An extension of the Lin-Yip estimator for continuous-time capture studies

A unified approach is suggested to estimate the population size for continuous time capture studies with possible removals during the capture process. It extends and improves the Lin-Yip estimator. The usual recaptureand removal models can be shown to be particular cases of the general formulation. A Horvitz-Thompson procedure is used to estimate the population size based on the estimated capture probabilities. The resultant estimators for the regression parameters and population size are consistent and asymptotically normal underappropriateregularity conditions. We assess the properties of the proposed estimators through Monte Carlo simulation. Two examples are given.     

Estimation of population size using open capture-recapture models

One of the most important needs for wildlife managers is an accurate estimate of population size. Yet, for many species, including most marine species and large mammals, accurate and precise estimation of numbers is one of the most difficult of all research challenges. Open-population capture-recapture models have proven useful in many situations to estimate survival probabilities but typically have not been used to estimate population size. We show that open-population models can be used to estimate population size by developing a Horvitz-Thompson-type estimate of population size and an estimator of its variance. Our population size estimate keys on the probability of capture at each trap occasion and therefore is quite general and can be made a function of external covariates measured during the study. Here we define the estimator and investigate its bias, variance, and variance estimator via computer simulation. Computer simulations make extensive use of real data taken from a study of polar bears (Ursus maritimus) in the Beaufort Sea. The population size estimator is shown to be useful because it was negligibly biased in all situations studied. The variance estimator is shown to be useful in all situations, but caution is warranted in cases of extreme capture heterogeneity.     

A Varying Coefficients Model For Estimating Finite Population Totals: A Hierarchical Bayesian Approach

In some finite sampling situations, there is a primary variable that is sampled, and there are measurements on covariates for the entire population. A Bayesian hierarchical model for estimating totals for finite populations is proposed. A nonparametric linear model is assumed to explain the relationship between the dependent variable of interest and covariates. The regression coefficients in the linear model are allowed to vary as a function of a subset of covariates nonparametrically based on B-splines. The generality of this approach makes it robust and applicable to data collected using a variety of sampling techniques, provided the sample is representative of the finite population. A simulation study is carried out to evaluate the performance of the proposed model for the estimation of the population total. Results indicate accurate estimation of population totals using the approach. The modeling approach is used to estimate the total production of avocado for a large group of groves in Mexico.     

Polar bears in the Beaufort Sea: A 30-year mark-recapture case history

Knowledge of population size and trend is necessary to manage anthropogenic risks to polar bears (Ursus maritimus). Despite capturing over 1,025 females between 1967 and 1998, previously calculated estimates of the size of the southern Beaufort Sea (SBS) population have been unreliable. We improved estimates of numbers of polar bears by modeling heterogeneity in capture probability with covariates. Important covariates referred to the year of the study, age of the bear, capture effort, and geographic location. Our choice of best approximating model was based on the inverse relationship between variance in parameter estimates and likelihood of the fit and suggested a growth from ≈ 500 to over 1,000 females during this study. The mean coefficient of variation on estimates for the last decade of the study was 0.16—the smallest yet derived. A similar model selection approach is recommended for other projects where a best model is not identified by likelihood criteria alone.     

Full Open Population Capture–Recapture Models With Individual Covariates

Traditional analyses of capture–recapture data are based on likelihood functions that explicitly integrate out all missing data. We use a complete data likelihood (CDL) to show how a wide range of capture–recapture models can be easily fitted using readily available software JAGS/BUGS even when there are individual-specific time-varying covariates. The models we describe extend those that condition on first capture to include abundance parameters, or parameters related to abundance, such as population size, birth rates or lifetime. The use of a CDL means that any missing data, including uncertain individual covariates, can be included in models without the need for customized likelihood functions. This approach also facilitates modeling processes of demographic interest rather than the complexities caused by non-ignorable missing data. We illustrate using two examples, (i) open population modeling in the presence of a censored time-varying individual covariate in a full robust design, and (ii) full open population multi-state modeling in the presence of a partially observed categorical variable. Supplemental materials for this article are available online.     

