Estimation of Population Size When Capture Probability Depends on Individual States
Abstract
We develop a multistate model to estimate the size of a closed population from capture–recapture studies. We consider the case where capture–recapture data are not of a simple binary form, but where the state of an individual is also recorded upon every capture as a discrete variable. The proposed multistate model can be regarded as a generalisation of the commonly applied set of closed population models to a multistate form. The model allows for heterogeneity within the capture probabilities associated with each state while also permitting individuals to move between the different discrete states. A closedform expression for the likelihood is presented in terms of a set of sufficient statistics. The link between existing models for capture heterogeneity is established, and simulation is used to show that the estimate of population size can be biased when movement between states is not accounted for. The proposed unconditional approach is also compared to a conditional approach to assess estimation bias. The model derived in this paper is motivated by a real ecological data set on great crested newts, Triturus cristatus. Supplementary materials accompanying this paper appear online.
Keywords
Abundance Closed population Individual heterogeneity Transition probabilities1 Introduction
The models presented within this paper focus on the estimation of the size of a closed population along with capture and transition probabilities between discrete states using real ecological capture–recapture data on a population of great crested newts Triturus cristatus. An assumption often made in the modelling of capture–recapture data is that of homogeneity in the probability of capture. When estimating the size of a closed population, one in which the population being sampled remains constant across all capture occasions, violations of this assumption can lead to biased estimates of abundance (Seber 1982; Hwang and Huggins 2005). A number of models have been proposed that relax this homogeneity assumption. In particular, Pollock (1974) and Otis et al. (1978) proposed a set of eight closed population capture–recapture models. These models allow the probability of capture to be affected by three factors: time (capture probabilities vary by occasion); behaviour (probability of initial capture is different to all subsequent recaptures) and heterogeneity (each individual has a different capture probability). These models have been fitted using a variety of methods including maximum likelihood (Otis et al. 1978; Agresti 1994; Norris and Pollock 1996; Coull and Agresti 1999; Pledger 2000), the jackknife (Burnham and Overton 1978, 1979; Pollock and Otto 1983), moment methods based on sample coverage (Chao et al. 1992) and Bayesian methods (Casteldine 1981; Gazey and Staley 1986; Smith 1988, 1991; George and Robert 1992; Diebolt and Robert 1993; Ghosh and Norris 2005; King and Brooks 2008). To specifically address the problem of heterogeneity in the capture probabilities, a variety of models have been proposed, including finite mixtures (Diebolt and Robert 1993; Agresti 1994; Norris and Pollock 1996; Pledger 2000) and infinite mixtures (Coull and Agresti 1999; Dorazio and Royle 2003). A comparison of examples of the two types of mixture through simulation are presented in Pledger (2005) and Dorazio and Royle (2005). An issue that commonly arises when estimating the size of closed populations is that different individual heterogeneity models which may be deemed to fit the data equally well can give rise to very different estimates of the abundance (Link 2003, 2006). An extended mixture model which provides a convenient framework for model selection is presented in Morgan and Ridout (2008); see also Holzmann et al. (2006).
The model we present can be considered a generalisation of the timedependent multistate closed population model of Schwarz and Ganter (1995) to a form that additionally includes trap dependence and heterogeneity in the capture probabilities. It may also be seen as a closed population capture–recapture equivalent to the Arnason–Schwarz (AS) model (Arnason 1972, 1973; Brownie et al. 1993; Schwarz et al. 1993; King and Brooks 2003; Lebreton et al. 2009), for open capture–recapture data. Initially developed for multisite capture–recapture data, but more generally applicable to individuals captured in discrete states, the AS model is a multistate generalisation of the Cormack–Jolly–Seber (CJS) model (Cormack 1964; Jolly 1965; Seber 1965). The CJS and AS models allow for a time dependence in the capture probabilities, with the AS model additionally able to allow capture probabilities to be state dependent. However, these models condition on the first capture of an individual and so are unable to estimate the total population size directly. Dupuis and Schwarz (2007) consider a similar multistate extension for the Jolly–Seber model for estimating abundance in open populations, fitted within a Bayesian (data augmentation) framework.
We consider a similar AStype state dependence in the closed population scenario, within an explicit closedform likelihood framework where population size is estimated directly through the likelihood. In particular, we compare the performance of the new unconditional multistate model to the existing singlestate models which ignore the state information and do not include movement between states and a conditional approach where the population size is not estimated directly through the likelihood. We present the likelihood in terms of a set of minimal sufficient statistics which permits the fit of the model to be assessed using a Pearson chisquared test.
The motivation for developing this methodology relates to a study on great crested newts. A species with protected status in Europe, individuals within the study population are captured weekly throughout the breeding season, with the additional state information referring to the pond in which the individuals are captured. Originally consisting of four ponds, the study site was extended to a total of eight ponds in 2009 with the new ponds being first colonised in the 2010 breeding season. How these new ponds have been colonised, the effectiveness of the traps to capture individuals and whether there are differences in the capture probabilities between the old and new ponds are of particular interest. Ignoring any differences between the old and new ponds, for instance differing amounts of vegetation which may be affecting the probability of capture, may lead to poorer estimates of the total population size, and for this study, the total population size and the states themselves are of interest.
In Sect. 2, we review the construction of existing singlestate closed population models in terms of sets of sufficient statistics, before introducing the likelihood function for the multistate model and considering the timevarying population size for each state in Sect. 3. The performance of the multistate model is compared to a conditional approach and existing heterogeneity models using simulation in Sect. 4 with a particular focus on the bias and precision of the population size estimates and the ability of the new model to estimate state specific parameters and population size. The new model is applied to the data set of great crested newts in Sect. 5. The paper concludes with a general discussion in Sect. 6.
2 SingleState Closed Population Models

