Prior distributions for stratified capture-recapture models

We consider the Arnason-Schwarz model, usually used to estimate survival and movement probabilities from capture-recapture data. A missing data structure of this model is constructed which allows a clear separation of information relative to capture and relative to movement. Extensions of the Arnason-Schwarz model are considered. For example, we consider a model that takes into account both the individual migration history and the individual reproduction history. Biological assumptions of these extensions are summarized via a directed graph. Owing to missing data, the posterior distribution of parameters is numerically intractable. To overcome those computational difficulties we advocate a Gibbs sampling algorithm that takes advantage of the missing data structure inherent in capture-recapture models. Prior information on survival, capture and movement probabilities typically consists of a prior mean and of a prior 95% credible confidence interval. Dirichlet distributions are used to incorporate some prior information on capture, survival probabilities, and movement probabilities. Finally, the influence of the prior on the Bayesian estimates of movement probabilities is examined.     

Bias in transition-specific survival and movement probabilities estimated using capture-recapture data

Transition probabilities can be estimated when capture-recapture data are available from each stratum on every capture occasion using a conditional likelihood approach with the Arnason-Schwarz model. To decompose the fundamental transition probabilities into derived parameters, all movement probabilities must sum to 1 and all individuals in stratum r at time i must have the same probability of survival regardless of which stratum the individual is in at time i + 1. If movement occurs among strata at the end of a sampling interval, survival rates of individuals from the same stratum are likely to be equal. However, if movement occurs between sampling periods and survival rates of individuals from the same stratum are not the same, estimates of stratum survival can be confounded with estimates of movement causing both estimates to be biased. Monte Carlo simulations were made of a three-sample model for a population with two strata using SURVIV. When differences were created in transition-specific survival rates for survival rates from the same stratum, relative bias was <2% in estimates of stratum survival and capture rates but relative bias in movement rates was much higher and varied. The magnitude of the relative bias in the movement estimate depended on the relative difference between the transition-specific survival rates and the corresponding stratum survival rate. The direction of the bias in movement rate estimates was opposite to the direction of this difference. Increases in relative bias due to increasing heterogeneity in probabilities of survival, movement and capture were small except when survival and capture probabilities were positively correlated within individuals.     

Selection among open population capture-recapture models when capture probabilities are heterogeneous

Selection of a parsimonious model as a basis for statistical inference from capture-recapture data is critical, especially when using open models in the analysis of multiple, interrelated data sets (e.g. males and females, with two to three age classes, over three to five areas and 10-15 years). The global (i.e. most general) model for such data sets might contain hundreds of survival and recapture parameters. Here, we focus on a series of nested models of the Cormack-Jolly-Seber type wherein the likelihood arises from products of multinomial distributions whose cell probabilities are reparameterized in terms of survival ( phi ) and mean capture ( p ) probabilities. This paper presents numerical results on two information-theoretic methods for model selection when the capture probabilities are heterogeneous over individual animals: Akaike's Information Criterion (AIC) and a dimension-consistent criterion (CAIC), derived from a Bayesian viewpoint. Quality of model selection was evaluated based on the relative Euclidian distance between standardized theta and theta (parameter theta is vector-valued and contains the survival ( phi ) and mean capture ( p ) probabilities); this quantity (RSS = sigma{(theta i - theta i )/ theta i } 2 ) is a sum of squared bias and variance. Thus, the quality of inference (RSS) was judged by comparing the performance of the two information criteria and the use of the true model (used to generate the data), in relation to the model that provided the smallest RSS. We found that heterogeneity in the capture probabilities had a negligible effect on model selection using AIC or CAIC. Model size increased as sample size increased with both AIC- and CAIC-selected models.     

Abundance estimation with a categorical covariate subject to missing in continuous-time capture-recapture studies

Abstract

In continuous-time capture-recapture experiments, individual heterogeneity has a large effect on the capture probability. To account for the heterogeneity, we consider an individual covariate, which is categorical and subject to missing. In this article, we develop a general model to summarize three kinds of missing mechanisms, and propose a maximum likelihood estimator of the abundance. A likelihood ratio confidence interval of the abundance is also proposed. We illustrate the proposed methods by simulation studies and a real data example of a bird species prinia subflava in Hong Kong.     

