Distance Sampling Models Research Articles

Effects of Broad-Scale Conservation on Northern Bobwhite Populations in Agricultural Landscapes

Private land initiatives such as the Conservation Reserve Enhancement Program (CREP) are avenues for broad-scale northern Bobwhite (Colinus virginianus) conservation. The CREP in Kentucky established 40,468 ha of native prairie grasses and riparian corridors in the Green River Basin. Northern bobwhite responses to similar conservation measures at local scales (i.e., the site of implementation) have been positive; however, the geographic extent of the influence of private land initiatives on populations is less understood. Our objectives were to investigate landscape-scale effects of CREP on northern bobwhite populations. Using a stratified random sampling design, 254 roadside point counts were performed over 5 years throughout the Green River Basin along a gradient of landscape-scale CREP density. Local-scale (500 m radius) CREP density was held constant at monitoring points. We analyzed data using an openpopulation distance sampling model that included estimators of appropriate landscape scale and strength of density dependence. Population response to the CREP was positive and outweighed conservation footprint. Our results suggest that broad-scale conservation can influence wildlife populations outside of targeted areas. Concurrently, because the majority of land in the Eastern U.S. is privately owned, private land conservation initiatives present an effective strategy for promoting wildlife population recovery across large areas. Our future directions with this research include improving model estimators, determining mechanisms behind landscape-scale effects of CREP, and determining the influence of the spatial arrangement of landscape features on local populations.

Distance sampling detection functions: 2D or not 2D?

Conventional distance sampling (CDS) methods assume that animals are uniformly distributed in the vicinity of lines or points. But when animals move in response to observers before detection, or when lines or points are not located randomly, this assumption may fail. By formulating distance sampling models as survival models, we show that using time to first detection in addition to perpendicular distance (line transect surveys) or radial distance (point transect surveys) allows estimation of detection probability, and hence density, when animal distribution in the vicinity of lines or points is not uniform and is unknown. We also show that times to detection can provide information about failure of the CDS assumption that detection probability is 1 at distance zero. We obtain a maximum likelihood estimator of line transect survey detection probability and effective strip half-width using times to detection, and we investigate its properties by simulation in situations where animals are nonuniformly distributed and their distribution is unknown. The estimator is found to perform well when detection probability at distance zero is 1. It allows unbiased estimates of density to be obtained in this case from surveys in which there has been responsive movement prior to animals coming within detectable range. When responsive movement continues within detectable range, estimates may be biased but are likely less biased than estimates from methods that assuming no responsive movement. We illustrate by estimating primate density from a line transect survey in which animals are known to avoid the transect line, and a shipboard survey of dolphins that are attracted to it.

Predicting the offshore distribution and abundance of marine birds with a hierarchical community distance sampling model.

Proposed offshore wind energy development on the Atlantic Outer Continental Shelf has brought attention to the need for baseline studies of the distribution and abundance of marine birds. We compiled line transect data from 15 shipboard surveys (June 2012-April 2014), along with associated remotely sensed habitat data, in the lower Mid-Atlantic Bight off the coast of Delaware, Maryland, and Virginia, USA. We implemented a recently developed hierarchical community distance sampling model to estimate the seasonal abundance of 40 observed marine bird species. Treating each season separately, we included six oceanographic parameters to estimate seabird abundance: three static (distance to shore, slope, sediment grain size) and three dynamic covariates (sea surface temperature [SST], salinity, primary productivity). We expected that avian bottom-feeders would respond primarily to static covariates that characterize seafloor variability, and that surface-feeders would respond more to dynamic covariates that quantify surface productivity. We compared the variation in species-specific and community-level responses to these habitat features, including for rare species, and we predicted species abundance across the study area. While several protected species used the study area in summer during their breeding season, estimated abundance and observed diversity were highest for nonbreeding species in winter. Distance to shore was the most common significant predictor of abundance, and thus useful in estimating the potential exposure of marine birds to offshore development. In many cases, our expectations based on feeding ecology were confirmed, such as in the first winter season, when bottom-feeders associated significantly with the three static covariates (distance to shore, slope, and sediment grain size), and surface-feeders associated significantly with two dynamic covariates (SST, primary productivity). However, other cases revealed significant relationships between static covariates and surface-feeders (e.g., distance to shore) and between dynamic covariates and bottom-feeders (e.g., primary productivity during that same winter). More generally, we found wide interannual, seasonal, and interspecies variation in habitat relationships with abundance. These results show the importance of quantifying detection and determining the ecological drivers of a community's distribution and abundance, within and among species, for evaluating the potential exposure of marine birds to offshore development.

