The Petersen–Lincoln Estimator and its Extension to Estimate the Size of a Shared Population

The Petersen–Lincoln estimator has been used to estimate the size of a population in a single mark release experiment. However, the estimator is not valid when the capture sample and recapture sample are not independent. We provide an intuitive interpretation for “independence” between samples based on 2 × 2 categorical data formed by capture/non‐capture in each of the two samples. From the interpretation, we review a general measure of “dependence” and quantify the correlation bias of the Petersen–Lincoln estimator when two types of dependences (local list dependence and heterogeneity of capture probability) exist. An important implication in the census undercount problem is that instead of using a post enumeration sample to assess the undercount of a census, one should conduct a prior enumeration sample to avoid correlation bias. We extend the Petersen–Lincoln method to the case of two populations. This new estimator of the size of the shared population is proposed and its variance is derived. We discuss a special case where the correlation bias of the proposed estimator due to dependence between samples vanishes. The proposed method is applied to a study of the relapse rate of illicit drug use in Taiwan. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Double-Observer Line Transect Methods: Levels of Independence

Summary .  Double-observer line transect methods are becoming increasingly widespread, especially for the estimation of marine mammal abundance from aerial and shipboard surveys when detection of animals on the line is uncertain. The resulting data supplement conventional distance sampling data with two-sample mark–recapture data. Like conventional mark–recapture data, these have inherent problems for estimating abundance in the presence of heterogeneity. Unlike conventional mark–recapture methods, line transect methods use knowledge of the distribution of a covariate, which affects detection probability (namely, distance from the transect line) in inference. This knowledge can be used to diagnose unmodeled heterogeneity in the mark–recapture component of the data. By modeling the covariance in detection probabilities with distance, we show how the estimation problem can be formulated in terms of different levels of independence. At one extreme, full independence is assumed, as in the Petersen estimator (which does not use distance data); at the other extreme, independence only occurs in the limit as detection probability tends to one. Between the two extremes, there is a range of models, including those currently in common use, which have intermediate levels of independence. We show how this framework can be used to provide more reliable analysis of double-observer line transect data. We test the methods by simulation, and by analysis of a dataset for which true abundance is known. We illustrate the approach through analysis of minke whale sightings data from the North Sea and adjacent waters.     

Spatial Mark-Recapture Method in the Estimation of Crayfish Population Size

The mark-recapture method is considered for estimation of population size of slowly moving animals like crayfish. The Petersen type estimator for closed population is generalized for situations where recaptures are spatially dependent between the capture sites, and its variance approximation is derived using point processes as models for the population. The method of quadratic forms is suggested to be used as variance estimator. Finally, a trapping design is proposed where one trap at recapture is replaced by four adjacent traps. A simulation experiment is performed to explain the robusticity of the new trapping design against movements of animals.     

Validation of mark‐recapture population estimates for invasive common carp,Cyprinus carpio,in Lake Crescent,Tasmania

A mark‐recapture study based on the Petersen method was implemented in 1998 to estimate the abundance of the invasive common carp, Cyprinus carpio L., in Lake Crescent, Tasmania. Multiple gear types were employed to minimise capture bias, with multiple capture and recapture events providing an opportunity to compute and compare Petersen and Schnabel estimates. A single Petersen estimate on recapture data and two Schnabel estimates – one each on mark (forward‐Schnabel estimate) and recapture (reverse‐Schnabel estimate) data – were conducted. An independent long‐term double tag study facilitated estimation of the annual natural mortality. Subsequent fish‐down of the population suggests that, in all likelihood, the carp have been eradicated from the lake, providing an unprecedented opportunity to verify the forward population estimates carried out in 1998. Results suggest that all three estimates were close to the true population size, with the reverse‐Schnabel estimate being the most accurate and within 1% of the true population in this relatively large lake (～2365 ha). Greater accuracy of the reverse‐Schnabel approach can be attributed to either minimised fish behavioural (i.e. gear susceptibility or avoidance) or computational bias associated with the forward‐Schnabel and Petersen approaches, respectively. While the original estimates served as a guide in eradication of carp from the lake, the ultimate validation provides a reliable framework for abundance estimation of this invasive fish in relatively large water bodies elsewhere.     

