Continuous-time capture-recapture population estimation when capture probabilities vary over time

Closed capture-recapture (CR) estimators have been used extensively to estimate population size. Most closed CR approaches have been developed and evaluated for discrete-time models, but there has been little effort to evaluate their continuous-time counterparts. Continuous-time estimators — developed using maximum likelihood theory by Craig (1953) and Darroch (1958), and martingale theory by Becker (1984) — that allow capture probabilities to vary over time were evaluated using Monte Carlo simulation. Overall, the ML estimators had a smaller MSE. The estimators performed well when model assumptions were upheld, and were somewhat robust to heterogeneity in capture probabilities. However, the estimators were not robust to behavioural effects in the capture probabilities. Time lag effects (periods when animals might be unavailable for immediate recapture) on continuous-time estimates were also investigated and results indicated a positive bias which was greater for smaller populations. There was no gain in performance when using a continuous-time estimator versus a discrete-time estimator on the same simulated data. Usefulness of the continuous-time approach may be limited to study designs where animals are easier to sample using continuous-time methodology.     

A combination line transect and capture-recapture sampling model for multiple observers in aerial surveys

We present a robust sampling methodology to estimate population size using line transect and capture-recapture procedures for aerial surveys. Aerial surveys usually underestimate population density due to animals being missed. A combination of capture-recapture and line transect sampling methods with multiple observers allows violation of the assumption that all animals on the centreline are sighted from the air. We illustrate our method with an example of inanimate objects which shows evidence of failure of the assumption that all objects on the centreline have probability 1 of being detected. A simulation study is implemented to evaluate the performance of three variations of the Lincoln-Petersen estimator: the overall estimator, the stratified estimator, and the general stratified estimator based on the combined likelihood proposed in this paper. The stratified Lincoln-Petersen estimator based on the combined likelihood is found to be generally superior to the other estimators.     

On the choice of statistical models for estimating occurrence and extinction from animal surveys

In surveys of natural animal populations the number of animals that are present and available to be detected at a sample location is often low, resulting in few or no detections. Low detection frequencies are especially common in surveys of imperiled species; however, the choice of sampling method and protocol also may influence the size of the population that is vulnerable to detection. In these circumstances, probabilities of animal occurrence and extinction will generally be estimated more accurately if the models used in data analysis account for differences in abundance among sample locations and for the dependence between site-specific abundance and detection. Simulation experiments are used to illustrate conditions wherein these types of models can be expected to outperform alternative estimators of population site occupancy and extinction.     

Estimating Population Size of Elusive Animals with DNA from Hunter-Collected Feces: Four Methods for Brown Bears

Abstract:  Noninvasive genetic methods can be used to estimate animal abundances and offer several advantages over conventional methods. Few attempts have been made, however, to evaluate the accuracy and precision of the estimates. We compared four methods of estimating population size based on fecal sampling. Two methods used rarefaction indices and two were based on capture-mark-recapture (CMR) estimators, one combining genetic and field data. Volunteer hunters and others collected 1904 fecal samples over 2 consecutive years in a large area containing a well-studied population of brown bears ( Ursus arctos ). On our 49,000-km² study area in south-central Sweden, population size estimates ranged from 378 to 572 bears in 2001 and 273 to 433 bears in 2002, depending on the method of estimation used. The estimates from the best model in the program MARK appeared to be the most accurate, based on the minimum population size estimate from radio-marked bears in a subsection of our sampling area. In addition, MARK models included heterogeneity and temporal variation in detection probabilities, which appeared to be present in our samples. All methods, though, incorrectly suggested a biased sex ratio, probably because of sex differences in detection probabilities and low overall detection probabilities. The population size of elusive animals can be estimated reliably over large areas with noninvasive genetic methods, but we stress the importance of an adequate and well-distributed sampling effort. In cases of biased sampling, calibration with independent estimates may be necessary. We recommend that this noninvasive genetic approach, using the MARK models, be used in the future in areas where sufficient numbers of volunteers can be mobilized.     

Design based estimation for ranked set sampling in finite populations

In this paper, we consider design-based estimation using ranked set sampling (RSS) in finite populations. We first derive the first and second-order inclusion probabilities for an RSS design and present two Horvitz–Thompson type estimators using these inclusion probabilities. We also develop an alternate Hansen–Hurwitz type estimator and investigate its properties. In particular, we show that this alternate estimator always outperforms the usual Hansen–Hurwitz type estimator in the simple random sampling with replacement design with comparable sample size. We also develop formulae for ratio estimator for all three developed estimators. The theoretical results are augmented by numerical and simulation studies as well as a case study using a well known data set. These show that RSS design can yield a substantial improvement in efficiency over the usual simple random sampling design in finite populations.     