Ungulate distributions in a rangeland with competitors, predators and pastoralists

Summary African rangelands support diverse ungulate communities whose member species exhibit unique combinations of body morphology and behaviour that have evolved over millions of years to limit the effects of competition and predation on fitness, and more recently, to cope with people and livestock. The mechanisms by which native ungulates cope with the combined effects of competition, predation and human disturbance are poorly understood. Addressing this knowledge gap will help guide management and conservation plans for large mammal communities outside of strictly protected areas. We conducted animal counts on line transects and used a spatially explicit distance sampling model to test the influence of bottom–up effects (vegetation), top–down effects from African lion Panthera leo and spotted hyena Crocuta crocuta and human disturbance (livestock density, occupied human settlements) on native ungulate densities and distributions across a multiple‐use rangeland in Kenya. We examined five species that varied in body morphology and foraging strategy, including two large‐bodied grazers (zebra Equus quagga; wildebeest Connochaetes taurinus), two medium‐bodied, mixed‐feeders (Grant's gazelle Nanger granti; impala Aepyceros melampus) and one very large‐bodied browser (giraffe Giraffa camelopardalis tippelskirchi). Densities and distributions of all species varied across available land‐use types, with bottom–up effects (vegetation) and human disturbance having the strongest overall effects. Responses varied by ungulate foraging strategy. Distributions of grazers and the browser (giraffe) changed seasonally, while distributions of mixed‐feeders and a grazer (zebra) changed diurnally. At broad spatial scales, ungulates did not respond to long‐term variation in predation risk though they have been shown to respond behaviourally to short‐term variation in risk. Synthesis and applications. Our results revealed that native ungulates in this multiple‐use rangeland occur at densities comparable to many flagship protected area populations. We attribute observed densities to spatial heterogeneity across this landscape that included variation in human land use (community conservation area, seasonal settlements and livestock grazing, permanent settlements) and vegetation (grassland, bushland, woodland) that changed through time (seasonally, diurnally). This variation allowed ungulates to carve out ecological niches while reducing exposure to the potential limiting effects of competition and human disturbance. For some species, securing access to forage resources and avoiding human disturbance may be associated with costs through increased risk of predation. Our research reveals the trade‐offs native ungulates make to cope with changes in forage availability, human disturbance and predation risk, providing important insights to help guide native ungulate conservation efforts in multiple‐use rangelands.

A hierarchical distance sampling model to estimate abundance and covariate associations of species and communities

SummaryDistance sampling is a common survey method in wildlife studies, because it allows accounting for imperfect detection. The framework has been extended to hierarchical distance sampling (HDS), which accommodates the modelling of abundance as a function of covariates, but rare and elusive species may not yield enough observations to fit such a model.We integrate HDS into a community modelling framework that accommodates multi‐species spatially replicated distance sampling data. The model allows species‐specific parameters, but these come from a common underlying distribution. This form of information sharing enables estimation of parameters for species with sparse data sets that would otherwise be discarded from analysis. We evaluate the performance of the model under varying community sizes with different species‐specific abundances through a simulation study. We further fit the model to a seabird data set obtained from shipboard distance sampling surveys off the East Coast of the USA.Comparing communities comprised of 5, 15 or 30 species, bias of all community‐level parameters and some species‐level parameters decreased with increasing community size, while precision increased. Most species‐level parameters were less biased for more abundant species. For larger communities, the community model increased precision in abundance estimates of rarely observed species when compared to single‐species models. For the seabird application, we found a strong negative association of community and species abundance with distance to shore. Water temperature and prey density had weak effects on seabird abundance. Patterns in overall abundance were consistent with known seabird ecology.The community distance sampling model can be expanded to account for imperfect availability, imperfect species identification or other missing individual covariates. The model allowed us to make inference about ecology of species communities, including rarely observed species, which is particularly important in conservation and management. The approach holds great potential to improve inference on species communities that can be surveyed with distance sampling.

