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Rank and Redundancy of Multistate Mark-Recapture Models for Seabird Populations with Unobservable States

  • Chapter
Modeling Demographic Processes In Marked Populations

Part of the book series: Environmental and Ecological Statistics ((ENES,volume 3))

Abstract

Unobservable stages are common in many life cycles. Estimates of the vital rates, such as survival and breeding probabilities, of these stages are essential for demographic analysis but difficult to obtain. Explicit modeling of these states in multi-state mark-recapture methods can provide such estimates. However, models can be rank-deficient, meaning that not all parameters can be estimated. Determining whether a model is full rank is essential for interpretation of model selection and estimation results. Full rank models can be obtained by imposing biologically reasonable constraints on parameters. Developing such models requires an efficient way to assess model rank and determine which parameters, if any, are redundant. We introduce the use of automatic differentiation (AD) for this purpose. It generates the Jacobian matrix of the likelihood function in a way that is numerically stable, can accommodate large complicated models, and produces rank estimates accurate to machine precision. It reveals whether a model is full rank or rank-deficient (either intrinsically or for a particular data set), how many parameters or parameter combinations can be estimated, and which parameters are confounded. We use the method to explore three examples relevant to seabirds: a model with multiple breeding sites, a model distinguishing successful and failed breeders, and a model for pre-breeder survival and recruitment. We find a surprisingly large number of time-invariant and time-varying models to be of full rank, thus allowing estimation of all parameters, despite the unobservable states. We present a biological example for the Wandering Albatross (Diomedea exulans). Reliable assessment of model rank for multi-state mark-recapture models with unobservable stages will make it possible to use these methods in demographic applications.

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Notes

  1. 1.

    In sufficiently simple models, including extra information on capture probabilities obtained using Pollock’s robust design can sometimes resolve parameter redundancy (Kendall et al. 1997; Kendall and Nichols 2002). However, when the unobservable states are more richly structured, merely knowing capture probabilities in the observable states will generally not solve the problem.

  2. 2.

    Murota states this result for matrices whose entries are rational functions of the parameters, so it would apply directly to Jacobians calculated in terms of the probabilities (the identity link) rather than the logit transform of the probabilities. When calculated using the logit link, the entries of \({\bf J}\) are analytic, but not rational, functions of \({\boldsymbol \theta}\). However, analytic functions are given everywhere by their Taylor series, so the result applies equally to the logit link case (M. Golubitsky personal communication).

  3. 3.

    Note that this same argument is used independently by Rouan et al. (unpublished).

  4. 4.

    The number is slightly less than a simple combinatorial calculation would suggest because some possibilities (e.g., all \(\sigma_i\) equal, with additive time variation) are not possible.

References

  • Bailey LL, Kendall WL, Church DR, Wilbur HM (2004) Estimating survival and breeding probability for pond-breeding amphibians: a modified robust design. Ecology 85:2456–2466

    Article  Google Scholar 

  • Brownie C, Hines JE, Nichols JD, Pollock K, Hestbeck JB (1993) Capture–recapture studies for multiple strata including non-Markovian transitions. Biometrics 49:1173–1187

    Article  MATH  Google Scholar 

  • Cam E, Hines JE, Monnat J-Y, Nichols JD, Danchin E (1998) Are adult nonbreeders prudent parents? The kittiwake model. Ecology 79:2917–2930

    Article  Google Scholar 

  • Caswell H (2001) Matrix population models. Second edition. Sinauer, Sunderland, MA

    Google Scholar 

  • Caswell H, Fujiwara M (2004) Beyond survival estimation: mark-recapture, matrix population models, and population dynamics. Animal Biodiversity and Conservation 27:471–488

    Google Scholar 

  • Catchpole EA, Morgan BJT (1997) Detecting parameter redundancy. Biometrika 84:187–196

    Article  MATH  MathSciNet  Google Scholar 

  • Catchpole EA, Morgan BJT (2001) Deficiency of parameter-redundant models. Biometrika 88:593–598

    Article  MATH  MathSciNet  Google Scholar 

  • Catchpole EA, Freeman SN, Morgan BJT (1996) Steps to parameter redundancy in age-dependent recovery models. Journal of the Royal Statistical Society Series B- Methodological 58: 763–774

    MATH  MathSciNet  Google Scholar 

  • Catchpole EA, Morgan BJT, Freeman SN (1998) Estimation in parameter-redundant models. Biometrika 85:462–468

    Article  MATH  MathSciNet  Google Scholar 

  • Catchpole EA, Kgosi PM, Morgan BJT (2001) On the near-singularity of models for animal recovery data. Biometrics 57:720–726