\(M_{h(k)}\): individual capture probabilities come from a mixture model with k components (Pledger 2000);

\(M_{h(be)}\): individual capture probabilities specified to be from an underlying beta distribution (Burnham 1972; Dorazio and Royle 2003);

\(M_{h(bbe)}\): individual capture probabilities come from a mixture model with two components: one component simply has a fixed capture probability, while the other component is specified to be from some underlying beta distribution (Morgan and Ridout 2008).
The use of sufficient statistics allows for an efficient evaluation of the likelihood. In addition, they have the advantage of being able to be used to assess the performance of each of these models through the calculation of the Pearson chisquared statistic, since the likelihood of the data is of multinomial form.
3 Multistate Closed Population Model

\(p_t(r)\): the probability an individual is initially captured at time \(t=1,\dots ,T\) given that the individual is in state \(r \in \mathcal{R}\) at this time;

\(c_t(r)\): the probability an individual is recaptured at time \(t=2,\dots ,T\) given that the individual is in state \(r \in \mathcal{R}\) at this time;

\(\psi _t(r,s)\): probability an individual is in state \(s \in \mathcal{R}\) at time \(t+1\), given that an individual is in state \(r \in \mathcal{R}\) at time \(t=1,\dots ,T1\);

\(\alpha (r)\): probability an individual is in state \(r \in \mathcal{R}\) at time \(t=1\).
3.1 Likelihood Formulation
 (i)
\(z_t(r)\): the number of individuals that are observed for the first time at time \(t=1,\dots ,T\) in state \(r \in \mathcal{R}\);
 (ii)
\(n_{t_1,t_2}(r,s)\): the number of individuals that are observed at time \(t_1 =1,\dots ,T1\) in state \(r \in \mathcal{R}\) and next observed at time \(t_2 = t_1+1,\dots ,T\) in state \(s \in \mathcal{R}\);
 (iii)
\(v_t(r)\): the number of individuals that are observed for the last time at time \(t=1,\dots ,T1\) in state \(r \in \mathcal{R}\).

