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.     

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.     

Productivity indices and survival rate estimates from MAPS,a continent-wide programme of constant-effort mist-netting in North America

The Monitoring Avian Productivity and Survivorship (MAPS) programme is a cooperative effort to provide annual regional indices of adult population size and post-fledging productivity and estimates of adult survival rates from data pooled from a network of constant-effort mist-netting stations across North America. This paper provides an overview of the field and analytical methods currently employed by MAPS, a discussion of the assumptions underlying the use of these techniques, and a discussion of the validity of some of these assumptions based on data gathered during the first 5 years (1989-1993) of the programme, during which time it grew from 17 to 227 stations. Ageand species-specific differences in dispersal characteristics are important factors affecting the usefulness of the indices of adult population size and productivity derived from MAPS data. The presence of transients, heterogeneous capture probabilities among stations, and the large sample sizes required by models to deal effectively with these two considerations are important factors affecting the accuracy and precision of survival rate estimates derived from MAPS data. Important results from the first 5 years of MAPS are: (1) indices of adult population size derived from MAPS mist-netting data correlated well with analogous indices derived from point-count data collected at MAPS stations; (2) annual changes in productivity indices generated by MAPS were similar to analogous changes documented by direct nest monitoring and were generally as expected when compared to annual changes in weather during the breeding season; and (3) a model using between-year recaptures in Cormack-Jolly-Seber (CJS) mark-recapture analyses to estimate the proportion of residents among unmarked birds was found, for most tropical-wintering species sampled, to provide a better fit with the available data and more realistic and precise estimates of annual survival rates of resident birds than did standard CJS mark-recapture analyses. A detailed review of the statistical characteristics of MAPS data and a thorough evaluation of the field and analytical methods used in the MAPS programme are currently under way.     

An adaptive rule for stopping a discovery process under time censorship

Consider a finite population of large but unknown size of hidden objects. Consider searching for these objects for a period of time, at a certain cost, and receiving a reward depending on the sizes of the objects found. Suppose that the size and discovery time of the objects both have unknown distributions, but the conditional distribution of time given size is exponential with an unknown non-negative and non-decreasing function of the size as failure rate. The goal is to find an optimal way to stop the discovery process. Assuming that the above parameters are known, an optimal stopping time is derived and its asymptotic properties are studied. Then, an adaptive rule based on order restricted estimates of the distributions from truncated data is presented. This adaptive rule is shown to perform nearly as well as the optimal stopping time for large population size.     

Nonparametric Estimation of the Number of Drug Users in Hong Kong Using Repeated Multiple Lists

We update a previous approach to the estimation of the size of an open population when there are multiple lists at each time point. Our motivation is 35 years of longitudinal data on the detection of drug users by the Central Registry of Drug Abuse in Hong Kong. We develop a two‐stage smoothing spline approach. This gives a flexible and easily implemented alternative to the previous method which was based on kernel smoothing. The new method retains the property of reducing the variability of the individual estimates at each time point. We evaluate the new method by means of a simulation study that includes an examination of the effects of variable selection. The new method is then applied to data collected by the Central Registry of Drug Abuse. The parameter estimates obtained are compared with the well known Jolly–Seber estimates based on single capture methods.     

A SEMIPARAMETRIC MODEL FOR A FUNCTIONAL BEHAVIOURAL RESPONSE TO CAPTURE IN CAPTURE–RECAPTURE EXPERIMENTS

Capture–recapture experiments are commonly used to estimate the size of a closed population. However, the associated estimators of the population size are well known to be highly sensitive to misspecification of the capture probabilities. To address this, we present a general semiparametric framework for the analysis of capture–recapture experiments when the capture probability depends on individual characteristics, time effects and behavioural response. This generalizes well‐known general parametric capture–recapture models and extends previous semiparametric models in which there is no time dependence or behavioural response. The method is evaluated in simulations and applied to two real data sets.     

A BETA-BINOMIAL MODEL FOR ESTIMATING THE SIZE OF A HETEROGENEOUS POPULATION

This paper compares the properties of various estimators for a beta‐binomial model for estimating the size of a heterogeneous population. It is found that maximum likelihood and conditional maximum likelihood estimators perform well for a large population with a large capture proportion. The jackknife and the sample coverage estimators are biased for low capture probabilities. The performance of the martingale estimator is satisfactory, but it requires full capture histories. The Gibbs sampler and Metropolis‐Hastings algorithm provide reasonable posterior estimates for informative priors.     

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.     

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.     

On the Bayesian analysis of the mixture of power function distribution using the complete and the censored sample

The power function distribution is often used to study the electrical component reliability. In this paper, we model a heterogeneous population using the two-component mixture of the power function distribution. A comprehensive simulation scheme including a large number of parameter points is followed to highlight the properties and behavior of the estimates in terms of sample size, censoring rate, parameters size and the proportion of the components of the mixture. The parameters of the power function mixture are estimated and compared using the Bayes estimates. A simulated mixture data with censored observations is generated by probabilistic mixing for the computational purposes. Elegant closed form expressions for the Bayes estimators and their variances are derived for the censored sample as well as for the complete sample. Some interesting comparison and properties of the estimates are observed and presented. The system of three non-linear equations, required to be solved iteratively for the computations of maximum likelihood (ML) estimates, is derived. The complete sample expressions for the ML estimates and for their variances are also given. The components of the information matrix are constructed as well. Uninformative as well as informative priors are assumed for the derivation of the Bayes estimators. A real-life mixture data example has also been discussed. The posterior predictive distribution with the informative Gamma prior is derived, and the equations required to find the lower and upper limits of the predictive intervals are constructed. The Bayes estimates are evaluated under the squared error loss function.     