    Article  MATH  MathSciNet  Google Scholar 

  • Catchpole EA, Morgan BJT, Viallefont A (2002) Solving problems in parameter redundancy using computer algebra. Journal of Applied Statistics 29:625–636

    Article  MATH  MathSciNet  Google Scholar 

  • Choquet R, Reboulet AM, Pradel R, Gimenez O, Lebreton JD (2004) M-SURGE: new software specifically designed for multistate capture-recapture models. Animal Biodiversity and Conservation 27:207–215

    Google Scholar 

  • Clobert J, Lebreton J-D, Allaine D, Gaillard JM (1994) The estimation of age-specific breeding probabilities from recaptures or resightings in vertebrate populations: II. Longitudinal models. Biometrics 50:375–387

    Article  Google Scholar 

  • Crespin L, Harris MP, Lebreton J-D, Frederikesen M, Wanless S (2006) Recruitment to a seabird population depends on environmental factors and on population size. Journal of Animal Ecology 75:228–238

    Article  Google Scholar 

  • Croxall J, Prince P, Rothery P, Wood AG (1998) Population changes in albatrosses at South Georgia. In: Robertson G, Gales R (eds) Albatross biology and conservation. Surrey Beatty Sons Pty Ltd., Chipping Norton, NSW, pp 69–83

    Google Scholar 

  • Forth SA, Edvall MM (2006) User guide for MAD – a Matlab automatic differentiation toolbox. TOMLAB/MAD. Version 1.4. The forward mode. Tomlab Optimization Inc., 855 Beech St #121, San Diego, CA, USA

    Google Scholar 

  • Fujiwara M, Caswell H (2002a) A general approach to temporary emigration in mark-recapture analysis. Ecology 83:3266–3275

    Google Scholar 

  • Fujiwara M, Caswell H (2002b) Estimating population projection matrices from multi-stage mark-recapture data. Ecology 83:3257–3265

    Google Scholar 

  • Gimenez O, Choquet R, Lebreton J-D (2003) Parameter redundancy in multi-state capture-recapture models. Biometrical Journal 45:704–722

    Article  MathSciNet  Google Scholar 

  • Gimenez O, Viallefont A, Catchpole EA, Choquet R, Morgan BJT (2004) Methods for investigating parameter redundancy. Animal Biodiversity and Conservation 27:561–572

    Google Scholar 

  • Golub GH, Van Loan CF (1996) Matrix computations. Third edition. Johns Hopkins University Press, Baltimore, MD

    Google Scholar 

  • Griewank A (2000) Evaluating derivatives: applications of algorithmic differentiation. SIAM, Philadelphia, PA

    Google Scholar 

  • Griewank A (2003) A Mathematical view of automatic differentiation. Acta Numerica 12:321–398

    Article  MATH  MathSciNet  Google Scholar 

  • Hirsch MW, Smale S (1974) Differential equations, dynamical systems, and linear algebra. Academic Press, New York

    MATH  Google Scholar 

  • Hunter CM, Caswell H (2005) Selective harvest of sooty shearwater chicks: effects on population dynamics and sustainability. Journal of Animal Ecology 74:589–600

    Article  Google Scholar 

  • Hunter CM, Moller H, Fletcher D (2000) Parameter uncertainty and elasticity analyses of a population model: setting research priorities for shearwaters. Ecological Modelling 134:299–323

    Article  Google Scholar 

  • Kendall WL, Nichols JD (1995) On the use of secondary capture–recapture samples to estimate temporary emigration and breeding proportions. Journal of Applied Statistics 22:751–762

    Article  Google Scholar 

  • Kendall WL, Bjorkland R (2001) Using open robust design models to estimate temporary emigration from capture–recapture data. Biometrics 57:1113–1122

    Article  MATH  MathSciNet  Google Scholar 

  • Kendall WL, Nichols JD (2002) Estimating state-transition probabilities for unobservable states using capture–recapture/resighting data. Ecology 83:3276–3284

    Google Scholar 

  • Kendall WL, Nichols JD, Hines JE (1997) Estimating temporary emigration using capture–recapture data with Pollock’s robust design. Ecology 78:563–578

    Google Scholar 

  • Kery M, Gregg KB, Schaub M (2005) Demographic estimation methods for plants with unobservable life-states. Oikos 108:307–320

    Article  Google Scholar 

  • Lebreton J-D, Pradel R (2002) Multistate recapture models: modelling incomplete individual histories. Journal of Applied Statistics 29:353–369

    Article  MATH  MathSciNet  Google Scholar 

  • Lebreton J-D, Burnham KP, Clobert J, Anderson DR (1992) Modeling survival and testing biological hypotheses using marked animals: a unified approach with case studies. Ecological Monographs 62:67–118

