Semiparametric inference for open populations using the Jolly–Seber model: a penalized spline approach

When there are frequent capture occasions, both semiparametric and nonparametric estimators for the size of an open population have been proposed using kernel smoothing methods. While kernel smoothing methods are mathematically tractable, fitting them to data is computationally intensive. Here, we use smoothing splines in the form of P-splines to provide an alternate less computationally intensive method of fitting these models to capture–recapture data from open populations with frequent capture occasions. We fit the model to capture data collected over 64 occasions and model the population size as a function of time, seasonal effects and an environmental covariate. A small simulation study is also conducted to examine the performance of the estimators and their standard errors.     

On the estimation of homogeneous population size from a complex dual-record system

ABSTRACT

A dual-record system (DRS) (equivalently two sample capture–recapture experiments) model, with time and behavioural response variation, has attracted much attention specifically in the domain of official statistics and epidemiology, as the assumption of list independence often fails. The relevant model suffers from parameter identifiability problem, and suitable Bayesian methodologies could be helpful. In this article, we formulate population size estimation in DRS as a missing data problem and two empirical Bayes approaches are proposed along with the discussion of an existing Bayes treatment. Some features and associated posterior convergence for these methods are mentioned. Investigation through an extensive simulation study finds that our proposed approaches compare favourably with the existing Bayes approach for this complex model depending upon the availability of directional nature of underlying behavioural response effect. A real-data example is given to illustrate these methods.     

Estimation in capture–recapture models when covariates are subject to measurement errors and missing data

For capture–recapture models when covariates are subject to measurement errors and missing data, a set of estimating equations is constructed to estimate population size and relevant parameters. These estimating equations can be solved by an algorithm similar to the EM algorithm. The proposed method is also applicable to the situation when covariates with no measurement errors have missing data. Simulation studies are used to assess the performance of the proposed estimator. The estimator is also applied to a capture–recapture experiment on the bird species Prinia flaviventris in Hong Kong. The Canadian Journal of Statistics 37: 645–658; 2009 © 2009 Statistical Society of Canada     

APPLICATION OF SEMIPARAMETRIC REGRESSION MODELS IN THE ANALYSIS OF CAPTURE-RECAPTURE EXPERIMENTS

If the capture probabilities in a capture‐recapture experiment depend on covariates, parametric models may be fitted and the population size may then be estimated. Here a semiparametric model for the capture probabilities that allows both continuous and categorical covariates is developed. Kernel smoothing and profile estimating equations are used to estimate the nonparametric and parametric components. Analytic forms of the standard errors are derived, which allows an empirical bias bandwidth selection procedure to be used to estimate the bandwidth. The method is evaluated in simulations and is applied to a real data set concerning captures of Prinia flaviventris, which is a common bird species in Southeast Asia.     

Choosing Sampling Effort for Recapture Experiments with Behavioural Response

One of the main aims of a recapture experiment is to estimate the unknown size, N of a closed population. Under the so-called behavioural model, individual capture probabilities change after the first capture. Unfortunately, the maximum likelihood estimator given by Zippin (1956) may give an infinite result and often has poor precision. Chaiyapong & Lloyd (1997) have given formulae for the asymptotic bias and variance as well as for the probability that the estimate is infinite.
The purpose of this article is to tabulate the inversions of the above cited formulae so that practitioners can plan the required capture effort. This paper develops simple approximations for the minimum capture effort required to achieve (i) no more than a certain probability of breakdown, (ii) a given relative standard error.     

Transformed Logit Confidence Intervals for Small Populations in Single Capture–Recapture Estimation

The good performance of logit confidence intervals for the odds ratio with small samples is well known. This is true unless the actual odds ratio is very large. In single capture–recapture estimation the odds ratio is equal to 1 because of the assumption of independence of the samples. Consequently, a transformation of the logit confidence intervals for the odds ratio is proposed in order to estimate the size of a closed population under single capture–recapture estimation. It is found that the transformed logit interval, after adding .5 to each observed count before computation, has actual coverage probabilities near to the nominal level even for small populations and even for capture probabilities near to 0 or 1, which is not guaranteed for the other capture–recapture confidence intervals proposed in statistical literature. Thus, given that the .5 transformed logit interval is very simple to compute and has a good performance, it is appropriate to be implemented by most users of the single capture–recapture method.     