Combining survey methods to estimate abundance and transience of migratory birds among tropical nonbreeding habitats

ABSTRACT Estimating population abundance for territorial species is challenging because individuals often differ in behavior (e.g., transience, multiterritoriality), and thus in detectability. How well prevailing methods detect and quantify individuals using multiple strategies is rarely addressed. In our effort to efficiently quantify avian abundance and transience among diverse nonbreeding habitats, we combined ‘unmarked' (distance sampling) with traditional ‘marked' (territory mapping) survey methods using a migratory passerine, the American Redstart (Setophaga ruticilla), wintering in 3 habitats in Jamaica from 2010 to 2012. We predicted that the ‘unmarked' survey method would detect not only all known (marked) territorial individuals, but also individuals that were transient or nonterritorial in the same habitats. Comparisons of abundance estimates generated by our best distance sampling (unmarked) model with territory mapping data identified high proportions of transient individuals (up to 50%) in 2...

Surveying Two Endangered Primate Species (Alouatta palliata aequatorialis and Cebus aequatorialis) in the Pacoche Marine and Coastal Wildlife Refuge, West Ecuador

Accurate information on the distribution, demography, and conservation status of endangered populations in threatened habitats are essential to provide the basis for conservation actions and management plans. Neglect of western Ecuador by biologists has resulted in a paucity of information of primate populations in the region. Capuchins (Cebus spp.) and howlers (Alouatta sp.) occur in the Tumbes–Choco–Magdalena hotspot along the Pacific coast. We conducted the first primate survey in Pacoche Marine and Coastal Wildlife Refuge, one of the few protected areas in western Ecuador. Although the Pacoche refuge is protected, illegal activities inside the area include slash-and-burn agriculture and natural resource extraction. We surveyed 18 1-km transects within the protected area between April and July 2012 to evaluate the effects of anthropogenic disturbance and habitat characteristics on population densities of the Critically Endangered Ecuadorian white-fronted capuchin (Cebus aequatorialis) and the Vulnerable mantled howler (Alouatta palliata aequatorialis). We confirmed the presence of C. aequatorialis through direct observation on three occasions outside transects. The low detection rate of C. aequatorialis underscores the need for immediate conservation action for this species. Using a hierarchical distance sampling model, we predicted group size as a proxy of probability of detection and found that canopy cover explained group density of A. p. aequatorialis, following a half-normal distribution. We estimated the mean density of A. p. aequatorialis as 12.4 ind./km2 and the total population to be 621.5 individuals. The correlation between the density of A. p. aequatorialis and canopy cover underlines the need to preserve the remaining forest and its connectivity. Our results also highlight the importance of including vegetation structure in primate censuses.

A Unimodal Model for Double Observer Distance Sampling Surveys.

Distance sampling is a widely used method to estimate animal population size. Most distance sampling models utilize a monotonically decreasing detection function such as a half-normal. Recent advances in distance sampling modeling allow for the incorporation of covariates into the distance model, and the elimination of the assumption of perfect detection at some fixed distance (usually the transect line) with the use of double-observer models. The assumption of full observer independence in the double-observer model is problematic, but can be addressed by using the point independence assumption which assumes there is one distance, the apex of the detection function, where the 2 observers are assumed independent. Aerially collected distance sampling data can have a unimodal shape and have been successfully modeled with a gamma detection function. Covariates in gamma detection models cause the apex of detection to shift depending upon covariate levels, making this model incompatible with the point independence assumption when using double-observer data. This paper reports a unimodal detection model based on a two-piece normal distribution that allows covariates, has only one apex, and is consistent with the point independence assumption when double-observer data are utilized. An aerial line-transect survey of black bears in Alaska illustrate how this method can be applied.

An open-population hierarchical distance sampling model.