Modeling misidentification errors that result from use of genetic tags in capture–recapture studies

Misidentification of animals is potentially important when naturally existing features (natural tags) such as DNA fingerprints (genetic tags) are used to identify individual animals. For example, when misidentification leads to multiple identities being assigned to an animal, traditional estimators tend to overestimate population size. Accounting for misidentification in capture–recapture models requires detailed understanding of the mechanism. Using genetic tags as an example, we outline a framework for modeling the effect of misidentification in closed population studies when individual identification is based on natural tags that are consistent over time (non-evolving natural tags). We first assume a single sample is obtained per animal for each capture event, and then generalize to the case where multiple samples (such as hair or scat samples) are collected per animal per capture occasion. We introduce methods for estimating population size and, using a simulation study, we show that our new estimators perform well for cases with moderately high capture probabilities or high misidentification rates. In contrast, conventional estimators can seriously overestimate population size when errors due to misidentification are ignored.     

Effects of Multiple Levels of Social Organization on Survival and Abundance

Abstract: Identifying how social organization shapes individual behavior, survival, and fecundity of animals that live in groups can inform conservation efforts and improve forecasts of population abundance, even when the mechanism responsible for group‐level differences is unknown. We constructed a hierarchical Bayesian model to quantify the relative variability in survival rates among different levels of social organization (matrilines and pods) of an endangered population of killer whales (Orcinus orca). Individual killer whales often participate in group activities such as prey sharing and cooperative hunting. The estimated age‐specific survival probabilities and survivorship curves differed considerably among pods and to a lesser extent among matrilines (within pods). Across all pods, males had lower life expectancy than females. Differences in survival between pods may be caused by a combination of factors that vary across the population's range, including reduced prey availability, contaminants in prey, and human activity. Our modeling approach could be applied to demographic rates for other species and for parameters other than survival, including reproduction, prey selection, movement, and detection probabilities.     

Randomized graph sampling

Randomized graph sampling (RGS) is an approach for sampling populations associated with or describable as graphs, when the structure of the graph is known and the parameter of interest is the total weight of the graph. RGS is related to, but distinct from, other graph-based approaches such as snowball and network sampling. Graph elements are clustered into walks that reflect the structure of the graph, as well as operational constraints on sampling. The basic estimator in RGS can be constructed as a Horvitz-Thompson estimator. I prove it to be design-unbiased, and also show design-unbiasedness of an estimator of the sample variance when walks are sampled with replacement. Covariates can be employed for variance reduction either through improved assignment of selection probabilities to walks in the design step, or through the use of alternative estimators during analysis. The approach is illustrated with a trail maintenance example, which demonstrates that complicated approaches to assignment of selection probabilities can be counterproductive. I describe conditions under which RGS may be efficient in practice, and suggest possible applications.     

An estimating function approach to the inference of catch-effort models

A class of catch-effort models, which allows for heterogeneous removal probabilities, is proposed for closed populations. The model includes three types of removal probabilities: multiplicative, Poisson and logistic. The usual removal and generalized removal models then become special cases. The equivalence of the proposed model and a special type of capture-recapture model is discussed. A unified estimating function approach is used to estimate the initial population size. For the homogeneous model, the resulting population size estimator based on optimal estimating functions is asymptotically equivalent to the maximum likelihood estimator. One advantage for our approach is that it can be extended to handle the heterogeneous populations in which the maximum likelihood estimators do not exist. The bootstrap method is applied to construct variance estimators and confidence intervals. We illustrate the method by two real data examples. Results of a simulation study investigating the performance of the proposed estimation procedure are presented.     

Use of the superpopulation approach to estimate breeding population size: an example in asynchronously breeding birds

Many populations of animals are fluid in both space and time, making estimation of numbers difficult. Much attention has been devoted to estimation of bias in detection of animals that are present at the time of survey. However, an equally important problem is estimation of population size when all animals are not present on all survey occasions. Here, we showcase use of the superpopulation approach to capture-recapture modeling for estimating populations where group membership is asynchronous, and where considerable overlap in group membership among sampling occasions may occur. We estimate total population size of long-legged wading bird (Great Egret and White Ibis) breeding colonies from aerial observations of individually identifiable nests at various times in the nesting season. Initiation and termination of nests were analogous to entry and departure from a population. Estimates using the superpopulation approach were 47-382% larger than peak aerial counts of the same colonies. Our results indicate that the use of the superpopulation approach to model nesting asynchrony provides a considerably less biased and more efficient estimate of nesting activity than traditional methods. We suggest that this approach may also be used to derive population estimates in a variety of situations where group membership is fluid.     