One‐inflation and unobserved heterogeneity in population size estimation

We present the one‐inflated zero‐truncated negative binomial (OIZTNB) model, and propose its use as the truncated count distribution in Horvitz–Thompson estimation of an unknown population size. In the presence of unobserved heterogeneity, the zero‐truncated negative binomial (ZTNB) model is a natural choice over the positive Poisson (PP) model; however, when one‐inflation is present the ZTNB model either suffers from a boundary problem, or provides extremely biased population size estimates. Monte Carlo evidence suggests that in the presence of one‐inflation, the Horvitz–Thompson estimator under the ZTNB model can converge in probability to infinity. The OIZTNB model gives markedly different population size estimates compared to some existing truncated count distributions, when applied to several capture–recapture data that exhibit both one‐inflation and unobserved heterogeneity.     

Open Capture–Recapture Models with Heterogeneity: II. Jolly–Seber Model

Summary Estimation of abundance is important in both open and closed population capture–recapture analysis, but unmodeled heterogeneity of capture probability leads to negative bias in abundance estimates. This article defines and develops a suite of open population capture–recapture models using finite mixtures to model heterogeneity of capture and survival probabilities. Model comparisons and parameter estimation use likelihood‐based methods. A real example is analyzed, and simulations are used to check the main features of the heterogeneous models, especially the quality of estimation of abundance, survival, recruitment, and turnover. The two major advances in this article are the provision of realistic abundance estimates that take account of heterogenetiy of capture, and an appraisal of the amount of overestimation of survival arising from conditioning on the first capture when heterogeneity of survival is present.     

A Comparison of Population Size Estimators under the Truncated Count Model With and Without Allowance for Contaminations

The purpose of the study is to estimate the population size under a homogeneous truncated count model and under model contaminations via the Horvitz‐Thompson approach on the basis of a count capture‐recapture experiment. The proposed estimator is based on a mixture of zero‐truncated Poisson distributions. The benefit of using the proposed model is statistical inference of the long‐tailed or skewed distributions and the concavity of the likelihood function with strong results available on the nonparametric maximum likelihood estimator (NPMLE). The results of comparisons, for finding the appropriate estimator among McKendrick's, Mantel‐Haenszel's, Zelterman's, Chao's, the maximum likelihood, and the proposed methods in a simulation study, reveal that under model contaminations the proposed estimator provides the best choice according to its smallest bias and smallest mean square error for a situation of sufficiently large population sizes and the further results show that the proposed estimator performs well even for a homogeneous situation. The empirical examples, containing the cholera epidemic in India based on homogeneity and the heroin user data in Bangkok 2002 based on heterogeneity, are fitted with an excellent goodness‐of‐fit of the models and the confidence interval estimations may also be of considerable interest. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Density estimation in a wolverine population using spatial capture–recapture models

Classical closed-population capture–recapture models do not accommodate the spatial information inherent in encounter history data obtained from camera-trapping studies. As a result, individual heterogeneity in encounter probability is induced, and it is not possible to estimate density objectively because trap arrays do not have a well-defined sample area. We applied newly-developed, capture–recapture models that accommodate the spatial attribute inherent in capture–recapture data to a population of wolverines (Gulo gulo) in Southeast Alaska in 2008. We used camera-trapping data collected from 37 cameras in a 2,140-km² area of forested and open habitats largely enclosed by ocean and glacial icefields. We detected 21 unique individuals 115 times. Wolverines exhibited a strong positive trap response, with an increased tendency to revisit previously visited traps. Under the trap-response model, we estimated wolverine density at 9.7 individuals/1,000 km² (95% Bayesian CI: 5.9–15.0). Our model provides a formal statistical framework for estimating density from wolverine camera-trapping studies that accounts for a behavioral response due to baited traps. Further, our model-based estimator does not have strict requirements about the spatial configuration of traps or length of trapping sessions, providing considerable operational flexibility in the development of field studies. © 2011 The Wildlife Society.     