A gamma-shaped detection function for line-transect surveys with mark-recapture and covariate data

We have developed a procedure for estimating animal population size from aerial survey data collected simultaneously by two observers on the same sighting platform. We used a line transect sample design where transects follow elevation contours in mountainous terrain. Because our 10 data sets from aerial line transect surveys, conducted over a terrestrial environment, consistently show unimodal detection shapes, we chose a gamma-shaped detection function that is unimodal and admits covariates. We fit models separately to data from each observer, and then used all of the data to estimate the probabilities at the apex of the detection curves. We used a Horvitz-Thompson estimator to estimate the population size. We illustrate our procedure on a recently collected brown bear data set.     

A Bayesian Approach to Estimate OJD Prevalence From Pooled Fecal Samples of Variable Pool Size

This paper describes a Bayesian approach to prevalence estimation based on pooled samples that accommodates variation in pool size and adjusts for test imperfection. A logistic model was developed for pooled fecal culture (PFC) sensitivity as a function of pool size and a logistic mixed model for ovine Johne’s disease (OJD) prevalence as a function of covariates that were found significant in a recent OJD risk factor study conducted in Australia. Available data on these factors and prior information about prevalence and sensitivity were incorporated into a Bayesian model to estimate OJD prevalence from PFC data. Overall, posterior cohort OJD prevalence was estimated to be 0.16 (range of prevalences across cohorts 0.002 to 0.72). The average prevalence was higher in wethers than ewes. PFC sensitivities for pool sizes 10, 30 and 50 were estimated to be 0.91 (95% probability intervals 0.80, 0.96), 0.85 (0.80, 0.90) and 0.77 (0.65, 0.88), respectively. Posterior specificity of PFC was almost perfect though based primarily on the prior. Results suggest the Bayesian model successfully estimated the animal-level prevalence after accounting for variable pool size and imperfect test parameters. The method can be easily adapted for other conditions and diseases where pooled samples are collected. WinBugs code for the article is available online.     

A New Approach to Modelling the Relationship Between Annual Population Abundance Indices and Weather Data

Weather has often been associated with fluctuations in population sizes of species; however, it can be difficult to estimate the effects satisfactorily because population size is naturally measured by annual abundance indices whilst weather varies on much shorter timescales. We describe a novel method for estimating the effects of a temporal sequence of a weather variable (such as mean temperatures from successive months) on annual species abundance indices. The model we use has a separate regression coefficient for each covariate in the temporal sequence, and over-fitting is avoided by constraining the regression coefficients to lie on a curve defined by a small number of parameters. The constrained curve is the product of a periodic function, reflecting assumptions that associations with weather will vary smoothly throughout the year and tend to be repetitive across years, and an exponentially decaying term, reflecting an assumption that the weather from the most recent year will tend to have the greatest effect on the current population and that the effect of weather in previous years tends to diminish as the time lag increases. We have used this approach to model 501 species abundance indices from Great Britain and present detailed results for two contrasting species alongside an overall impression of the results across all species. We believe this approach provides an important advance to the challenge of robustly modelling relationships between weather and species population size.Supplementary materials accompanying this paper appear online.     

Evaluating estimators of the numbers of females with cubs-of-the-year in the yellowstone grizzly bear population

Current management of the grizzly bear (Ursus arctos) population in Yellowstone National Park and surrounding areas requires annual estimation of the number of adult female bears with cubs-of-the-year. We examined the performance of nine estimators of population size via simulation. Data were simulated using two methods for different combinations of population size, sample size, and coefficient of variation of individual sighting probabilities. We show that the coefficient of variation does not, by itself, adequately describe the effects of capture heterogeneity, because two different distributions of capture probabilities can have the same coefficient of variation. All estimators produced biased estimates of population size with bias decreasing as effort in creased. Based on the simulation results we recommend the Chao estimator for model M _h be used to estimate the number of the female bears with cubs of the year; however, the estimator of Chao and Shen may also be useful depending on the goals of the research.     

A Measurement Error Model for Heterogeneous Capture Probabilities in Mark-Recapture Experiments: An Estimating Equation Approach

Logistic models for capture probabilities that depend on covariates are effective if the covariates can be measured exactly. If there is measurement error so that a surrogate for the covariate is observed rather than the covariate itself, simple adjustments may be made if the parameters of joint distribution of the covariate and the surrogate are known. Here we consider the case when a surrogate is observed whenever an individual is captured and the parameters must also be estimated from the data. An estimating equation regression calibration approach is developed and it is illustrated on a real dataset where the surrogate is an individual bird’s wing-length, which varies from occasion to occasion.     