The use of multi-state capture-recapture models to address questions in evolutionary ecology

Multi-state capture-recapture models can be used to estimate survival rates in populations that are stratified by location or by state variables associated with individual animals. In populations stratified by location, movement probabilities can be estimated and used to test hypotheses relevant to population genetics and evolutionary ecology. When the interest is in state variables, these models permit estimation and testing of hypotheses about state-specific survival probabilities. If the state variable of interest is reproductive activity or success, then the multi-state modeling approach can be used to test hypotheses about life history trade-offs and a possible cost of reproduction.     

Multistate recapture models: Modelling incomplete individual histories

Multistate capture-recapture models are a natural generalization of the usual one-site recapture models. Similarly, individuals are sampled on discrete occasions, at which they may be captured or not. However, contrary to the one-site case, the individuals can move within a finite set of states between occasions. The growing interest in spatial aspects of population dynamics presently contributes to making multistate models a very promising tool for population biology. We review first the interest and the potential of multistate models, in particular when they are used with individual states as well as geographical sites. Multistate models indeed constitute canonical capture-recapture models for individual categorical covariates changing over time, and can be linked to longitudinal studies with missing data and models such as hidden Markov chains. Multistate models also provide a promising tool for handling heterogeneity of capture, provided states related to capturability can be defined and used. Such an approach could be relevant for population size estimation in closed populations. Multistate models also constitute a natural framework for mixtures of information in individual history data. Presently, most models can be fit using program MARK. As an example, we present a canonical model for multisite accession to reproduction, which fully generalizes a classical one-site model. In the generalization proposed, one can estimate simultaneously age-dependent rates of accession to reproduction, natal and breeding dispersal. Finally, we discuss further generalizations - such as a multistate generalization of growth rate models and models for data where the state in which an individual is detected is known with uncertainty - and prospects for software development.     

Comparison of Akaike information criterion and consistent Akaike information criterion for model selection and statistical inference from capture-recapture studies

SUMMARY We compare properties of parameter estimators under Akaike information criterion (AIC) and 'consistent' AIC (CAIC) model selection in a nested sequence of open population capture-recapture models. These models consist of product multinomials, where the cell probabilities are parameterized in terms of survival ( ) and capture ( p ) i i probabilities for each time interval i . The sequence of models is derived from 'treatment' effects that might be (1) absent, model H ; (2) only acute, model H ; or (3) acute and 0 2 p chronic, lasting several time intervals, model H . Using a 35 factorial design, 1000 3 repetitions were simulated for each of 243 cases. The true number of parameters ranged from 7 to 42, and the sample size ranged from approximately 470 to 55 000 per case. We focus on the quality of the inference about the model parameters and model structure that results from the two selection criteria. We use achieved confidence interval coverage as an integrating metric to judge what constitutes a 'properly parsimonious' model, and contrast the performance of these two model selection criteria for a wide range of models, sample sizes, parameter values and study interval lengths. AIC selection resulted in models in which the parameters were estimated with relatively little bias. However, these models exhibited asymptotic sampling variances that were somewhat too small, and achieved confidence interval coverage that was somewhat below the nominal level. In contrast, CAIC-selected models were too simple, the parameter estimators were often substantially biased, the asymptotic sampling variances were substantially too small and the achieved coverage was often substantially below the nominal level. An example case illustrates a pattern: with 20 capture occasions, 300 previously unmarked animals are released at each occasion, and the survival and capture probabilities in the control group on each occasion were 0.9 and 0.8 respectively using model H . There was a strong acute treatment effect 3 on the first survival ( ) and first capture probability ( p ), and smaller, chronic effects 1 2 on the second and third survival probabilities ( and ) as well as on the second capture 2 3 probability ( p ); the sample size for each repetition was approximately 55 000. CAIC 3 selection led to a model with exactly these effects in only nine of the 1000 repetitions, compared with 467 times under AIC selection. Under CAIC selection, even the two acute effects were detected only 555 times, compared with 998 for AIC selection. AIC selection exhibited a balance between underfitted and overfitted models (270 versus 263), while CAIC tended strongly to select underfitted models. CAIC-selected models were overly parsimonious and poor as a basis for statistical inferences about important model parameters or structure. We recommend the use of the AIC and not the CAIC for analysis and inference from capture-recapture data sets.     