\(M_0^R\): \(p_t(r) = c_t(r) = p\), for all \(r \in \mathcal{R}\) and \(t=1,\dots ,T\);

\(M_t^R\): \(p_t(r) = c_t(r) = p_t\), for all \(r \in \mathcal{R}\) and \(t=1,\dots ,T\);

\(M_b^R\): \(p_t(r) = p\) and \(c_t(r) = c\), for all \(r \in \mathcal{R}\) and \(t=1,\dots ,T\);

\(M_h^R\): \(p_t(r) = c_t(r) = p(r)\), for all \(r \in \mathcal{R}\) and \(t=1,\dots ,T\);
Evaluating the likelihood through the sufficient statistics uses recursions similar to those in hidden Markov models (HMMs) but in more efficient forms. In the HMM framework, the likelihood considers each individual encounter history in turn. By using more efficient sufficient statistics, we are able to reduce the number of operations required to calculate the likelihood. This is achieved by using the probabilities associated with each of the sufficient statistics for multiple partial histories.
The likelihood estimates the total population size N. This is the number of individuals in the population on each capture occasion (since a closed population remains constant). The number of individuals in each state on each occasion can be estimated using a forward–backwardtype algorithm. Typically applied to hidden Markov models (HMMs) the forward–backward algorithm calculates the conditional state probabilities on each occasion for a given observation sequence. We use these conditional probabilities calculated for the observed capture histories and the estimated total population size to obtain estimates of statedependent abundance. Our approach differs from the typical HMM application since the states are partially observed (uncertainty in the state of an individual only occurs when they are not captured). Further details on this approach are presented in the online supplementary material Appendix A and are demonstrated in the simulation study and newt application below.
3.2 Conditional and Unconditional Approaches
4 Simulation Study
For the true model, in all cases considered, the estimates of N do not show any bias. In the twostate scenario, the remaining model parameters are estimated well with little difference in variation between low and high mobilities. In the threestate scenario, the remaining model parameters are estimated well in the case of low mobility. In the scenario of high mobility for the threestate case, some of the remaining model parameter estimates appear to exhibit some bias and there is very large variation in all of the parameter estimates. This appears to be due to an “averaging” or “mixing” effect across the states where there is greater uncertainty about the state of an individual when they are not captured leading to greater uncertainty in the parameter estimates. The traditional models without any individual heterogeneity, \(M_0\), \(M_t\) and \(M_b\), indicate a strong negative bias for the case of low mobility for both the two and threestate scenarios. For these three models, the variability in the estimates of N is similar for low and high mobilities for both two and three states. For the lowmobility scenarios, the heterogeneity models \(M_{h(2)}\), \(M_{h(3)}\), \(M_{h(be)}\) and \(M_{h(bbe)}\) all estimate N well, but there are a large number of extremely large estimates. For the highmobility scenario, the heterogeneity models \(M_{h(2)}\), \(M_{h(3)}\), \(M_{h(be)}\) and \(M_{h(bbe)}\) appear to be positively biased. This is due to underestimation of the capture or mixture probabilities or both caused by the mixing effect described above. When there are three states, the heterogeneity models have higher precision in estimating N when there is high mobility. In comparison with the existing heterogeneity models, the new multistate model has the greatest precision of all the heterogeneity models considered for low and high mobility in both the two and threestate cases. The conditional model displays very similar results to the unconditional approach, which appears to agree with the findings of Fewster and Jupp (2009). The new multistate model shows no bias in estimating the population size in each state on each occasion in all cases except the threestate highmobility scenario. In the case of low mobility, the estimated number in states with higher capture probabilities shows less variation, and this is expected since more individuals in the state are captured leading to less uncertainty about the number in the state. For the threestate highmobility case, the estimates are biased to varying degrees, and this is again due to the mixing issue described above and the high uncertainty of the model parameters. Plots of the bias of this statedependent population size are given in Appendix B of the online supplementary material.
5 Application: Great Crested Newts
These data are collected from a study site on the University of Kent campus and are included as electronic supplementary material to this article. Data have been collected on the population of newts breeding at the site since 2002. Within this study, capture occasions occur weekly throughout the breeding season, with individuals being identified through unique physical markings. The additional state information corresponds to the pond in which the newts are captured. Originally the site consisted of four ponds but was extended in 2009 to a total of eight ponds, four “old” ponds (state 1) and four “new” ponds (state 2), with the new ponds first being colonised in 2010. Of specific interest is whether there were any differences between the old and new ponds in terms of capture and transition probabilities when they were first colonised and whether any differences have remained. In order to assess this, we compare results from the 2010 and 2013 data sets. In order to make the assumption of closure reasonable, we take a subset of six weeks (\(T=6\)) from the middle of each of the 2010 and 2013 breeding seasons, during which it can be assumed that all breeding newts have arrived at the breeding ponds and have not yet started to leave the area.
MLEs and 95% nonparametric bootstrap confidence intervals for the parameters of the \(M_h^2\) model for the great crested newt study 2010 and 2013 data.
Parameter  2010 data  2013 data  

