EURING PROCEEDINGS Model comparison and assessment for multi-state capture–recapture–recovery models Rachel S. McCrea • Byron J. T. Morgan • Thomas Bregnballe Received: 12 September 2009 / Revised: 2 October 2010 / Accepted: 1 November 2010 / Published online: 1 December 2010 Ó Dt. Ornithologen-Gesellschaft e.V. 2010 Abstract The work of this paper is motivated by a study of Great Cormorants, Phalacrocorax carbo sinensis, in Denmark. The dataset is complex, involving birds in dif- ferent states living in and moving between neighbouring colonies. As a consequence, the set of probability models that might describe the data is large. In order to choose between the models, we present a score test approach for moving efficiently between the members of a model set with many members. We then provide a new measure for testing the absolute goodness-of-fit of the selected model to the data. This measure may be used when a model is multi- state/multi-site, and involves age- and time-dependence, as well as integrated recovery and recapture data, which is needed for the application. An illustration is provided by data from a single colony only, but with two breeding states, and an additional emigrated state. Keywords Goodness-of-fit  Great Cormorants  Integrated recovery and recapture data  Multi-state models  Score tests Introduction Tests for multi-state models Multi-state models for capture–recapture data are evidently important, but it is unfortunately often difficult in practice to choose between alternative versions of such models. The reason for this is that there are frequently very many models to choose between, and also some of these models can be hard to fit. In extreme cases, models can be parameter redundant, which occurs for certain model pa- rameterisations and means that not all parameters can be estimated (Catchpole and Morgan 1997). Irrespective of how many data are collected, parameter-redundant models result in flat likelihoods, due to the fact that it is not pos- sible to estimate all the model parameters; for instance, this can occur if parameters are confounded and only appear together as a product. These issues are more severe for multi-state models; see, for example, Cole (2011). Methods of symbolic algebra now exist that can be used to detect parameter redundancy (Catchpole and Morgan 1997). Score tests have been proposed as a means of choosing between alternative capture–recovery models by Catchpole and Morgan (1996), and used for variable-selection in such models by Catchpole et al. (1999). Score tests are due to Rao (1948), and when the hypothesis being tested is true, they result in test statistics that are asymptotically equiv- alent to likelihood-ratio tests. However, they posses the advantage that, when one is comparing nested models, only the simpler of the two models needs to be fitted in order to conduct the test. By contrast, for likelihood-ratio tests, both models have to be fitted. Thus, in order to compare models using either likelihood-ratio tests or the Akaike information criterion (AIC), it is necessary to fit all the models in a model set. As stated above, this can be very difficult in the Communicated by M. Schaub. R. S. McCrea  B. J. T. Morgan (&) National Centre for Statistical Ecology, School of Mathematics, Statistics and Actuarial Science, University of Kent, Canterbury CT2 7NF, UK e-mail: B.J.T.Morgan@kent.ac.uk R. S. McCrea e-mail: R.S.McCrea@kent.ac.uk T. Bregnballe Department of Wildlife Ecology and Biodiversity, National Environmental Research Institute, Aarhus University, Kalø, Grena˚vej 14, 8410 Rønde, Denmark e-mail: tb@dmu.dk 123 J Ornithol (2012) 152 (Suppl 2):S293–S303 DOI 10.1007/s10336-010-0611-z