Separation of survival and movement rates in multi-state tag-return and capture-recapture models

There has been growing interest in the estimation of transition probabilities among stages (Hestbeck et al. , 1991; Brownie et al. , 1993; Schwarz et al. , 1993) in tag-return and capture-recapture models. This has been driven by the increasing interest in meta-population models in ecology and the need for parameter estimates to use in these models. These transition probabilities are composed of survival and movement rates, which can only be estimated separately when an additional assumption is made (Brownie et al. , 1993). Brownie et al. (1993) assumed that movement occurs at the end of the interval between time i and i + 1. We generalize this work to allow different movement patterns in the interval for multiple tag-recovery and capture-recapture experiments. The time of movement is a random variable with a known distribution. The model formulations can be viewed as matrix extensions to the model formulations of single open population capturerecapture and tag-recovery experiments (Jolly, 1965; Seber, 1965; Brownie et al. , 1985). We also present the results of a small simulation study for the tag-return model when movement time follows a beta distribution, and later another simulation study for the capture-recapture model when movement time follows a uniform distribution. The simulation studies use a modified program SURVIV (White, 1983). The Relative Standard Errors (RSEs) of estimates according to high and low movement rates are presented. We show there are strong correlations between movement and survival estimates in the case that the movement rate is high. We also show that estimators of movement rates to different areas and estimators of survival rates in different areas have substantial correlations.     

Multistate recapture models: modelling incomplete individual histories

Multistate capture-recapture models are a natural generalization of the usual one-site recapture models. Similarly, individuals are sampled on discrete occasions, at which they may be captured or not. However, contrary to the one-site case, the individuals can move within a finite set of states between occasions. The growing interest in spatial aspects of population dynamics presently contributes to making multistate models a very promising tool for population biology. We review first the interest and the potential of multistate models, in particular when they are used with individual states as well as geographical sites. Multistate models indeed constitute canonical capture-recapture models for individual categorical covariates changing over time, and can be linked to longitudinal studies with missing data and models such as hidden Markov chains. Multistate models also provide a promising tool for handling heterogeneity of capture, provided states related to capturability can be defined and used. Such an approach could be relevant for population size estimation in closed populations. Multistate models also constitute a natural framework for mixtures of information in individual history data. Presently, most models can be fit using program MARK. As an example, we present a canonical model for multisite accession to reproduction, which fully generalizes a classical one-site model. In the generalization proposed, one can estimate simultaneously age-dependent rates of accession to reproduction, natal and breeding dispersal. Finally, we discuss further generalizations - such as a multistate generalization of growth rate models and models for data where the state in which an individual is detected is known with uncertainty - and prospects for software development.     

Model selection with missing covariates for policy considerations in fox enclosures

Foxhound training enclosures are facilities where wild-trapped foxes are placed into large fenced areas for dog training purposes. Although the purpose of these facilities is to train dogs without harming foxes, dog-related mortality has been reported to be an issue in some enclosures. Using data from a fox enclosure in Virginia, we investigate factors that influence fox survival in these dog training facilities and propose a set of policies to improve fox survival. In particular, a Bayesian hierarchical model is formulated to compute fox survival probabilities based on a fox's time in the enclosure and the number of dogs allowed in the enclosure at one time. These calculations are complicated by missing information on the number of dogs in the enclosure for many days during the study. We elicit expert knowledge for a prior on the number of dogs to account for the uncertainty in the missing data. Reversible jump Markov Chain Monte Carlo is used for model selection in the presence of missing covariates. We then use our model to examine possible changes to foxhound training enclosure policy and what effect those changes may have on fox survival.     