MLE  95% CI  MLE  95% CI  
N  33.95  (33.00, 36.33)  45.96  (44.00, 49.42) 
p(1)  0.82  (0.39, 0.99)  0.36  (0.21, 0.48) 
p(2)  0.33  (0.22, 0.54)  0.41  (0.30, 0.56) 
\(\psi (1,2)\)  0.31  (0.07, 0.48)  0.05  (0.00, 0.15) 
\(\psi (2,1)\)  0.03  (0.00, 0.09)  0.08  (0.02, 0.19) 
\(\alpha (1)\)  0.48  (0.31, 0.67)  0.33  (0.16, 0.54) 
The MLEs of the capture probabilities indicate that in 2010 the old ponds had a higher capture probability than for the new ponds. However, by 2013 the higher capture probability for the old ponds seems to have disappeared with similar capture probabilities for both old and new ponds (see below for discussion of model selection). The old ponds had more vegetation around the traps which meant that the newts had a greater chance of entering them than in the new ponds, where traps were more exposed. In addition, for 2010 the transition probabilities indicate a general movement trend away from the old ponds to the new ponds, but once a newt reaches the new pond demonstrates high fidelity to the new ponds. This movement can be clearly seen in Fig. 5 (lefthand plot). Previous analyses suggested that new recruits (first time breeders) used the new ponds more frequently than newts returning to the ponds (Lewis 2010). By 2013, the newts show high fidelity to both the old and new ponds. Finally, we note that in 2010 the newts appear to be evenly distributed between the two ponds at the beginning of the study period, but by 2013, the proportion of newts increases in the new ponds (though the confidence intervals are reasonably wide).
Interestingly, the results imply that only a single individual was missed during the study period in 2010 and two were missed during the 2013 study period. These estimates are in keeping with the ecological understanding of the population. It is believed that capture probability over the breeding season as a whole is very high. This was confirmed in 2005 when a drift fence was set up confirming that all individuals had been captured at least once. A period of six weeks has been selected here in the central part of the breeding season, to accommodate the assumption of closure. Outside of the selected six week period, in 2010, 7 individuals were seen before the selected period, but not during the study and one individual after but not during the study period. No newts were captured both before and after. Of those seen only before, all were seen quite early in the season, while the one individual seen after the study period is seen immediately after the 6 week period. In 2013, 5 individuals were seen before, but not during, the closed period (all were seen very early in the season) and one individual is recaptured before and after the study period, but not during.
\(\Delta \)AIC values, MLEs and 95% nonparametric bootstrap CIs for N (denoted \(\hat{N}\)) and corresponding chisquared goodnessoffit parametric bootstrap pvalues for four multistate models for the great crested newt study 2010 and 2013 data.
Model  2010 data  2013 data  

\(\Delta \)AIC  \(\hat{N}\)  95% CI  pvalue  \(\Delta \)AIC  \(\hat{N}\)  95% CI  pvalue  
\(M_0^2\)  8.40  33.1  (33.0, 34.2)  0.004  6.77  44.0  (44.0, 48.3)  0.175 
\(M_h^2\)  0  33.9  (33.0, 36.3)  0.022  6.67  46.0  (44.0, 49.4)  0.203 
\(M_t^2\)  5.34  33.0  (33.0, 33.8)  0.007  0  45.6  (44.0, 47.8)  0.794 
\(M_{th}^2\)  1.70  33.0  (33.0, 35.4)  0.039  1.69  45.7  (44.0, 48.9)  0.803 
MLEs and 95% nonparametric bootstrap confidence intervals for the parameters of the \(M_t^2\) model for the great crested newt study 2013 data.
Parameter  MLE  95% CI 