Estimating the hidden number of scrapie affected holdings in Great Britain using a simple,truncated count model allowing for heterogeneity

None of the current surveillance streams monitoring the presence of scrapie in Great Britain provide a comprehensive and unbiased estimate of the prevalence of the disease at the holding level. Previous work to estimate the under-ascertainment adjusted prevalence of scrapie in Great Britain applied multiple-list capture-recapture methods. The enforcement of new control measures on scrapie-affected holdings in 2004 has stopped the overlapping between surveillance sources and, hence, the application of multiple-list capture-recapture models. Alternative methods, still under the capture-recapture methodology, relying on repeated entries in one single list have been suggested in these situations. In this article, we apply one-list capture-recapture approaches to data held on the Scrapie Notifications Database to estimate the undetected population of scrapie-affected holdings with clinical disease in Great Britain for the years 2002, 2003, and 2004. For doing so, we develop a new diagnostic tool for indication of heterogeneity as well as a new understanding of the Zelterman and Chao’s lower bound estimators to account for potential unobserved heterogeneity. We demonstrate that the Zelterman estimator can be viewed as a maximum likelihood estimator for a special, locally truncated Poisson likelihood equivalent to a binomial likelihood. This understanding allows the extension of the Zelterman approach by means of logistic regression to include observed heterogeneity in the form of covariates—in case studied here, the holding size and country of origin. Our results confirm the presence of substantial unobserved heterogeneity supporting the application of our two estimators. The total scrapie-affected holding population in Great Britain is around 300 holdings per year. None of the covariates appear to inform the model significantly.     

Models for Jointly Estimating Abundances of Two Unmarked Site-Associated Species Subject to Imperfect Detection

In ecological field surveys, it is often of interest to estimate the abundance of species. It is frequently the case that unmarked animals are counted on different sites over several time occasions. A natural starting point to model these data, while accounting for imperfect detection, is by using Royle’s N-mixture model (Biometrics 60:108–115, 2004). Subsequently, many multivariate extensions have been proposed to model communities as a whole. However, these approaches are used to study species richness and other community-level variables and do not focus on the relationship between two site-associated species. Here, we extend the N-mixture modelling framework to model two site-associated species abundances jointly and propose to measure the influence of one species’ abundance on the populations of the other and study how this changes over time and space. By including a new parameter in the abundance distribution of one of the species, linking it to abundance of the other, our proposed model treats extra variability as an effect induced by an associated species’ abundance and allows one to study how environmental covariates may affect this. Using results from simulation studies, we show that the model is able to recover true parameter estimates. We illustrate our approach using data from bald eagles and mallards obtained in the 2015 survey of the North American Breeding Bird Survey. By using the joint model, we were able to separate overdispersion from mallard-induced variability and hence what would be accounted for with a dispersion parameter in the univariate framework for the eagles was explained by covariates related to mallard abundance in the joint model. Our approach represents an attractive, yet simple, way of modelling site-associated species populations jointly. Conservation ecologists can use the approach to devise management strategies based on the strength of association between species, which may be due to direct interactions and/or environmental effects affecting both species’ populations. Also, mathematical ecologists can use this framework to develop tools for studying population dynamics under different scenarios. Supplementary materials accompanying this paper appear on-line.     

Determining minimum sample size for the conditioned Latin hypercube sampling algorithm