The use of auxiliary variables in capture-recapture modelling: An overview

I review the use of auxiliary variables in capture-recapture models for estimation of demographic parameters (e.g. capture probability, population size, survival probability, and recruitment, emigration and immigration numbers). I focus on what has been done in current research and what still needs to be done. Typically in the literature, covariate modelling has made capture and survival probabilities functions of covariates, but there are good reasons also to make other parameters functions of covariates as well. The types of covariates considered include environmental covariates that may vary by occasion but are constant over animals, and individual animal covariates that are usually assumed constant over time. I also discuss the difficulties of using time-dependent individual animal covariates and some possible solutions. Covariates are usually assumed to be measured without error, and that may not be realistic. For closed populations, one approach to modelling heterogeneity in capture probabilities uses observable individual covariates and is thus related to the primary purpose of this paper. The now standard Huggins-Alho approach conditions on the captured animals and then uses a generalized Horvitz-Thompson estimator to estimate population size. This approach has the advantage of simplicity in that one does not have to specify a distribution for the covariates, and the disadvantage is that it does not use the full likelihood to estimate population size. Alternately one could specify a distribution for the covariates and implement a full likelihood approach to inference to estimate the capture function, the covariate probability distribution, and the population size. The general Jolly-Seber open model enables one to estimate capture probability, population sizes, survival rates, and birth numbers. Much of the focus on modelling covariates in program MARK has been for survival and capture probability in the Cormack-Jolly-Seber model and its generalizations (including tag-return models). These models condition on the number of animals marked and released. A related, but distinct, topic is radio telemetry survival modelling that typically uses a modified Kaplan-Meier method and Cox proportional hazards model for auxiliary variables. Recently there has been an emphasis on integration of recruitment in the likelihood, and research on how to implement covariate modelling for recruitment and perhaps population size is needed. The combined open and closed 'robust' design model can also benefit from covariate modelling and some important options have already been implemented into MARK. Many models are usually fitted to one data set. This has necessitated development of model selection criteria based on the AIC (Akaike Information Criteria) and the alternative of averaging over reasonable models. The special problems of estimating over-dispersion when covariates are included in the model and then adjusting for over-dispersion in model selection could benefit from further research.     

The use of auxiliary variables in capture-recapture modelling: an overview

I review the use of auxiliary variables in capture-recapture models for estimation of demographic parameters (e.g. capture probability, population size, survival probability, and recruitment, emigration and immigration numbers). I focus on what has been done in current research and what still needs to be done. Typically in the literature, covariate modelling has made capture and survival probabilities functions of covariates, but there are good reasons also to make other parameters functions of covariates as well. The types of covariates considered include environmental covariates that may vary by occasion but are constant over animals, and individual animal covariates that are usually assumed constant over time. I also discuss the difficulties of using time-dependent individual animal covariates and some possible solutions. Covariates are usually assumed to be measured without error, and that may not be realistic. For closed populations, one approach to modelling heterogeneity in capture probabilities uses observable individual covariates and is thus related to the primary purpose of this paper. The now standard Huggins-Alho approach conditions on the captured animals and then uses a generalized Horvitz-Thompson estimator to estimate population size. This approach has the advantage of simplicity in that one does not have to specify a distribution for the covariates, and the disadvantage is that it does not use the full likelihood to estimate population size. Alternately one could specify a distribution for the covariates and implement a full likelihood approach to inference to estimate the capture function, the covariate probability distribution, and the population size. The general Jolly-Seber open model enables one to estimate capture probability, population sizes, survival rates, and birth numbers. Much of the focus on modelling covariates in program MARK has been for survival and capture probability in the Cormack-Jolly-Seber model and its generalizations (including tag-return models). These models condition on the number of animals marked and released. A related, but distinct, topic is radio telemetry survival modelling that typically uses a modified Kaplan-Meier method and Cox proportional hazards model for auxiliary variables. Recently there has been an emphasis on integration of recruitment in the likelihood, and research on how to implement covariate modelling for recruitment and perhaps population size is needed. The combined open and closed 'robust' design model can also benefit from covariate modelling and some important options have already been implemented into MARK. Many models are usually fitted to one data set. This has necessitated development of model selection criteria based on the AIC (Akaike Information Criteria) and the alternative of averaging over reasonable models. The special problems of estimating over-dispersion when covariates are included in the model and then adjusting for over-dispersion in model selection could benefit from further research.     