Estimating population size by spatially explicit capture–recapture

The number of animals in a population is conventionally estimated by capture–recapture without modelling the spatial relationships between animals and detectors. Problems arise with non‐spatial estimators when individuals differ in their exposure to traps or the target population is poorly defined. Spatially explicit capture–recapture (SECR) methods devised recently to estimate population density largely avoid these problems. Some applications require estimates of population size rather than density, and population size in a defined area may be obtained as a derived parameter from SECR models. While this use of SECR has potential benefits over conventional capture–recapture, including reduced bias, it is unfamiliar to field biologists and no study has examined the precision and robustness of the estimates. We used simulation to compare the performance of SECR and conventional estimators of population size with respect to bias and confidence interval coverage for several spatial scenarios. Three possible estimators for the sampling variance of realised population size all performed well. The precision of SECR estimates was nearly the same as that of the null‐model conventional population estimator. SECR estimates of population size were nearly unbiased (relative bias 0–10%) in all scenarios, including surveys in randomly generated patchy landscapes. Confidence interval coverage was near the nominal level. We used SECR to estimate the population of a species of skink Oligosoma infrapunctatum from pitfall trapping. The estimated number in the area bounded by the outermost traps differed little between a homogeneous density model and models with a quadratic trend in density or a habitat effect on density, despite evidence that the latter models fitted better. Extrapolation of trend models to a larger plot may be misleading. To avoid extrapolation, a large region of interest should be sampled throughout, either with one continuous trapping grid or with clusters of traps dispersed widely according to a probability‐based and spatially representative sampling design.     

Estimation of capture probabilities using generalized estimating equations and mixed effects approaches

Modeling individual heterogeneity in capture probabilities has been one of the most challenging tasks in capture–recapture studies. Heterogeneity in capture probabilities can be modeled as a function of individual covariates, but correlation structure among capture occasions should be taking into account. A proposed generalized estimating equations (GEE) and generalized linear mixed modeling (GLMM) approaches can be used to estimate capture probabilities and population size for capture–recapture closed population models. An example is used for an illustrative application and for comparison with currently used methodology. A simulation study is also conducted to show the performance of the estimation procedures. Our simulation results show that the proposed quasi‐likelihood based on GEE approach provides lower SE than partial likelihood based on either generalized linear models (GLM) or GLMM approaches for estimating population size in a closed capture–recapture experiment. Estimator performance is good if a large proportion of individuals are captured. For cases where only a small proportion of individuals are captured, the estimates become unstable, but the GEE approach outperforms the other methods.     

Population size estimates based on the frequency of genetically assigned parent–offspring pairs within a subsample

Estimating population density as precise as possible is a key premise for managing wild animal species. This can be a challenging task if the species in question is elusive or, due to high quantities, hard to count. We present a new, mathematically derived estimator for population size, where the estimation is based solely on the frequency of genetically assigned parent–offspring pairs within a subsample of an ungulate population. By use of molecular markers like microsatellites, the number of these parent–offspring pairs can be determined. The study's aim was to clarify whether a classical capture–mark–recapture (CMR) method can be adapted or extended by this genetic element to a genetic‐based capture–mark–recapture (g‐CMR). We numerically validate the presented estimator (and corresponding variance estimates) and provide the R‐code for the computation of estimates of population size including confidence intervals. The presented method provides a new framework to precisely estimate population size based on the genetic analysis of a one‐time subsample. This is especially of value where traditional CMR methods or other DNA‐based (fecal or hair) capture–recapture methods fail or are too difficult to apply. The DNA source used is basically irrelevant, but in the present case the sampling of an annual hunting bag is to serve as data basis. In addition to the high quality of muscle tissue samples, hunting bags provide additional and essential information for wildlife management practices, such as age, weight, or sex. In cases where a g‐CMR method is ecologically and hunting‐wise appropriate, it enables a wide applicability, also through its species‐independent use.     