In digital soil mapping (DSM), a fundamental assumption is that the spatial variability of the target variable can be explained by the predictors or environmental covariates. Strategies to adequately sample the predictors have been well documented, with the conditioned Latin hypercube sampling (cLHS) algorithm receiving the most attention in the DSM community. Despite advances in sampling design, a critical gap remains in determining the number of samples required for DSM projects. We propose a simple workflow and function coded in R language to determine the minimum sample size for the cLHS algorithm based on histograms of the predictor variables using the Freedman-Diaconis rule for determining optimal bin width. Data preprocessing was included to correct for multimodal and non-normally distributed data, as these can affect sample size determination from the histogram. Based on a user-selected quantile range (QR) for the sample plan, the densities of the histogram bins at the upper and lower bounds of the QR were used as a scaling factor to determine minimum sample size. This technique was applied to a field-scale set of environmental covariates for a well-sampled agricultural study site near Guelph, Ontario, Canada, and tested across a range of QRs. The results showed increasing minimum sample size with an increase in the QR selected. Minimum sample size increased from 44 to 83 when the QR increased from 50% to 95% and then increased exponentially to 194 for the 99% QR. This technique provides an estimate of minimum sample size that can be used as an input to the cLHS algorithm.     

Comparing mixture estimates by parametric bootstrapping likelihood ratios

Wildlife managers and researchers often need to estimate the relative contributions of distinct populations in a miture of organisms. Increasingly, there is interest in comparing these mixture contributions across space or time. Comparisons usually just check for overlap in the interval estimates for each population contribution from each mixture. This method inflates Type I error rates, has limited power due to its focus on marginal comparisons, and employs a fundamentally inappropriate measure of mixture difference. Given the difficulty of defining an appropriate measure of mixture difference, a powerful alternative is to compare mixtures using a likelihood ratio test. In applications where the standard asymptotic theory does not hold, the null reference distribution can be obtained through parametric bootstrapping. In addition to testing simple hypotheses, a likelihood ratio framework encourages modeling the change in mixture contributions as a function of covariates. The method is demonstrated with an analysis of potential sampling bias in the estimation of population contributions to the commercial sockeye, salmon (Oncorhynchus nerka) fishery in Upper Cook Inlet, Alaska.     

Predicting potential habitat and population size for reintroduction of the Far Eastern leopards in the Russian Far East

The Far Eastern Leopard (Panthera pardus orientalis; Schlegel, 1857) is perhaps the world’s most endangered large felid subspecies occurring in a single population of ?30 adults, and faces immediate risk of extinction unless additional populations can be established within its historical range in the Russian Far East. We used locations of leopard tracks (and their ungulate prey) collected from snow track surveys from 1997 to 2007 to develop resource selection functions (RSF) to identify potential habitat for reintroduction. We compared models that include prey versus those based on landscape covariates, and also included covariates related to human-induced mortality. To estimate potential population size, we used a habitat-based population estimate based on the ratio of population size and RSF value of occupied range. Far Eastern leopards selected for areas with high ungulate density, lower-elevation Korean pine forests on southwest facing slopes, and in areas far from human activity. Using this RSF model, we identified a total of 10,648 km² in eight patches >500 km² of potential Far Eastern leopard habitat that could harbor a potential population of 105.3 (57.9–147.2) adults. In combination with the existing population, successful reintroductions could result in a total of 139.2 (76.5–194.6) adult leopards, a 3–4-fold increase in population size. Our habitat models assist the reintroduction planning process by identifying factors that predict presence and potential suitable habitat. Identifying the highest quality, most connected patches, in combination with appropriate selection and training of released animals, is recommended for successfully reintroducing Far Eastern leopards, and potentially other endangered carnivores into the wild.     

Identifying QTLs and Epistasis in Structured Plant Populations Using Adaptive Mixed LASSO

Association analysis in important crop species has generated heightened interest for its potential in dissecting complex traits by utilizing diverse mapping populations. However, the mixed linear model approach is currently limited to single marker analysis, which is not suitable for studying multiple QTL effects, epistasis and gene by environment interactions. In this paper, we propose the adaptive mixed LASSO method that can incorporate a large number of predictors (genetic markers, epistatic effects, environmental covariates, and gene by environment interactions) while simultaneously accounting for the population structure. We show that the adaptive mixed LASSO estimator possesses the oracle property of adaptive LASSO. Algorithms are developed to iteratively estimate the regression coefficients and variance components. Our results demonstrate that the adaptive mixed LASSO method is very promising in modeling multiple genetic effects when a large number of markers are available and the population structure cannot be ignored. It is expected to be a powerful tool for studying the architecture of complex traits in important plant species. Supplemental materials for this article are available from the journal website.     

