State-space models are widely used in ecology. However, it is well known that in practice it can be difficult to estimate both the process and observation variances that occur in such models. We consider this issue for integrated population models, which incorporate state-space models for population dynamics. To some extent, the mechanism of integrated population models protects against this problem, but it can still arise, and two illustrations are provided, in each of which the observation variance is estimated as zero. In the context of an extended case study involving data on British Grey herons, we consider alternative approaches for dealing with the problem when it occurs. In particular, we consider penalised likelihood, a method based on fitting splines and a method of pseudo-replication, which is undertaken via a simple bootstrap procedure. For the case study of the paper, it is shown that when it occurs, an estimate of zero observation variance is unimportant for inference relating to the model parameters of primary interest. This unexpected finding is supported by a simulation study.
This is a preview of subscription content, log in to check access.
Buy single article
Instant access to the full article PDF.
Price includes VAT for USA
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
This is the net price. Taxes to be calculated in checkout.
Abadi, F., Gimenez, O., Arlettaz, R., Schaub, M.: An assessment of integrated population models: bias, accuracy, and the violation of the assumption of independence. Ecology 91, 7–14 (2010)
Barry, S.C., Brooks, S.P., Catchpole, E.A., Morgan, B.J.T.: The analysis of ring-recovery data using random effects. Biometrics 59, 54–65 (2003)
Bengtsson, T., Cavanaugh, J.E.: An improved Akaike information criterion for state-space model selection. Comput. Stat. Data Anal. 50, 2635–2654 (2006)
Besbeas, P., Morgan, B.J.T.: Kalman filter initialisation for integrated population modelling. Appl. Stat. 61, 151–162 (2011)
Besbeas, P., Morgan, B.J.T.: A threshold model for heron productivity. J. B. Agric. Environ. Stat. 17, 128–141 (2012)
Besbeas, P., Morgan, B.J.T.: Goodness of fit of integrated population models using calibrated simulation. Methods Ecol. Evol. 5, 1373–1382 (2014)
Besbeas, P., Freeman, S.N., Morgan, B.J.T., Catchpole, E.A.: Integrating mark–recapture–recovery and census data to estimate animal abundance and demographic parameters. Biometrics 58, 540–547 (2002)
Besbeas, P., Lebreton, J.-D., Morgan, B.J.T.: The efficient integration of abundance and demographic data. Appl. Stat. 52, 95–102 (2003)
Besbeas, P., Borysiewicz, R.S., Morgan, B.J.T.: Completing the ecological jigsaw. In: D.L. Thomson, E.G. Cooch, and M. J. Conroy (Eds.) Modelling Demographic Processes in Marked Populations. Springer Series: Environmental and Ecological Statistics, vol. 3, pp. 513–540. Springer, Berlin (2009)
Besbeas, P., McCrea, R.S., Morgan, B.J.T.: Integrated population model selection in ecology. University of Kent Technical Report. https://kar.kent.ac.uk/id/eprint/48039 (2015)
Brooks, S.P., King, R., Morgan, B.J.T.: A Bayesian approach to combining animal abundance and demographic data. Anim. Biodivers. Conserv. 27, 515–529 (2004)
Burnham, K.P., Rexstad, E.A.: Modeling heterogeneity in survival rates of banded waterfowl. Biometrics 49, 1194–1208 (1993)
Chandler, R., Clark, J.: Spatially explicit integrated population models. Methods Ecol. Evol. 5, 1351–1360 (2014)
Dennis, B., Ponciano, J.M., Lele, S.R., Taper, M.L., Staples, D.F.: Estimating density dependence, process noise and observation error. Ecol. Monogr. 76, 323–341 (2006)
Dennis, B., Ponciano, J.M., Taper, M.L.: Replicated sampling increases efficiency in monitoring biological populations. Ecology 91, 610–620 (2010)
de Valpine, P., Hastings, A.: Fitting population models incorporating process noise and observation error. Ecol. Monogr. 72, 57–76 (2002)
de Valpine, P., Hilborn, R.: State-space likelihoods for nonlinear fisheries time-series. Can. J. Fish. Aquat. Sci. 62, 1937–1952 (2005)
Durbin, J., Koopman, S.J.: Time Series Analysis by State Space Methods. Oxford University Press, Oxford (2001)