N  45.63  (44.00, 47.76) 
\(p_1\)  0.39  (0.26, 0.55) 
\(p_2\)  0.53  (0.38, 0.68) 
\(p_3\)  0.35  (0.21, 0.50) 
\(p_4\)  0.18  (0.07, 0.29) 
\(p_5\)  0.46  (0.31, 0.61) 
\(p_6\)  0.44  (0.30, 0.59) 
\(\psi (1,2)\)  0.05  (0.00, 0.15) 
\(\psi (2,1)\)  0.07  (0.02, 0.14) 
\(\alpha (1)\)  0.32  (0.16, 0.49) 
All models fitted to the data suggest high fidelity to the old ponds in 2013 and the new ponds in both years with an increase in the proportion of individuals arriving at the new ponds in 2013 with similar estimates for the total population size. The difference in choice of pond on arrival can be seen in Fig. 5. In comparison with 2010 (model \(M_h^2\)), model \(M_t^2\) for the 2013 data shows the distribution of newts between the old and new ponds converging to a near equal split between the two. The Pearson’s chisquared goodnessoffit test (with parametric bootstrap pvalues from 9999 bootstraps) does not indicate a lack of fit for the models fitted to the 2013 data. However, for the 2010 data, it is suggestive of a lack of fit. In conducting the goodnessoffit test we do not pool small cells together. Fewer individuals are observed in 2010 leading to an increase in the number of small cells observed.
MLEs and 95% nonparametric bootstrap confidence intervals for N (denoted \(\hat{N}\)) for seven singlestate models for the great crested newt study 2010 and 2013 data.
Model  2010 data  2013 data  