Estimating the number of active and successful bald eagle nests:an application of the dual frame method

Monitoring the number of animals in a wildlife population is an important task. In this paper, we estimate the number of active and successful bald eagle nests in a specific region using dual frame sampling techniques. The dual frame method combines independent samples from an incomplete list frame and an area frame that is assumed to be complete. Hartley's screening estimator is used to combine sample information from both frames. This methodology will be useful for monitoring any wildlife population where the breeding individuals have highly visible nesting territories which tend to be stable over many years.     

Quantifying the probability of detection of wild ungulates with the Judas technique

The Judas technique is often used in control or eradication of particular vertebrate pests. The technique exploits the tendency of individuals to form social groups. A radio collar is affixed to an individual and its subsequent monitoring facilitates the detection of other conspecifics. Efficacy of this technique would be improved if managers could estimate the probability that a Judas individual would detect conspecifics. To calculate this probability, we estimated association rates of Judas individuals with other Judas individuals, given the length of time the Judas has been deployed. We developed a simple model of space-use for individual Judas animals and constrained detection probabilities to those specific areas. We then combined estimates for individual Judas animals to infer the probability that a wild individual could be detected in an area of interest via Judas surveillance. We illustrated the method by using data from a feral goat eradication program on Isla Santiago, Galápagos, and a feral pig eradication program on Santa Cruz Island, California. Association probabilities declined as the proximity between individual areas of use of a Judas pair decreased. Unconditional probabilities of detection within individual areas of use averaged 0.09 per month for feral pigs and 0.11 per month for feral goats. Probabilities that eradication had been achieved, given no detections of wild conspecifics, and an uninformative prior probability of eradication were 0.79 (90% CI 0.22–0.99) for feral goats and 0.87 (90% CI 0.44–1.0) for feral pigs. We envisage several additions to the analyses used that could improve estimates of Judas detection probability. Analyses such as these can help managers increase the efficacy of eradication efforts, leading to more effective effects to restore native biodiversity.     

Multiple data sources improve DNA-based mark-recapture population estimates of grizzly bears.

A fundamental challenge to estimating population size with mark-recapture methods is heterogeneous capture probabilities and subsequent bias of population estimates. Confronting this problem usually requires substantial sampling effort that can be difficult to achieve for some species, such as carnivores. We developed a methodology that uses two data sources to deal with heterogeneity and applied this to DNA mark-recapture data from grizzly bears (Ursus arctos). We improved population estimates by incorporating additional DNA "captures" of grizzly bears obtained by collecting hair from unbaited bear rub trees concurrently with baited, grid-based, hair snag sampling. We consider a Lincoln-Petersen estimator with hair snag captures as the initial session and rub tree captures as the recapture session and develop an estimator in program MARK that treats hair snag and rub tree samples as successive sessions. Using empirical data from a large-scale project in the greater Glacier National Park, Montana, USA, area and simulation modeling we evaluate these methods and compare the results to hair-snag-only estimates. Empirical results indicate that, compared with hair-snag-only data, the joint hair-snag-rub-tree methods produce similar but more precise estimates if capture and recapture rates are reasonably high for both methods. Simulation results suggest that estimators are potentially affected by correlation of capture probabilities between sample types in the presence of heterogeneity. Overall, closed population Huggins-Pledger estimators showed the highest precision and were most robust to sparse data, heterogeneity, and capture probability correlation among sampling types. Results also indicate that these estimators can be used when a segment of the population has zero capture probability for one of the methods. We propose that this general methodology may be useful for other species in which mark-recapture data are available from multiple sources.     