Modeling population dynamics while accounting for imperfect detection is essential to monitoring programs. Distance sampling allows estimating population size while accounting for imperfect detection, but existing methods do not allow for estimation of demographic parameters. We develop a model that uses temporal correlation in abundance arising from underlying population dynamics to estimate demographic parameters from repeated distance sampling surveys. Using a simulation study motivated by designing a monitoring program for Island Scrub-Jays (Aphelocoma insularis), we investigated the power of this model to detect population trends. We generated temporally autocorrelated abundance and distance sampling data over six surveys, using population rates of change of 0.95 and 0.90. We fit the data generating Markovian model and a mis-specified model with a log-linear time effect on abundance, and derived post hoc trend estimates from a model estimating abundance for each survey separately. We performed these analyses for varying numbers of survey points. Power to detect population changes was consistently greater under the Markov model than under the alternatives, particularly for reduced numbers of survey points. The model can readily be extended to more complex demographic processes than considered in our simulations. This novel framework can be widely adopted for wildlife population monitoring.

A Unifying Model for Capture–Recapture and Distance Sampling Surveys of Wildlife Populations

A fundamental problem in wildlife ecology and management is estimation of population size or density. The two dominant methods in this area are capture–recapture (CR) and distance sampling (DS), each with its own largely separate literature. We develop a class of models that synthesizes them. It accommodates a spectrum of models ranging from nonspatial CR models (with no information on animal locations) through to DS and mark-recapture distance sampling (MRDS) models, in which animal locations are observed without error. Between these lie spatially explicit capture–recapture (SECR) models that include only capture locations, and a variety of models with less location data than are typical of DS surveys but more than are normally used on SECR surveys. In addition to unifying CR and DS models, the class provides a means of improving inference from SECR models by adding supplementary location data, and a means of incorporating measurement error into DS and MRDS models. We illustrate their utility by comparing inference on acoustic surveys of gibbons and frogs using only capture locations, using estimated angles (gibbons) and combinations of received signal strength and time-of-arrival data (frogs), and on a visual MRDS survey of whales, comparing estimates with exact and estimated distances. Supplementary materials for this article are available online.

Estimating sperm whale (Physeter macrocephalus) daily abundance from a shore-based survey within the Kaikoura submarine canyon, New Zealand

The Kaikoura submarine canyon (New Zealand) is one of the few places in the world where male sperm whales (Physeter macrocephalus) can be found year round close to the shore. The objective of this study was to estimate the daily abundance of sperm whale within the study area. Positions of sperm whales were measured from a shore-based station over a 2-year study (2010–2012) which included information from 4613 sperm whale sightings. Distance sampling and mark-resight models were used to estimate daily abundance. Results showed a mean of four (SEM = 0.13) individuals present in the study area at any given time. These results differ from, and are much lower than, those from previous studies. A decline in the daily number of individuals suggests that a cautious management approach is necessary. Our study illustrated the application of distance sampling and mark-resight methods from a shore-based station.

Bayesian Methods for Hierarchical Distance Sampling Models

The few distance sampling studies that use Bayesian methods typically consider only line transect sampling with a half-normal detection function. We present a Bayesian approach to analyse distance sampling data applicable to line and point transects, exact and interval distance data and any detection function possibly including covariates affecting detection probabilities. We use an integrated likelihood which combines the detection and density models. For the latter, densities are related to covariates in a log-linear mixed effect Poisson model which accommodates correlated counts. We use a Metropolis-Hastings algorithm for updating parameters and a reversible jump algorithm to include model selection for both the detection function and density models. The approach is applied to a large-scale experimental design study of northern bobwhite coveys where the interest was to assess the effect of establishing herbaceous buffers around agricultural fields in several states in the US on bird densities. Results were compared with those from an existing maximum likelihood approach that analyses the detection and density models in two stages. Both methods revealed an increase of covey densities on buffered fields. Our approach gave estimates with higher precision even though it does not condition on a known detection function for the density model.