Assessment of mark–recapture models to estimate the abundance of a humpback whale feeding aggregation in Southeast Alaska

Aim The aim of this study was to use photographs of the unique pattern on the ventral surface of the flukes to estimate the abundance of humpback whales (Megaptera novaeangliae) in a discrete feeding aggregation in northern Southeast Alaska. Location The study was located in northern Southeast Alaska, USA, in the eastern North Pacific Ocean. Methods This study evaluated mark–recapture models, ranging from the simpler models (pooled and stratified, closed Petersen estimators) to more complex multi‐strata models (closed Darroch and open Hilborn). The Akaike Information Criterion, corrected (AICc) was used as a model comparison statistic. Results Our best estimate of whale abundance in northern Southeast Alaska in 2000 is 961 whales [95% confidence interval (657, 1076)]. This estimate comes from the Hilborn open, multi‐strata approach with constant migration over time, time‐dependent capture probabilities by area, and a fixed survival rate of 0.98. The simpler models were problematic owing to several aspects of whale behaviour, including that (1) the whales did not mix randomly throughout the study area, (2) some whales emigrated temporarily outside the study area and were not available for capture, and (3) whales were not equally identifiable because they did not behave in the same way when they showed their flukes upon diving. This led to heterogeneity in capture probability and a bias in the estimates. The more complex models stratified by area, and using migration movements among areas, compensated for some of these issues when estimating population size. Main conclusions We believe that the Hilborn open, multi‐strata model produced the best estimate because: (1) it incorporated the best information about survival, (2) it used detailed information about the various release groups, (3) the analysis provided an integrated environment in which parameters such as migration and capture probabilities are shared, (4) the three strata encompassed a large portion of the areas used by whales, and (5) the Hilborn model selected was superior in terms of model selection criteria and biological realism. These data provide valuable insights into the numbers and movements of humpback whales in three areas of Southeast Alaska.     

On the estimation of species richness based on the accumulation of previously unrecorded species

Estimation of species richness of local communities has become an important topic in community ecology and monitoring. Investigators can seldom enumerate all the species present in the area of interest during sampling sessions. If the location of interest is sampled repeatedly within a short time period, the number of new species recorded is typically largest in the initial sample and decreases as sampling proceeds, but new species may be detected if sampling sessions are added. The question is how to estimate the total number of species. The data collected by sampling the area of interest repeatedly can be used to build species accumulation curves: the cumulative number of species recorded as a function of the number of sampling sessions (which we refer to as “species accumulation data”). A classic approach used to compute total species richness is to fit curves to the data on species accumulation with sampling effort. This approach does not rest on direct estimation of the probability of detecting species during sampling sessions and has no underlying basis regarding the sampling process that gave rise to the data. Here we recommend a probabilistic, nonparametric estimator for species richness for use with species accumulation data. We use estimators of population size that were developed for capture‐recapture data, but that can be used to estimate the size of species assemblages using species accumulation data. Models of detection probability account for the underlying sampling process. They permit variation in detection probability among species. We illustrate this approach using data from the North American Breeding Bird Survey (BBS). We describe other situations where species accumulation data are collected under different designs (e.g., over longer periods of time, or over spatial replicates) and that lend themselves to of use capture‐recapture models for estimating the size of the community of interest. We discuss the assumptions and interpretations corresponding to each situation.     

Spatial capture–recapture models allowing Markovian transience or dispersal

Spatial capture–recapture (SCR) models are a relatively recent development in quantitative ecology, and they are becoming widely used to model density in studies of animal populations using camera traps, DNA sampling and other methods which produce spatially explicit individual encounter information. One of the core assumptions of SCR models is that individuals possess home ranges that are spatially stationary during the sampling period. For many species, this assumption is unlikely to be met and, even for species that are typically territorial, individuals may disperse or exhibit transience at some life stages. In this paper we first conduct a simulation study to evaluate the robustness of estimators of density under ordinary SCR models when dispersal or transience is present in the population. Then, using both simulated and real data, we demonstrate that such models can easily be described in the BUGS language providing a practical framework for their analysis, which allows us to evaluate movement dynamics of species using capture–recapture data. We find that while estimators of density are extremely robust, even to pathological levels of movement (e.g., complete transience), the estimator of the spatial scale parameter of the encounter probability model is confounded with the dispersal/transience scale parameter. Thus, use of ordinary SCR models to make inferences about density is feasible, but interpretation of SCR model parameters in relation to movement should be avoided. Instead, when movement dynamics are of interest, such dynamics should be parameterized explicitly in the model.     