A three-sample multiple-recapture approach to census population estimation with heterogeneous catchability

"A central assumption in the standard capture-recapture approach to the estimation of the size of a closed population is the homogeneity of the 'capture' probabilities. In this article we develop an approach that allows for varying susceptibility to capture through individual parameters using a variant of the Rasch model from psychological measurement situations. Our approach requires an additional recapture. In the context of census undercount estimation, this requirement amounts to the use of a second independent sample or alternative data source to be matched with census and Post-Enumeration Survey (PES) data.... We illustrate [our] models and their estimation using data from a 1988 dress-rehearsal study for the 1990 census conducted by the U.S. Bureau of the Census, which explored the use of administrative data as a supplement to the PES. The article includes a discussion of extensions and related models."     

Estimating turnover,movements and capture parameters of resting passerines in standardized capture-recapture studies

Jolly-Seber models A, B, D and 2 were used to investigate capture-recapture data. The standard Jolly-Seber model A (time-dependent survival phi and capture probability p ) fits capture-recapture data of migrating passerines. Captures from a long-term mist-netting study (Mettnau Peninsula, SW Germany) at a stop-over site were used to estimate stop-over length from survival rate between days and capture probability. For some data, model 2 could be used, indicating a termporary reduction in 'survival' rate. Application of models B and D gave poor results. The total number of birds stopping over, i.e. population size, was estimated from captures of 1-5 line transects of nets in the spatial trapping design. Behaviour, movements within the stop-over site, catchability and ecophysiological covariables such as moult, fat deposition and climatic parameters are likely to have strong influence on the estimation of capture parameters.     

On the estimation of the extreme value and normal distribution parameters based on progressive type-II hybrid-censored data

A progressive hybrid censoring scheme is a mixture of type-I and type-II progressive censoring schemes. In this paper, we mainly consider the analysis of progressive type-II hybrid-censored data when the lifetime distribution of the individual item is the normal and extreme value distributions. Since the maximum likelihood estimators (MLEs) of these parameters cannot be obtained in the closed form, we propose to use the expectation and maximization (EM) algorithm to compute the MLEs. Also, the Newton–Raphson method is used to estimate the model parameters. The asymptotic variance–covariance matrix of the MLEs under EM framework is obtained by Fisher information matrix using the missing information and asymptotic confidence intervals for the parameters are then constructed. This study will end up with comparing the two methods of estimation and the asymptotic confidence intervals of coverage probabilities corresponding to the missing information principle and the observed information matrix through a simulation study, illustrated examples and real data analysis.     

Capture-recapture and evolutionary ecology: A difficult wedding?

Most of the questions raised by evolutionary ecologists require the estimation of demographic parameters such as survival probabilities, breeding propensity, age at maturity, etc. These parameters are usually obtained by capturing, marking and recapturing or resighting (CMR) individuals during their lives. Because exhaustivity cannot be achieved in the field, it is necessary to have an estimate of sampling intensity (capture/resighting probabilities). Statisticians and bio-statisticians have recently developed a wide variety of models devoted to the analysis of capture-recapture data. For a number of reasons, these models are not widely used by evolutionary ecologists. This paper describes the problems that can be encountered when ignoring the measure of sampling intensity. The potential for applying CMR models to current questions in evolutionary ecology is reviewed, particularly with respect to measuring the cost of reproduction and trade-offs. Some future model developments are also sketched which are needed to meet fully the requirements of evolutionary ecologists.     