\(\hat{N}\)  95% CI  \(\hat{N}\)  95% CI  
\(M_0\)  33.1  (33.0, 33.4)  44.0  (44.0, 48.5) 
\(M_t\)  33.0  (33.0, 33.8)  45.6  (44.0, 47.8) 
\(M_b\)  33.0  (33.0, 33.1)  45.3  (44.0, 50.0) 
\(M_{h(2)}\)  41.5  (33.0, 10,000.0)  46.5  (44.8, 64.9) 
\(M_{h(3)}\)  41.5  (33.0, 10,000.0)  46.5  (44.0, 64.1) 
\(M_{h(be)}\)  34.7  (33.0, 47.8)  46.7  (44.0, 55.3) 
\(M_{h(bbe)}\)  41.5  (33.0, 10,000.0)  46.5  (44.0, 62.7) 
6 Discussion
We have focussed on deriving multistate closed capture–recapture models, where additional individual timevarying discrete covariates are observed. The models derived can be viewed as a closed population analogy to the AS model, assuming a firstorder Markovian process for the transitions between states. The construction of an explicit closedform (unconditional) likelihood via a set of sufficient statistics permits efficient evaluation of the likelihood and standard goodnessoffit techniques, in the form of Pearson’s chisquared tests, to be applied. This can lead to generally small cell entries in the goodnessoffit test, with different approaches for pooling cells and their interpretation a focus of current research. Similarities of the closed multistate model to the AS model also permit other extensions to be immediately applied. For example, in many cases, state may be only partially observed, including failure to observe a state when an individual is observed, or observing states with error (King and McCrea 2014). In the case where no states are known with certainty, the model reduces to a multieventtype model (Pradel 2005) corresponding to a finite mixture model which allows for transitions between states. Conditional on the observed number of individuals, these multievent models can be fitted within ESURGE (Choquet et al. 2009) to estimate the model parameters (though this package does not have the associated Horvitz–Thompsonlike estimator incorporated into it). We note that the limiting case where no states are observed upon recapture and there are no transitions between states, the model reduces to the mixture models proposed by Pledger (2000). Further, the modelling approach can be applied to the case of continuous individual timevarying covariates by considering an approximate (discretised) likelihood of multistate form (Langrock and King 2013). The movement between the different states can also be generalised by removing the firstorder Markov assumption, where the dwelltime distribution (the time spent in each state) is geometric, and instead imposing a more general dwelltime distribution, for example a shifted Poisson or negativebinomial distribution (King and Langrock 2016).
The proposed multistate closed population model shows better accuracy and precision in estimating N compared to competing models where the additional discrete state information is ignored. Further, additional insight can be obtained with regard to the states, which may themselves be of interest. Most notably, transition probabilities can be estimated (and hence the stable equilibrium distribution of the population over the states) and the relationship between state and capture probabilities evaluated. For the newt data analysis conducted, particular interest lays in the potential transition of newts from the old ponds to the new ponds installed in 2009 with a interest also in the total population size, not least with regard to the completeness of the data collection process and observing all individuals present. The analyses concluded that the data survey collection process appears to be close to a complete census of individuals present at the site which is unusual for capture–recapture studies. Further, there was a general transition of newts from the old ponds to the new ponds between 2010 and 2013, but with little movement within the season. Finally, it appeared that there were initial differences between the capture probabilities in new and old ponds, in 2010, but once the new ponds had become established, the state dependence was no longer significant by 2013.
In the presence of an underlying multistate system process for closed populations, an unconditional likelihood can be derived and MLEs of the model parameters obtained, extending the previous conditional approaches. In the absence of the observed discrete covariate data, existing heterogeneity models appear to perform adequately; however, including the covariate information does improve the precision of the population estimate, as would be expected. The model developed here can be extended to the open population case, permitting the estimation of both recruitment and departure times from the study population along with statedependent capture and transition probabilities, i.e. to stopover models where departure times can additionally depend on time since recruitment. Developing these models using classical methods is a focus of current research.
Notes
Acknowledgements