    Article  Google Scholar 

  • Lebreton J-D, Almeras T, Pradel R (1999) Competing events, mixtures of information, and multistratum recapture models. Bird Study 46:S39–S46

    Article  Google Scholar 

  • Lindberg MS, Kendall WL, Hines JE, Anderson MG (2001) Combining band recovery data and Pollock’s robust design to model temporary and permanent emigration. Biometrics 57: 273–282

    Article  MATH  MathSciNet  Google Scholar 

  • Magnus JR, Neudecker H (1988) Matrix differential calculus with applications in statistics and econometrics. John Wiley & Sons, New York

    MATH  Google Scholar 

  • Morgan BJT, Freeman SN (1989) A model with first-year variation for ring-recovery data. Biometrics 45:1087–1101

    Article  MATH  Google Scholar 

  • Murota K (2003) Matrices and matroids for ststems analysis. Springe-Verlag, New York

    Google Scholar 

  • Nichols JD, Sauer JR, Pollock KH, Hestbeck JB (1992) Estimating transition probabilities for stage-based population projection matrices using capture–recapture data. Ecology 73:306–312

    Article  Google Scholar 

  • Pradel R, Lebreton J-D (1999) Comparison of different approaches to the study of local recruitment of breeders. Bird Study 46(Supplement):74–81

    Google Scholar 

  • Rouan L, Choquet R, Pradel R (unpublished) A general framework for modeling memory in capture–recapture data. Submitted for publication.

    Google Scholar 

  • Schwarz CJ, Arnason AN (2000) Estimation of age-specific breeding probabilities from capture-recapture data. Biometrics 56:59–64

    Article  MATH  Google Scholar 

  • Schwarz CJ, Stobo WT (1997) Estimating temporary migration using the robust design. Biometrics 53:178–194

    Article  MATH  Google Scholar 

  • Shampine LF, Ketzscher R, Forth SA (2005) Using AD to solve BVPs in MATLAB. ACM Transactions on Mathematical Software 31:79–94

    Article  MATH  MathSciNet  Google Scholar 

  • Spendelow JA, Nichols JD, Hines JE, Lebreton J-D, Pradel R (2002) Modeling post-fledging survival and age-specific breeding probabilities in species with delayed maturity: a case study of Roseate terns at Faulkner Island, Connecticut. Journal of Applied Statistics 29:385–405

    Article  MATH  MathSciNet  Google Scholar 

  • Stewart GW (1991) Perturbation theory for the singular value decomposition. in Vaccaro RJ (ed) SVD and signal processing II: algorithms, analysis, and implementation. Elsevier, Amsterdam

    Google Scholar 

  • Stewart GW (1992) Determining rank in the presence of error. In: Moonen MS, Golub GH, de Moor BL (eds) Linear algebra for large scale and real-time applications. Kluwer, Dordrecth

    Google Scholar 

  • Weimerskirch H, Clobert J, Jouventin P (1987) Survival in five southern albatrosses and its relationship with their life history. Journal of Animal Ecology 56:1043–1055

    Article  Google Scholar 

  • Wonham WM (1985) Linear multivariate control: a geometric approach. Third edition. Springer, New York

    Google Scholar 

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Authors and Affiliations

  1. Department of Biology and Wildlife Institute of Arctic Biology, University of Alaska Fairbanks, PO Box 757000, Fairbanks, AK 99775, USA

    Christine M. Hunter

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  1. Christine M. Hunter
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  2. Hal Caswell
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Correspondence to Christine M. Hunter .

Editor information

Editors and Affiliations

  1. Max Planck Institute for Demographic Research, Konrad-Zuse Strasse 1, D18057 Rostock, Germany

    David L Thomson

  2. Netherlands Institute of Ecology Centre for Terrestrial Ecology, Boterhoeksestraat 48, Postbus 40, 6666 ZG Heteren, Netherlands

    David L Thomson

  3. School of Biological Sciences, University of Hong Kong, Pokfulam, Hong Kong

    David L Thomson

  4. Department of Natural Resources, Cornell University, Ithaca, NY, USA

    Evan G. Cooch

  5. D.B. Warnell School of Forestry and Natural Resources University of Georgia, Athens, GA, USA

    Michael J. Conroy

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Hunter, C.M., Caswell, H. (2009). Rank and Redundancy of Multistate Mark-Recapture Models for Seabird Populations with Unobservable States. In: Thomson, D.L., Cooch, E.G., Conroy, M.J. (eds) Modeling Demographic Processes In Marked Populations. Environmental and Ecological Statistics, vol 3. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-78151-8_37

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