown index using data augmentation. J Comput Graph Stat 16:67–85CrossRefGoogle Scholar
  49. Royle JA, Dorazio RM (2008) Hierarchical modeling and inference in ecology: the analysis of data from populations, metapopulations and communities. Academic, San DiegoGoogle Scholar
  50. Royle JA, Gardner B (2010) Hierarchical spatial capture–recapture models for estimating density from trapping arrays. In: O'Connell AF, Nichols JD, Karanth KU (eds) Camera traps in animal ecology: methods and analyses. Springer, Berlin Google Scholar
  51. Royle JA, Young KV (2008) A hierarchical model for spatial capture–recapture data. Ecology 89:2281–2289PubMedCrossRefGoogle Scholar
  52. Royle JA, Karanth KU, Gopalaswamy AM, Kumar NS (2009) Bayesian inference in camera trap studies using a class of spatial capture–recapture models. Ecology 90:3233–3244PubMedCrossRefGoogle Scholar
  53. Schwarz C, Arnason A (1996) A general methodology for the analysis of capture–recapture experiments in open populations. Biometrics 52:860–873CrossRefGoogle Scholar
  54. Schofield MR, Barker RJ (2008) A unified capture–recapture framework. J Agric Biol Environ Stat 13:458–477CrossRefGoogle Scholar
  55. Seber G (1965) A note on the multiple-recapture census. Biometrika 52:249–59PubMedGoogle Scholar
  56. Tanner MA (1996) Tools for statistical inference: methods for the exploration of posterior distributions and likelihood functions, 3rd edn. Springer, New YorkGoogle Scholar
  57. Tanner MA, Wong WH (1987) The calculation of posterior distributions by data augmentation. J Am Stat Assoc 82:528–540CrossRefGoogle Scholar
  58. Williams BK, Nichols JD, Conroy MJ (2002) Analysis and management of animal populations. Academic, San DiegoGoogle Scholar
  59. Wright JA, Barker RJ, Schofield MR, Frantz AC, Byrom AE, Gleeson DM (2009) Incorporating genotype uncertainty into mark-recapture–type models for estimating abundance using DNA samples. Biometrics 65:833–840PubMedCrossRefGoogle Scholar

Copyright information

© Dt. Ornithologen-Gesellschaft e.V. (outside the USA) 2010

Authors and Affiliations

  1. 1.USGS Patuxent Wildlife Research CenterLaurelUSA
  2. 2.USGS Southeast Ecological Science CenterGainesvilleUSA
  3. 3.Department of StatisticsUniversity of FloridaGainesvilleUSA

Personalised recommendations