A robust design capture-recapture model with multiple age classes augmented with population assignment data

The relative contribution of in situ reproduction versus immigration to the recruitment process is important to ecologists. Here we consider a robust design superpopulation capture-recapture model for a population with two age classes augmented with population assignment data. We first use age information to estimate the entry probabilities of new animals originating via in situ reproduction and immigration separately for all except the first period. Then we combine age and population assignment information with the capture-recapture model, which enables us to estimate the entry probability of in situ births and the entry probability of immigrants separately for all sampling periods. Further, this augmentation of age specific capture-recapture data with population assignment data greatly improves the estimators’ precision. We apply our new model to a capture-recapture data set with genetic information for banner-tailed kangaroo rats in Southern Arizona. We find that many more individuals are born in situ than are immigrants for all time periods. Young animals have lower survival probabilities than adults born in situ. Adult animals born in situ have higher survival probabilities than adults that were immigrants.     

Comparison of distribution function estimators for forestry applications following variable probability sampling

Practical considerations often motivate employing variable probability sampling designs when estimating characteristics of forest populations. Three distribution function estimators, the Horvitz-Thompson estimator, a difference estimator, and a ratio estimator, are compared following variable probability sampling in which the inclusion probabilities are proportional to an auxiliary variable, X. Relative performance of the estimators is affected by several factors, including the distribution of the inclusion probabilities, the correlation () between X and the response Y, and the position along the distribution function being estimated. Both the ratio and difference estimators are superior to the Horvitz-Thompson estimator. The difference estimator gains better precision than the ratio estimator toward the upper portion of the distribution function, but the ratio estimator is superior toward the lower end of the distribution function. The point along the distribution function at which the difference estimator becomes more precise than the ratio estimator depends on the sampling design, as well as the coefficient of variation of X and . A simple confidence interval procedure provides close to nominal coverage for intervals constructed from both the difference and ratio estimators, with the exception that coverage may be poor for the lower tail of the distribution function when using the ratio estimator.     

State-specific detection probabilities and disease prevalence.

Investigations of disease dynamics in wild animal populations often use estimated prevalence or incidence as a measure of true disease frequency. Such indices, almost always based solely on raw counts of infected and uninfected individuals, are often used as the basis for analysis of temporal and spatial dynamics of diseases. Generally, such studies do not account for potential differences in observer detection probabilities of host individuals stratified by biotic and/or abiotic factors. We demonstrate the potential effects of heterogeneity in state-specific detection probabilities on estimated disease prevalence using mark-recapture data from previous work in a House Finch (Carpodacus mexicanus) and Mycoplasma gallisepticum system. In this system, detection probabilities of uninfected finches were generally higher than infected individuals. We show that the magnitude and seasonal pattern of variation in estimated prevalence, corrected for differences in detection probabilities, differed markedly from uncorrected (apparent) prevalence. When the detection probability of uninfected individuals is higher than infected individuals (as in our study), apparent prevalence is negatively biased, and vice versa. In situations where state-specific detection probabilities strongly interact over time, we show that the magnitude and pattern of apparent prevalence can change dramatically; in such cases, observed variations in prevalence may be completely spurious artifacts of variation in detection probability, rather than changes in underlying disease dynamics. Accounting for differential detection probabilities in estimates of disease frequency removes a potentially confounding factor in studies seeking to identify biotic and/or abiotic drivers of disease dynamics. Given that detection probabilities of different groups of individuals are likely to change temporally and spatially in most field studies, our results underscore the importance of estimating and incorporating detection probabilities in estimated disease prevalence (specifically), and more generally, any ecological index used to estimate some parameter of interest. While a mark-recapture approach makes it possible to estimate detection probabilities, it is not always practical, especially at large scales. We discuss several alternative approaches and categorize the assumptions under which analysis of uncorrected prevalence may be acceptable.     

A generalized estimating equations approach for capture–recapture closed population models

The estimation of population density animal population parameters, such as capture probability, population size, or population density, is an important issue in many ecological applications. Capture–recapture data may be considered as repeated observations that are often correlated over time. If these correlations are not taken into account then parameter estimates may be biased, possibly producing misleading results. We propose a generalized estimating equations (GEE) approach to account for correlation over time instead of assuming independence as in the traditional closed population capture–recapture studies. We also account for heterogeneity among observed individuals and over-dispersion, modelling capture probabilities as a function of covariates. The GEE versions of all closed population capture–recapture models and their corresponding estimating equations are proposed. We evaluate the effect of accounting for correlation structures on capture–recapture model selection based on the quasi-likelihood information criterion (QIC). An example is used for an illustrative application and for comparison to currently used methodology. A Horvitz–Thompson-like estimator is used to obtain estimates of population size based on conditional arguments. A simulation study is conducted to evaluate the performance of the GEE approach in capture-recapture studies. The GEE approach performs well for estimating population parameters, particularly when capture probabilities are high. The simulation results also reveal that estimated population size varies on the nature of the existing correlation among capture occasions.     