Worthington was funded by the Carnegie Trust for the Universities of Scotland and part funded by EPSRC/NERC grant EP/10009171/1 and McCrea by NERC fellowship grant NE/J018473/1. We would like to thank Amy Wright, Brett Lewis and the volunteer newt surveyors for collecting and collating the field data. We would also like to thank Byron Morgan and Roland Langrock for helpful discussions regarding this work.
Supplementary material
References
 Agresti, A. (1994) Simple capturerecapture models permitting unequal catchability and variable sampling effort. Biometrics 50, 494–500.CrossRefGoogle Scholar
 Arnason, A.N. (1972) Parameter estimates from markrecapturerecovery experiments on two populations subject to migration and death. Researches on Population Ecology 13, 97–113.CrossRefGoogle Scholar
 Arnason, A.N. (1973) The estimation of population size, migration rates, and survival in a stratified population. Researches on Population Ecology 15, 1–8.Google Scholar
 Bishop, Y.M.M., Fienberg, S.E. and Holland, P.W. (1975) Discrete Multivariate Analysis: Theory and Practice. MIT Press, Cambridge, MA, USA.zbMATHGoogle Scholar
 Brownie, C., Hines, J.E., Nichols, J.D., Pollock, K.H. and Hestbeck, J.B. (1993) Capturerecapture studies for multiple strata including nonMarkovian transitions. Biometrics 49, 1173–1187.CrossRefzbMATHGoogle Scholar
 Burnham, K.P. (1972) Estimation of population size in multiple capturerecapture studies when capture probabilities vary among animals. PhD thesis, Oregon State University, Cornwallis.Google Scholar
 Burnham, K.P. and Overton, W.S. (1978) Estimation of the size of a closed population when capture probabilities vary among animals. Biometrika 65, 625–633.CrossRefzbMATHGoogle Scholar
 Burnham, K.P. and Overton, W.S. (1979) Robust estimation of population size when capture probabilities vary among animals. Ecology 60, 927–936.Google Scholar
 Casteldine, B.J. (1981) A Bayesian analysis of multiple capturerecapture sampling for a closed population. Biometrika 67, 197–210.CrossRefGoogle Scholar
 Chao, A. (2001) An overview of closed capturerecapture models. Journal of Agricultural, Biological, and Environmental Statistics 6, 158–175.CrossRefGoogle Scholar
 Chao, A., Lee, S.M. and Jeng, S.L. (1992) Estimating population size for capturerecapture data when capture probabilities vary by time and individual animal. Biometrics 48, 201–216.CrossRefzbMATHGoogle Scholar
 Choquet, R., Rouan, L. and Pradel, R. (2009) Program ESURGE: a software application for fitting multievent models. Environmental and Ecological Statistics 3, 845–865.Google Scholar
 Cormack, R.M. (1964) Estimates of survival from the sighting of marked animals. Biometrika 51, 429–438.CrossRefzbMATHGoogle Scholar
 Coull, B.A. and Agresti, A. (1999) The use of mixed logit models to reflect heterogeneity in capturerecapture studies. Biometrics 55, 294–301.CrossRefzbMATHGoogle Scholar
 Diebolt, J. and Robert, C.P. (1993) Estimation of finite mixture distributions through Bayesian sampling. Journal of the Royal Statistical Society: Series B 56, 363–375.MathSciNetzbMATHGoogle Scholar
 Dorazio, R.M. and Royle, J.A. (2003) Mixture models for estimating the size of a closed population when capture rates vary among individuals. Biometrics 59, 351–364.MathSciNetCrossRefzbMATHGoogle Scholar
 Dorazio, R.M. and Royle, J.A. (2005) Rejoinder to “The performance of mixture models in heterogeneous closed population capturerecapture”. Biometrics 61, 874–876.Google Scholar
 Dupuis, J.A. and Schwarz, C.J. (2007) A Bayesian approach to the multistate JollySeber capturerecapture model. Biometrics 63, 1015–1022.MathSciNetCrossRefzbMATHGoogle Scholar
 Fewster, R.M. and Jupp, P.E. (2009) Inference on population size in binomial detectability models. Biometrika 96, 805–820.MathSciNetCrossRefzbMATHGoogle Scholar
 Gazey, W.J. and Staley, M.J. (1986) Population estimation from markrecapture experiments using a sequential Bayes algorithm. Ecology 67, 941–951.CrossRefGoogle Scholar
 George, E.I. and Robert, C.P. (1992) Capturerecapture estimation via Gibbs sampling. Biometrika 79, 677–683.MathSciNetzbMATHGoogle Scholar
 Ghosh, S.K. and Norris, J.L. (2005) Bayesian capturerecapture analysis and model selection allowing for heterogeneity and behavioural effects. Journal of Agricultural, Biological, and Environmental Statistics 10, 35–49.CrossRefGoogle Scholar
 Holzmann, H., Munk, A. and Zucchini, W. (2006) On identifiability in capturerecapture models. Biometrics 62, 934–939.MathSciNetCrossRefGoogle Scholar
 Huggins, R.M. (1989) On the statistical analysis of capture experiments. Biometrika 76, 133–140.MathSciNetCrossRefzbMATHGoogle Scholar
 Huggins, R.M. (1991) Some practical aspects of a conditional likelihood approach to capture experiments. Biometrics 47, 725–732.CrossRefGoogle Scholar
 Hwang, W.H. and Huggins, R. (2005) An examination of the effect of heterogeneity on the estimation of population size using capturerecapture data. Biometrika 92, 229–233.MathSciNetCrossRefzbMATHGoogle Scholar