Estimating abundance of mountain lions from unstructured spatial sampling

Mountain lions (Puma concolor) are often difficult to monitor because of their low capture probabilities, extensive movements, and large territories. Methods for estimating the abundance of this species are needed to assess population status, determine harvest levels, evaluate the impacts of management actions on populations, and derive conservation and management strategies. Traditional mark–recapture methods do not explicitly account for differences in individual capture probabilities due to the spatial distribution of individuals in relation to survey effort (or trap locations). However, recent advances in the analysis of capture–recapture data have produced methods estimating abundance and density of animals from spatially explicit capture–recapture data that account for heterogeneity in capture probabilities due to the spatial organization of individuals and traps. We adapt recently developed spatial capture–recapture models to estimate density and abundance of mountain lions in western Montana. Volunteers and state agency personnel collected mountain lion DNA samples in portions of the Blackfoot drainage (7,908 km²) in west-central Montana using 2 methods: snow back-tracking mountain lion tracks to collect hair samples and biopsy darting treed mountain lions to obtain tissue samples. Overall, we recorded 72 individual capture events, including captures both with and without tissue sample collection and hair samples resulting in the identification of 50 individual mountain lions (30 females, 19 males, and 1 unknown sex individual). We estimated lion densities from 8 models containing effects of distance, sex, and survey effort on detection probability. Our population density estimates ranged from a minimum of 3.7 mountain lions/100 km² (95% CI 2.3–5.7) under the distance only model (including only an effect of distance on detection probability) to 6.7 (95% CI 3.1–11.0) under the full model (including effects of distance, sex, survey effort, and distance × sex on detection probability). These numbers translate to a total estimate of 293 mountain lions (95% CI 182–451) to 529 (95% CI 245–870) within the Blackfoot drainage. Results from the distance model are similar to previous estimates of 3.6 mountain lions/100 km² for the study area; however, results from all other models indicated greater numbers of mountain lions. Our results indicate that unstructured spatial sampling combined with spatial capture–recapture analysis can be an effective method for estimating large carnivore densities. Published 2012. This article is a U.S. Government work and is in the public domain in the USA.     

Understanding implications of detection heterogeneity in wildlife abundance estimation

Negative bias in mark-recapture abundance estimators due to heterogeneity in detection (capture) probability is a well-known problem, but we believe most biologists do not understand why heterogeneity causes bias and how bias can be reduced. We demonstrate how heterogeneity creates dependence and bias in mark-recapture approaches to abundance estimation. In comparison, heterogeneity, and hence estimator bias, is not as problematic for distance sampling and mark-resight methods because both techniques estimate detection probabilities based on a known quantity. We show how the introduction of a known number of individuals planted into a study population prior to a mark-recapture survey can reduce bias from heterogeneity in detection probability. We provide examples with simulation and an analysis of motion-sensitive camera data from a study population of introduced eastern wild turkeys (Meleagris gallopavo silvestris) of known size with a subset of telemetered birds. In choosing a method for abundance estimation, careful consideration should be given to assumptions and how heterogeneity in detection probability can be accommodated for each application.     

Inference from single occasion capture experiments using genetic markers

Accurate estimation of the size of animal populations is an important task in ecological science. Recent advances in the field of molecular genetics researches allow the use of genetic data to estimate the size of a population from a single capture occasion rather than repeated occasions as in the usual capture–recapture experiments. Estimating the population size using genetic data also has sometimes led to estimates that differ markedly from each other and also from classical capture–recapture estimates. Here, we develop a closed form estimator that uses genetic information to estimate the size of a population consisting of mothers and daughters, focusing on estimating the number of mothers, using data from a single sample. We demonstrate the estimator is consistent and propose a parametric bootstrap to estimate the standard errors. The estimator is evaluated in a simulation study and applied to real data. We also consider maximum likelihood in this setting and discover problems that preclude its general use.     