On the integrated maximum likelihood estimators for a closed population capture–recapture model with unequal capture probabilities

Nuisance parameter elimination is a central problem in capture–recapture modelling. In this paper, we consider a closed population capture–recapture model which assumes the capture probabilities varies only with the sampling occasions. In this model, the capture probabilities are regarded as nuisance parameters and the unknown number of individuals is the parameter of interest. In order to eliminate the nuisance parameters, the likelihood function is integrated with respect to a weight function (uniform and Jeffrey's) of the nuisance parameters resulting in an integrated likelihood function depending only on the population size. For these integrated likelihood functions, analytical expressions for the maximum likelihood estimates are obtained and it is proved that they are always finite and unique. Variance estimates of the proposed estimators are obtained via a parametric bootstrap resampling procedure. The proposed methods are illustrated on a real data set and their frequentist properties are assessed by means of a simulation study.     

Approximate estimations in capture–recapture model with heteroscedastic measurement errors

In a capture–recapture experiment, the number of measurements for individual covariates usually equals the number of captures. This creates a heteroscedastic measurement error problem and the usual surrogate condition does not hold in the context of a measurement error model. This study adopts a small measurement error assumption to approximate the conventional estimating functions and the population size estimator. This study also investigates the biases of the resulting estimators. In addition, modifications for two common approximation methods, regression calibration and simulation extrapolation, to accommodate heteroscedastic measurement error are also discussed. These estimation methods are examined through simulations and illustrated by analysing a capture–recapture data set.     

A Semiparametric Model for Recapture Experiments

Abstract. A two-step procedure based on the conditional likelihood is proposed to estimate the population size of a closed population using a semiparametric model for recapture studies. An asymptotic variance estimate and numerical results are presented. The method is applied to a bird banding dataset in Hong Kong.     

Some properties of a nonparametric estimator of the size of an open population

It has been demonstrated in the literature that local polynomial models may be used to estimate the size of an open population using capture–recapture data. However, very little is known about their properties. Here we develop a setting in which the properties of nonparametric estimators of the size of an open population using capture–recapture data can be examined and establish conditions under which expressions for the bias and variance may be determined.     

ESTIMATING POPULATION SIZE FROM CAPTURE-RECAPTURE EXPERIMENTS IN CONTINUOUS TIME

Estimates for the size of a closed population are given for multiple recapture studies in continuous time. The estimates are derived by a method of moments for martingales. An estimate and associated standard error of the population size are derived for a homogeneous population when the capture rates are permitted to depend on time in an unspecified manner. Corresponding results are obtained when the capture rates vary among individuals as well. Explicit expressions are given for these estimates and standard errors which involve only simple computation.     

IMPROVED INFERENCE FROM RECAPTURE EXPERIMENTS WITH BEHAVIOURAL RESPONSE THROUGH MODELLING and AUXILIARY EXPERIMENTATION

Allowing for behavioural response in a recapture experiment involves a large reduction in the precision of estimating the unknown population size. Unless the number of individuals captured is very large, the model is of little use in practice. This paper studies the extent to which this efficiency loss can be reduced by modelling the behavioural response. The most dramatic improvements in efficiency are obtained by estimating the rate of the behavioural response from an intensive study of a small subset of the population. In many practical situations this may be a cost- and time-effective alternative to intensively sampling the entire population.     

Improved inference on capture recapture models with behavioural effects

In the context of capture-recapture modeling for estimating the unknown size of a finite population it is often required a flexible framework for dealing with a behavioural response to trapping. Many alternative settings have been proposed in the literature to account for the variation of capture probability at each occasion depending on the previous capture history. Inference is typically carried out relying on the so-called conditional likelihood approach. We highlight that such approach may, with positive probability, lead to inferential pathologies such as unbounded estimates for the finite size of the population. The occurrence of such likelihood failures is characterized within a very general class of behavioural effect models. It is also pointed out that a fully Bayesian analysis overcomes the likelihood failure phenomenon. The overall improved performance of alternative Bayesian estimators is investigated under different non-informative prior distributions verifying their comparative merits with both simulated and real data.     

A generalised likelihood framework for partially observed capture–recapture–recovery models

We provide a closed form likelihood expression for multi-state capture–recapture–recovery data when the state of an individual may be only partially observed. The corresponding sufficient statistics are presented in addition to a matrix formulation which facilitates an efficient calculation of the likelihood. This likelihood framework provides a consistent and unified framework with many standard models applied to capture–recapture–recovery data as special cases.     