 Jolly, G.M. (1965) Explicit estimates from capturerecapture data with both death and immigrationstochastic model. Biometrika 52, 225–247.MathSciNetCrossRefzbMATHGoogle Scholar
 King, R. and Brooks, S.P. (2003) Closed form likelihoods for ArnasonSchwarz models. Biometrika 90, 435–444.MathSciNetCrossRefzbMATHGoogle Scholar
 King, R. and Brooks, S.P. (2008) On the Bayesian estimation of a closed population size in the presence of heterogeneity and model uncertainty. Biometrics 64, 816–824.MathSciNetCrossRefzbMATHGoogle Scholar
 King, R. and Langrock, R. (2016) SemiMarkov ArnasonSchwarz models. Biometrics 72, 619–628.MathSciNetCrossRefzbMATHGoogle Scholar
 King, R. and McCrea, R.S. (2014) A generalised likelihood framework for partially observed capturerecapturerecovery models. Statistical Methodology 17, 30–45.MathSciNetCrossRefGoogle Scholar
 King, R. and McCrea, R.S. (2019) Capturerecapture: Methods and Models. Handbook of Statistics 40, Elsevier – in press.Google Scholar
 Langrock, R. and King, R. (2013) Maximum likelihood estimation of markrecapturerecovery models in the presence of continuous covariates. Annals of Applied Statistics 7, 1709–1732.MathSciNetCrossRefzbMATHGoogle Scholar
 Lebreton, J.D., Almeras, T. and Pradel, R. (2009) Competing events, mixtures of information and multistratum recapture models. Bird Study 46, S39–S46.CrossRefGoogle Scholar
 Lewis, B. (2012) An evaluation of mitigation actions for great crested newts at development sites. PhD Thesis, University of Kent.Google Scholar
 Link, W.A. (2003) Nonidentifiability of population size from capturerecapture data with heterogeneous detection probabilities. Biometrics 59, 1123–1130.MathSciNetCrossRefzbMATHGoogle Scholar
 Link, W.A. (2006) Rejoinder to “On identifiability in capturerecapture models”. Biometrics 62, 936–939.MathSciNetCrossRefGoogle Scholar
 McCrea, R.S. and Morgan, B.J.T. (2014) Analysis of CaptureRecapture Data. Chapman and Hall, CRC Press.CrossRefzbMATHGoogle Scholar
 Morgan, B.J.T. and Ridout, M.S. (2008) A new mixture model for capture heterogeneity. Journal of the Royal Statistical Society: Series C 57, 433–446.MathSciNetCrossRefzbMATHGoogle Scholar
 Norris J.L. and Pollock K.H. (1996) Nonparametric MLE under two closed capturerecapture models with heterogeneity. Biometrics 52, 639–649.CrossRefzbMATHGoogle Scholar
 Otis, D.L., Burnham, K.P., White, G.C. and Anderson, D.R. (1978) Statistical inference from capture data on closed animal populations. Wildlife Monographs 62, 3–135.zbMATHGoogle Scholar
 Pledger, S. (2000) Unified maximum likelihood estimates for closed capturerecapture models using mixtures. Biometrics 56, 434–442.CrossRefzbMATHGoogle Scholar
 Pledger, S. (2005) The performance of mixture models in heterogeneous closed population capturerecapture. Biometrics 61, 868–873.MathSciNetCrossRefGoogle Scholar
 Pollock, K.H. (1974) The assumption of equal catchability of animals in tagrecapture experiments. PhD thesis, Cornell University, Ithaca, New York.Google Scholar
 Pollock, K.H. and Otto, M.C. (1983) Robust estimation of population size in closed animal populations from capturerecapture experiments. Biometrics 39, 1035–1049.CrossRefzbMATHGoogle Scholar
 Pradel, R. (2005) Multievent: An extension of multistate capturerecapture models to uncertain states. Biometrics 61, 442–447.MathSciNetCrossRefzbMATHGoogle Scholar
 Sanathanan, L. (1972) Estimating the size of a multinomial population. Annals of Mathematical Statistics 43, 142–152.MathSciNetCrossRefzbMATHGoogle Scholar
 Schnabel, Z.E. (1938) The estimation of the total fish population of a lake. American Mathematical Monthly 45, 348–352.MathSciNetzbMATHGoogle Scholar
 Schwarz, C.G. and Ganter B. (1995) Estimating the movement among staging areas of the barnacle goose (Branta leucopsis). Journal of Applied Statistics 22, 711–724.CrossRefGoogle Scholar
 Schwarz, C.G., Schweigert, J.F. and Arnason, A.N. (1993) Estimating migration rates using tag recovery data. Biometrics 59, 291–318.Google Scholar
 Seber, G.A.F. (1965) A note on the multiplerecapture census. Biometrics 52, 249–259.MathSciNetCrossRefzbMATHGoogle Scholar
 Seber, G.A.F. (1982) The Estimation of Animal Abundance and Related Parameters. New York: MacMillan.zbMATHGoogle Scholar
 Smith, P.J. (1988) Bayesian methods for multiple capturerecapture surveys. Biometrics 44, 1177–1189.CrossRefzbMATHGoogle Scholar
 Smith, P.J. (1991) Bayesian analyses for a multiple capturerecapture model. Biometrika 78, 399–407.MathSciNetCrossRefGoogle Scholar
 Stoklosa, J., Dann, P. and Huggins, R. (2012) Inference on partially observed quasistationary Markov chains with applications to multistate population models. Journal of Agricultural, Biological and Environmental Statistics 17, 52–67.MathSciNetCrossRefzbMATHGoogle Scholar
Copyright information
OpenAccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.