Heterogeneous capture rates in low density populations and consequences for capture-recapture analysis of camera-trap data

Closed population capture-recapture analysis of camera-trap data has become the conventional method for estimating the abundance of individually recognisable cryptic species living at low densities, such as large felids. Often these estimates are the only information available to guide wildlife managers and conservation policy. Capture probability of the target species using camera traps is commonly heterogeneous and low. Published studies often report overall capture probabilities as low as 0.03 and fail to report on the level of heterogeneity in capture probability. We used simulations to study the effects of low and heterogeneous capture probability on the reliability of abundance estimates using the M_h jack-knife estimator within a closed-population capture-recapture framework. High heterogeneity in capture probability was associated with under- and over-estimates of true abundance. The use of biased abundance estimates could have serious conservation management consequences. We recommend that studies present capture frequencies of all sampled individuals so that policy makers can assess the reliability of the abundance estimates.     

Estimating the abundance of rare and elusive carnivores from photographic-sampling data when the population size is very small

Conservation and management agencies require accurate and precise estimates of abundance when considering the status of a species and the need for directed actions. Due to the proliferation of remote sampling cameras, there has been an increase in capture–recapture studies that estimate the abundance of rare and/or elusive species using closed capture–recapture estimators (C–R). However, data from these studies often do not meet necessary statistical assumptions. Common attributes of these data are (1) infrequent detections, (2) a small number of individuals detected, (3) long survey durations, and (4) variability in detection among individuals. We believe there is a need for guidance when analyzing this type of sparse data. We highlight statistical limitations of closed C–R estimators when data are sparse and suggest an alternative approach over the conventional use of the Jackknife estimator. Our approach aims to maximize the probability individuals are detected at least once over the entire sampling period, thus making the modeling of variability in the detection process irrelevant, estimating abundance accurately and precisely. We use simulations to demonstrate when using the unconditional-likelihood M ₀ (constant detection probability) closed C–R estimator with profile-likelihood confidence intervals provides reliable results even when detection varies by individual. If each individual in the population is detected on average of at least 2.5 times, abundance estimates are accurate and precise. When studies sample the same species at multiple areas or at the same area over time, we suggest sharing detection information across datasets to increase precision when estimating abundance. The approach suggested here should be useful for monitoring small populations of species that are difficult to detect.     

Capture–recapture models with heterogeneity to study survival senescence in the wild

Detecting senescence in wild populations and estimating its strength raise three challenges. First, in the presence of individual heterogeneity in survival probability, the proportion of high‐survival individuals increases with age. This increase can mask a senescence‐related decrease in survival probability when the probability is estimated at the population level. To accommodate individual heterogeneity we use a mixture model structure (discrete classes of individuals). Second, the study individuals can elude the observers in the field, and their detection rate can be heterogeneous. To account for detectability issues we use capture–mark–recapture (CMR) methodology, mixture models and data that provide information on individuals’ detectability. Last, emigration to non‐monitored sites can bias survival estimates, because it can occur at the end of the individuals’ histories and mimic earlier death. To model emigration we use Markovian transitions to and from an unobservable state. These different model structures are merged together using hidden Markov chain CMR models, or multievent models. Simulation studies illustrate that reliable evidence for survival senescence can be obtained using highly heterogeneous data from non site‐faithful individuals. We then design a tailored application for a dataset from a colony of black‐headed gull Chroicocephalus ridibundus. Survival probabilities do not appear individually variable, but evidence for survival senescence becomes significant only when accounting for other sources of heterogeneity. This result suggests that not accounting for heterogeneity leads to flawed inference and/or that emigration heterogeneity mimics survival heterogeneity and biases senescence estimates.