Francis, R.I.C.C.: Data weighting in statistical fisheries stock assessment models. Can. J. Fish. Aquat. Sci. 68, 1124–1138 (2011)
Freckleton, R.P., Watkinson, A.R., Green, R.E., Sutherland, W.J.: Census error and the detection of density dependence. J. Anim. Ecol. 75, 837–851 (2006)
Gonçalves, S., Politis, D.: Discussion: Bootstrap methods for dependent data: a review. J. Korean Stat. Soc. 40, 383–386 (2011)
Green, P., Silverman, B.: Nonparametric Regression and Generalized Linear Models: A Roughness Penalty Approach. Chapman & Hall/CRC Press, Boca Raton (1994)
Kéry, M., Schaub, M.: Bayesian Population Analysis using WinBUGS: A Hierarchical Perspective. Academic Press, Cambridge (2012)
King, R.: A review of Bayesian state-space modelling of capture–recapture–recovery data. Interface Focus 2, 190–204 (2012)
King, R.: Statistical ecology. Ann. Rev. Stat. Appl. 1, 401–426 (2014)
Knape, J.: Estimability of density dependence in models of time series data. Ecology 89, 2994–3000 (2008)
Knape, J., Korner-Nievergelt, F.: Estimates from non-replicated population surveys rely on critical assumptions. Methods Ecol. Evol. (2015). doi:10.1111/2041-210X.12329
Knape, J., Besbeas, P., de Valpine, P.: Using uncertainty estimates in analyses of population time series. Ecology 94, 2097–2107 (2013)
McCrea, R.S., Morgan, B.J.T.: Analysis of Capture–Recapture Data. CRC Chapman & Hall, Boca Raton (2014)
McCrea, R.S., Morgan, B.J.T., Gimenez, O., Besbeas, P., Bregnballe, T., Lebreton, J.-D.: Multi-site integrated population modelling. J. Biol. Agric. Environ. Stat. 15, 539–561 (2010)
Maunder, M.N., Deriso, R.B., Hanson, C.H.: Use of state-space population dynamics models in hypothesis testing: advantages over simple log-linear regressions for modeling survival, illustrated with application to longfin smelt (Spirinchus thaleichthys). Fish. Res. 164, 102–111 (2015)
Mazzettta, C., Morgan, B.J.T., Coulson, T.: A state-space modelling approach to population size estimation. Technical report, University of Kent Technical Report: UKC/SMSAS/10/025 (2010)
Newman, K.B., Buckland, S.T., Morgan, B.J.T., King, R., Borchers, D.L., Cole, D.J., Besbeas, P.T., Gimenez, O., Thomas, L.: Modelling Population Dynamics: Model Formulation, Fitting and Assessment using State-Space Methods. Springer, New York (2014)
Patterson, T.A., Parton, A., Langrock, R., Blackwell, P.G., Thomas, L., King. R.: Statistical modelling of individual animal movement: an overview of key methods and a discussion of practical challenges. arXiv:1603.07511v3 [stat.AP] (2017)
Pollock, K.H., Raveling, D.G.: Assumptions of modern band-recovery models, with emphasis on heterogeneous survival rates. J. Wildl. Manag. 46, 88–98 (1982)
Rice, J.: Bandwidth choice for nonparametric regression. Ann. Stat. 12, 1215–1230 (1984)
Schaub, M., Abadi, F.: Integrated population models: a novel analysis framework for deeper insights into population dynamics. J. Ornithol. 152, 227–237 (2011)
Searle, S.R.: Matrix Algebra Useful for Statistics. Wiley, New York (1982)
Tavecchia, G., Besbeas, P., Coulson, T., Morgan, B.J.T., Clutton-Brock, T.H.: Estimating population size and hidden demographic parameters with state-space modelling. Am Nat. 173, 722–733 (2009)
Wang, J.-P., Lindsay, B.G.: A penalized nonparametric maximum likelihood approach to species richness estimation. J. Am. Stat. Assoc. 100, 942–959 (2005)
We thank the Associate Editor, Roland Langrock, two anonymous referees, Stephen Freeman, Mark Maunder, Leo Polanski and Martin Ridout for their very helpful comments.
Electronic supplementary material
Below is the link to the electronic supplementary material.
Here, we give the short R programme for fitting a cubic spline, using cross-validation, and evaluating the standard deviation of the jacknife residuals. In the programme, x is a vector of the census years and y is a vector of the census values.
About this article
Cite this article
Besbeas, P., Morgan, B.J.T. Variance estimation for integrated population models. AStA Adv Stat Anal 101, 439–460 (2017). https://doi.org/10.1007/s10182-017-0304-5
- Cubic splines
- Grey heron
- Mark–recovery–recapture data
- Penalised likelihood
- Plug-in method
- Process/observation error estimation
- State-space models
- Time-dependent parameters