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 Note on the Inverse Birthday Problem With Applications

The classical birthday problem considers the probability that at least two people in a group of size N share the same birthday. The inverse birthday problem considers the estimation of the size N of a group given the number of different birthdays in the group. In practice, this problem is analogous to estimating the size of a population from occurrence data only. The inverse problem can be solved via two simple approaches including the method of moments for a multinominal model and the maximum likelihood estimate of a Poisson model, which we present in this study. We investigate properties of both methods and show that they can yield asymptotically equivalent Wald-type interval estimators. Moreover, we show that these methods estimate a lower bound for the population size when birth rates are nonhomogenous or individuals in the population are aggregated. A simulation study was conducted to evaluate the performance of the point estimates arising from the two approaches and to compare the performance of seven interval estimators, including likelihood ratio and log-transformation methods. We illustrate the utility of these methods by estimating: (1) the abundance of tree species over a 50-hectare forest plot, (2) the number of Chlamydia infections when only the number of different birthdays of the patients is known, and (3) the number of rainy days when the number of rainy weeks is known. Supplementary materials for this article are available online.     

Bayesian population estimation for small sample capture-recapture data using noninformative priors

One critical issue in the Bayesian approach is choosing the priors when there is not enough prior information to specify hyperparameters. Several improper noninformative priors for capture-recapture models were proposed in the literature. It is known that the Bayesian estimate can be sensitive to the choice of priors, especially when sample size is small to moderate. Yet, how to choose a noninformative prior for a given model remains a question. In this paper, as the first step, we consider the problem of estimating the population size for _Mt$M_{\mathrm{t}}$ model using noninformative priors. The _Mt$M_{\mathrm{t}}$ model has prodigious application in wildlife management, ecology, software liability, epidemiological study, census under-count, and other research areas. Four commonly used noninformative priors are considered. We find that the choice of noninformative priors depends on the number of sampling occasions only. The guidelines on the choice of noninformative priors are provided based on the simulation results. Propriety of applying improper noninformative prior is discussed. Simulation studies are developed to inspect the frequentist performance of Bayesian point and interval estimates with different noninformative priors under various population sizes, capture probabilities, and the number of sampling occasions. The simulation results show that the Bayesian approach can provide more accurate estimates of the population size than the MLE for small samples. Two real-data examples are given to illustrate the method.     

On the use of secondary capture-recapture samples to estimate temporary emigration and breeding proportions

The use of the Cormack-Jolly-Seber model under a standard sampling scheme of one sample per time period, when the Jolly-Seber assumption that all emigration is permanent does not hold, leads to the confounding of temporary emigration probabilities with capture probabilities. This biases the estimates of capture probability when temporary emigration is a completely random process, and both capture and survival probabilities when there is a temporary trap response in temporary emigration, or it is Markovian. The use of secondary capture samples over a shorter interval within each period, during which the population is assumed to be closed (Pollock's robust design), provides a second source of information on capture probabilities. This solves the confounding problem, and thus temporary emigration probabilities can be estimated. This process can be accomplished in an ad hoc fashion for completely random temporary emigration and to some extent in the temporary trap response case, but modelling the complete sampling process provides more flexibility and permits direct estimation of variances. For the case of Markovian temporary emigration, a full likelihood is required.     

On the use of secondary capture-recapture samples to estimate temporary emigration and breeding proportions

The use of the Cormack-Jolly-Seber model under a standard sampling scheme of one sample per time period, when the Jolly-Seber assumption that all emigration is permanent does not hold, leads to the confounding of temporary emigration probabilities with capture probabilities. This biases the estimates of capture probability when temporary emigration is a completely random process, and both capture and survival probabilities when there is a temporary trap response in temporary emigration, or it is Markovian. The use of secondary capture samples over a shorter interval within each period, during which the population is assumed to be closed (Pollock's robust design), provides a second source of information on capture probabilities. This solves the confounding problem, and thus temporary emigration probabilities can be estimated. This process can be accomplished in an ad hoc fashion for completely random temporary emigration and to some extent in the temporary trap response case, but modelling the complete sampling process provides more flexibility and permits direct estimation of variances. For the case of Markovian temporary emigration, a full likelihood is required.     

COVARIANCES FOR FIXED INTERVAL SMOOTHED KALMAN FILTER PARAMETER ESTIMATES

Given time series data for fixed interval t= 1,2,…, M with non-autocorrelated innovations, the regression formulae for the best linear unbiased parameter estimates at each time t are given by the Kalman filter fixed interval smoothing equations. Formulae for the variance of such parameter estimates are well documented. However, formulae for covariance between these fixed interval best linear parameter estimates have previously been derived only for lag one. In this paper more general formulae for covariance between fixed interval best linear unbiased estimates at times t and t - l are derived for t= 1,2,…, M and l= 0,1,…, t - 1. Under Gaussian assumptions, these formulae are also those for the corresponding conditional covariances between the fixed interval best linear unbiased parameter estimates given the data to time M. They have application, for example, in determination via the expectation-maximisation (EM) algorithm of exact maximum likelihood parameter estimates for ARMA processes expressed in statespace form when multiple observations are available at each time point.     

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.     

Bayesian analysis of a multiple-recapture model

In the Bayesian analysis of a multiple-recapture census, different diffuse prior distributions can lead to markedly different inferences about the population size N. Through consideration of the Fisher information matrix it is shown that the number of captures in each sample typically provides little information about N. This suggests that if there is no prior information about capture probabilities, then knowledge of just the sample sizes and not the number of recaptures should leave the distribution of Nunchanged. A prior model that has this property is identified and the posterior distribution is examined. In particular, asymptotic estimates of the posterior mean and variance are derived. Differences between Bayesian and classical point and interval estimators are illustrated through examples.     

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.