From distance sampling to spatial capture–recapture

Distance sampling and capture–recapture are the two most widely used wildlife abundance estimation methods. capture–recapture methods have only recently incorporated models for spatial distribution and there is an increasing tendency for distance sampling methods to incorporated spatial models rather than to rely on partly design-based spatial inference. In this overview we show how spatial models are central to modern distance sampling and that spatial capture–recapture models arise as an extension of distance sampling methods. Depending on the type of data recorded, they can be viewed as particular kinds of hierarchical binary regression, Poisson regression, survival or time-to-event models, with individuals’ locations as latent variables and a spatial model as the latent variable distribution. Incorporation of spatial models in these two methods provides new opportunities for drawing explicitly spatial inferences. Areas of likely future development include more sophisticated spatial and spatio-temporal modelling of individuals’ locations and movements, new methods for integrating spatial capture–recapture and other kinds of ecological survey data, and methods for dealing with the recapture uncertainty that often arise when “capture” consists of detection by a remote device like a camera trap or microphone.     

Proportional trapping-removal models with contaminated data

A continuous time proportional trapping-removal model for the estimation of the size of an animal population with consideration of disturbance from non-target animal is studied. Maximum likelihood estimates and corresponding standard errors for the targeted population are derived. Their large sample properties can be obtained using martingale limit theory. Simulations are conducted, and a comparison is done when ignoring the disturbance in the model. An example on a small mammal capture–recapture study on deer mouse is presented.     

A case study of green tree frog population size estimation by repeated capture–mark–recapture method with individual tagging: a parametric bootstrap method vs Jolly–Seber method

This paper deals with estimation of a green tree frog population in an urban setting using repeated capture–mark–recapture (CMR) method over several weeks with an individual tagging system which gives rise to a complicated generalization of the hypergeometric distribution. Based on the maximum likelihood estimation, a parametric bootstrap approach is adopted to obtain interval estimates of the weekly population size which is the main objective of our work. The method is computation-based; and programming intensive to implement the algorithm for re-sampling. This method can be applied to estimate the population size of any species based on repeated CMR method at multiple time points. Further, it has been pointed out that the well-known Jolly–Seber method, which is based on some strong assumptions, produces either unrealistic estimates, or may have situations where its assumptions are not valid for our observed data set.     

Uncertainty estimation in heterogeneous capture–recapture count data

The Conway–Maxwell–Poisson estimator is considered in this paper as the population size estimator. The benefit of using the Conway–Maxwell–Poisson distribution is that it includes the Bernoulli, the Geometric and the Poisson distributions as special cases and, furthermore, allows for heterogeneity. Little emphasis is often placed on the variability associated with the population size estimate. This paper provides a deep and extensive comparison of bootstrap methods in the capture–recapture setting. It deals with the classical bootstrap approach using the true population size, the true bootstrap, and the classical bootstrap using the observed sample size, the reduced bootstrap. Furthermore, the imputed bootstrap, as well as approximating forms in terms of standard errors and confidence intervals for the population size, under the Conway–Maxwell–Poisson distribution, have been investigated and discussed. These methods are illustrated in a simulation study and in benchmark real data examples.     

Parametric regression models for continuous time removal and recapture studies

We use a class of parametric counting process regression models that are commonly employed in the analysis of failure time data to formulate the subject-specific capture probabilities for removal and recapture studies conducted in continuous time. We estimate the regression parameters by modifying the conventional likelihood score function for left-truncated and right-censored data to accommodate an unknown population size and missing covariates on uncaptured subjects, and we subsequently estimate the population size by a martingale-based estimating function. The resultant estimators for the regression parameters and population size are consistent and asymptotically normal under appropriate regularity conditions. We assess the small sample properties of the proposed estimators through Monte Carlo simulation and we present an application to a bird banding exercise.     

A new mixture model for capture heterogeneity

Summary.  We propose a mixture of binomial and beta–binomial distributions for estimating the size of closed populations. The new mixture model is applied to several real capture–recapture data sets and is shown to provide a convenient, objective framework for model selection. The new model is compared with three alternative models in a simulation study, and the results shed light on the general performance of models in this area. The new model provides a robust flexible analysis, which automatically deals with small capture probabilities.