A Population-Size Model for Protein Spot Detection in Proteomic Studies

In proteomic studies, a population of proteins are often examined on a gel using a technique called two-dimensional gel eletrophoresis. The technique separates the protein population into individual protein spots on a two-dimensional gel by isoelectric charge and molecular weight. The resulting gel images are then processed by a software system for spot detection and subsequent analysis. The performance of a spot-detection program is evaluated by the total number of spots that are detected. A popular spot-detection program uses the “master–slave” approach, where all spots on “slave images” are subsets of the spots on the “master image.” We argue that this approach potentially misses a large proportion of proteins and propose a model that quantifies the lack of performance. We provide nonparametric estimators for the protein population size and the expected number of proteins to be detected if a “fusion-gel” approach was used. Using the data from a rat liver proteome study, we estimate that more than half of the protein population is missed by the master–slave approach.     

Estimating the minimum viable population size of kaka (Nestor meridionalis), a potential surrogate species in New Zealand lowland forest

While conservation management is increasingly turning towards an ecosystem-level framework, the focus on a small subset of surrogate species has recognised merit given insufficient time, resources, and expertise. The kaka (Nestor meridionalis), a large threatened New Zealand parrot, is an iconic, visible species in lowland forests. As kaka populations are sensitive to mustelid predation and habitat loss, kaka can act as both a flagship and indicator species for healthy lowland forest ecosystems in New Zealand. To ensure the sustained protection of kaka over a sufficient area, this research aims to estimate the minimum viable population (MVP) size of kaka in the Eglinton Valley, Fiordland, and the management required for population persistence. A post-breeding census, stochastic, age structured Leslie matrix model was developed to estimate the population size having a 95% probability of persistence over 100 years. Scenarios modeling current and alternate management regimes, uncertain life-history traits, and environmental unpredictability were run. The most ‘realistic’ scenario resulted in an MVP size of 258 kaka (155 adults). Maintaining current levels of predator control appears essential to ensure kaka population persistence. An area of >500 km² is proposed to maintain the MVP of kaka based on detailed information on home range size and territory overlap derived from radio-tracking studies. As one of a group of surrogate species in lowland forest ecosystems, kaka may be used to guide management decisions regarding large-scale mustelid trapping and the delineation of habitat area requiring protection in the face of proposed human developments in the region.     

Confidence intervals for expected abundance of rare species

In many ecological research studies, abundance data are skewed and contain more zeros than might be expected. Often, the aim is to model abundance in terms of covariates, and to estimate expected abundance for a given set of covariate values. An approach that has been advocated recently involves the use of a conditional model. This allows one to separately model presence and abundance given presence, which should lead to a more complete understanding as to how the covariates influence abundance. The focus of this article is on the calculation of confidence intervals for expected abundance given particular values of the covariates. The standard Wald confidence interval is symmetric, and therefore unlikely to be of much use for skewed data, where reliable confidence intervals for abundance will generally be asymmetric. The purpose of this article is to show how to calculate a profile likelihood confidence interval for expected abundance using a conditional model.     

A Bayesian Approach to Fitting Gibbs Processes with Temporal Random Effects

We consider spatial point pattern data that have been observed repeatedly over a period of time in an inhomogeneous environment. Each spatial point pattern can be regarded as a ??snapshot?? of the underlying point process at a series of times. Thus, the number of points and corresponding locations of points differ for each snapshot. Each snapshot can be analyzed independently, but in many cases there may be little information in the data relating to model parameters, particularly parameters relating to the interaction between points. Thus, we develop an integrated approach, simultaneously analyzing all snapshots within a single robust and consistent analysis. We assume that sufficient time has passed between observation dates so that the spatial point patterns can be regarded as independent replicates, given spatial covariates. We develop a joint mixed effects Gibbs point process model for the replicates of spatial point patterns by considering environmental covariates in the analysis as fixed effects, to model the heterogeneous environment, with a random effects (or hierarchical) component to account for the different observation days for the intensity function. We demonstrate how the model can be fitted within a Bayesian framework using an auxiliary variable approach to deal with the issue of the random effects component. We apply the methods to a data set of musk oxen herds and demonstrate the increased precision of the parameter estimates when considering all available data within a single integrated analysis.