Comparison between Hansen-Hurwitz and Horvitz-Thompson estimators for adaptive cluster sampling

Thompson (1990) introduced the adaptive cluster sampling design. This sampling design has been shown to be a useful sampling method for parameter estimation of a clustered and scattered population (Roesch, 1993; Smith et al., 1995; Thompson and Seber, 1996). Two estimators, the modified Hansen-Hurwitz (HH) and Horvitz-Thompson (HT) estimators, are available to estimate the mean or total of a population. Empirical results from previous researches indicate that the modified HT estimator has smaller variance than the modified HH estimator. We analytically compare the properties of these two estimators. Some results are obtained in favor of the modified HT estimator so that practitioners are strongly recommended to use the HT estimator despite easiness of computations for the HH estimator.     

Mark-recapture estimators for dual frame population size of prominent nesting structures: the effect of uncertain detection probability

The combined mark-recapture and line transect sampling methodology proposed by Alpizar-Jara and Pollock [Journal of Environmental and Ecological Statistics, 3(4), 311–327, 1996; In Marine Mammal Survey and Assessment Methods Symposium. G.W. Garner, S.C. Amstrup, J.L. Laake, B.F.J. Manly, L.L. McDonald, and D.C. Robertson (Eds.), A.A. Balkema, Rotterdam, Netherlands, pp. 99–114, 1999] is used to illustrate the estimation of population size for populations with prominent nesting structures (i.e., bald eagle nests). In the context of a bald eagle population, the number of nests in a list frame corresponds to a pre-marked sample of nests, and an area frame corresponds to a set of transect strips that could be regularly monitored. Unlike previous methods based on dual frame methodology using the screening estimator [Haines and Pollock (Journal of Environmental and Ecological Statistics, 5, 245–256, 1998a; Survey Methodology, 24(1), 79–88, 1998b)], we no longer need to assume that the area frame is complete (i.e., all the nests in the sampled sites do not need to be seen). One may use line transect sampling to estimate the probability of detection in a sampled area. Combining information from list and area frames provides more efficient estimators than those obtained by using data from only one frame. We derive an estimator for detection probability and generalize the screening estimator. A simulation study is carried out to compare the performance of the Chapman modification of the Lincoln–Petersen estimator to the screening estimator. Simulation results show that although the Chapman estimator is generally less precise than the screening estimator, the latter can be severely biased in presence of uncertain detection. The screening estimator outperforms the Chapman estimator in terms of mean squared error when detection probability is near 1 wheareas the Chapman estimator outperforms the screening estimator when detection probability is lower than a certain threshold value depending on particular scenarios.     

Factors affecting aural detections of songbirds.

Many factors affect the number of birds detected on point count surveys of breeding songbirds. The magnitude and importance of these factors are not well understood. We used a bird song simulation system to quantify the effects of detection distance, singing rate, species differences, and observer differences on detection probabilities of birds detected by ear. We simulated 40 point counts consisting of 10 birds per count for five primary species (Black-and-white Warbler Mniotilta varia, Black-throated Blue Warbler Dendroica caerulescens, Black-throated Green Warbler Dendroica virens, Hooded Warbler Wilsonia citrina, and Ovenbird Seiurus aurocapillus) over a range of 15 distances (34-143 m). Songs were played at low (two songs per count) and high (13-21 songs per count) singing rates. Detection probabilities averaged across observers ranged from 0.60 (Black-and-white Warbler) to 0.83 (Hooded Warbler) at the high singing rate and 0.41 (Black-and-white Warbler) to 0.67 (Hooded Warbler) at the low singing rate. Logistic regression analyses indicated that species, singing rate, distance, and observer were all significant factors affecting detection probabilities. Singing rate x species and singing rate X distance interactions were also significant. Simulations of expected counts, based on the best logistic model, indicated that observers detected between 19% (for the worst observer, lowest singing rate, and least detectable species) and 65% (for the best observer, highest singing rate, and most detectable species) of the true population. Detection probabilities on actual point count surveys are likely to vary even more because many sources of variability were controlled in our experiments. These findings strongly support the importance of adjusting measures of avian diversity or abundance from auditory point counts with direct estimates of detection probability.