Inferential properties of an individual-based survival model using release-recapture data: Sample size,validity and power

A model for analyzing release-recapture data is presented that generalizes a previously existing individual covariate model to include multiple groups of animals. As in the previous model, the generalized version includes selection parameters that relate individual covariates to survival potential. Significance of the selection parameters was equivalent to significance of the individual covariates. Simulation studies were conducted to investigate three inferential properties with respect to the selection parameters: (1) sample size requirements, (2) validity of the likelihood ratio test (LRT) and (3) power of the LRT. When the survival and capture probabilities ranged from 0.5 to 1.0, a total sample size of 300 was necessary to achieve a power of 0.80 at a significance level of 0.1 when testing the significance of the selection parameters. However, only half that (a total of 150) was necessary for the distribution of the maximum likelihood estimators of the selection parameters to approximate their asymptotic distributions. In general, as the survival and capture probabilities decreased, the sample size requirements increased. The validity of the LRT for testing the significance of the selection parameters was confirmed because the LRT statistic was distributed as theoretically expected under the null hypothesis, i.e. like a chi 2 random variable. When the baseline survival model was fully parameterized with population and interval effects, the LRT was also valid in the presence of unaccounted for random variation. The power of the LRT for testing the selection parameters was unaffected by over-parameterization of the baseline survival and capture models. The simulation studies showed that for testing the significance of individual covariates to survival the LRT was remarkably robust to assumption violations.     

Inferential properties of an individual-based survival model using release-recapture data: sample size, validity and power

A model for analyzing release-recapture data is presented that generalizes a previously existing individual covariate model to include multiple groups of animals. As in the previous model, the generalized version includes selection parameters that relate individual covariates to survival potential. Significance of the selection parameters was equivalent to significance of the individual covariates. Simulation studies were conducted to investigate three inferential properties with respect to the selection parameters: (1) sample size requirements, (2) validity of the likelihood ratio test (LRT) and (3) power of the LRT. When the survival and capture probabilities ranged from 0.5 to 1.0, a total sample size of 300 was necessary to achieve a power of 0.80 at a significance level of 0.1 when testing the significance of the selection parameters. However, only half that (a total of 150) was necessary for the distribution of the maximum likelihood estimators of the selection parameters to approximate their asymptotic distributions. In general, as the survival and capture probabilities decreased, the sample size requirements increased. The validity of the LRT for testing the significance of the selection parameters was confirmed because the LRT statistic was distributed as theoretically expected under the null hypothesis, i.e. like a chi 2 random variable. When the baseline survival model was fully parameterized with population and interval effects, the LRT was also valid in the presence of unaccounted for random variation. The power of the LRT for testing the selection parameters was unaffected by over-parameterization of the baseline survival and capture models. The simulation studies showed that for testing the significance of individual covariates to survival the LRT was remarkably robust to assumption violations.     

Confidence Bands for Comparison of Transition Probabilities in a Markov Chain Model

We consider graphs, confidence procedures and tests that can be used to compare transition probabilities in a Markov chain model with intensities specified by a Cox proportional hazard model. Under assumptions of this model, the regression coefficients provide information about the relative risks of covariates in one–step transitions, however, they cannot in general be used to to assess whether or not the covariates have a beneficial or detrimental effect on the endpoint events. To alleviate this problem, we consider graphical tests based on confidence procedures for a generalized Q–Q plot and for the difference between transition probabilities. The procedures are illustrated using data of the International Bone Marrow Transplant Registry.     

Spike-and-slab type variable selection in the Cox proportional hazards model for high-dimensional features

In this paper, we develop a variable selection framework with the spike-and-slab prior distribution via the hazard function of the Cox model. Specifically, we consider the transformation of the score and information functions for the partial likelihood function evaluated at the given data from the parameter space into the space generated by the logarithm of the hazard ratio. Thereby, we reduce the nonlinear complexity of the estimation equation for the Cox model and allow the utilization of a wider variety of stable variable selection methods. Then, we use a stochastic variable search Gibbs sampling approach via the spike-and-slab prior distribution to obtain the sparsity structure of the covariates associated with the survival outcome. Additionally, we conduct numerical simulations to evaluate the finite-sample performance of our proposed method. Finally, we apply this novel framework on lung adenocarcinoma data to find important genes associated with decreased survival in subjects with the disease.