Occupancy Patterns of Regionally Declining Grassland Sparrow Populations in a Forested Pennsylvania Landscape

Organisms can be affected by processes in the surrounding landscape outside the boundary of habitat areas and by local vegetation characteristics. There is substantial interest in understanding how these processes affect populations of grassland birds, which have experienced substantial population declines. Much of our knowledge regarding patterns of occupancy and density stem from prairie systems, whereas relatively little is known regarding how occurrence and abundance of grassland birds vary in reclaimed surface mine grasslands. Using distance sampling and single-season occupancy models, we investigated how the occupancy probability of Grasshopper (Ammodramus savannarum) and Henslow's Sparrows (A. henslowii) on 61 surface mine grasslands (1591 ha) in Pennsylvania changed from 2002 through 2011 in response to landscape, grassland, and local vegetation characteristics . A subset (n = 23; 784 ha) of those grasslands were surveyed in 2002, and we estimated changes in sparrow density and vegetation across 10 years. Grasshopper and Henslow's Sparrow populations declined 72% and 49%, respectively from 2002 to 2011, whereas overall woody vegetation density increased 2.6 fold. Henslow's Sparrows avoided grasslands with perimeter-area ratios ≥0.141 km/ha and woody shrub densities ≥0.04 shrubs/m(2). Both species occupied grasslands ≤13 ha, but occupancy probability declined with increasing grassland perimeter-area ratio and woody shrub density. Grassland size, proximity to nearest neighboring grassland (x = 0.2 km), and surrounding landscape composition at 0.5, 1.5, and 3.0 km were not parsimonious predictors of occupancy probability for either species. Our results suggest that reclaimed surface mine grasslands, without management intervention, are ephemeral habitats for Grasshopper and Henslow's Sparrows. Given the forecasted decline in surface coal production for Pennsylvania, it is likely that both species will continue to decline in our study region for the foreseeable future.

Calibrating indices of avian density from non‐standardized survey data: making the most of a messy situation

SummaryThe analysis of large heterogeneous data sets of avian point‐count surveys compiled across studies is hindered by a lack of analytical approaches that can deal with detectability and variation in survey protocols.We reformulated removal models of avian singing rates and distance sampling models of the effective detection radius (EDR) to control for the effects of survey protocol and temporal and environmental covariates on detection probabilities.We estimated singing rates andEDRfor 75 boreal forest songbird species and found that survey protocol, especially point‐count radius, explained most of the variation in detectability. However, environmental and temporal covariates (date, time, vegetation) affected singing rates andEDRfor 73% and 59% of species, respectively.Unadjusted survey counts increased by an average of 201% from a 5‐min, 50‐m radius survey to a 10‐min, 100‐m radius survey (n= 75 species). This variability was decreased to 8·5% using detection probabilities estimated from a combination of removal and distance sampling models.Our modelling approach reduced computation when fitting complex models to large data sets and can be used with a wide range of statistical techniques for inference and prediction of avian densities.

Markov-Modulated Nonhomogeneous Poisson Processes for Modeling Detections in Surveys of Marine Mammal Abundance

We consider Markov-modulated nonhomogeneous Poisson processes for modeling sightings of marine mammals in shipboard or aerial surveys. In such surveys, detection of an animal is possible only when it surfaces, and with some species a substantial proportion of animals is missed because they are diving and thus not available for detection. This needs to be adequately accounted for to avoid biased abundance estimates. The tendency of surfacing events of marine mammals to occur in clusters motivates consideration of the flexible class of Markov-modulated Poisson processes in this context. We embed these models in distance sampling models, introducing nonhomogeneity in the process to account for the fact that the observer's probability of detecting an animal decreases with increasing distance to the animal. We derive approximate expressions for the likelihood of Markov-modulated nonhomogeneous Poisson processes that enable us to estimate the model parameters through numerical maximum likelihood. The performance of the approach is investigated in an extensive simulation study, and applications to pilot and beaked whale tag data as well as to minke whale tag and survey data demonstrate its relevance in abundance estimation.

Hierarchical distance‐sampling models to estimate population size and habitat‐specific abundance of an island endemic

Population size and habitat-specific abundance estimates are essential for conservation management. A major impediment to obtaining such estimates is that few statistical models are able to simultaneously account for both spatial variation in abundance and heterogeneity in detection probability, and still be amenable to large-scale applications. The hierarchical distance-sampling model of J. A. Royle, D. K. Dawson, and S. Bates provides a practical solution. Here, we extend this model to estimate habitat-specific abundance and rangewide population size of a bird species of management concern, the Island Scrub-Jay (Aphelocoma insularis), which occurs solely on Santa Cruz Island, California, USA. We surveyed 307 randomly selected, 300 m diameter, point locations throughout the 250-km2 island during October 2008 and April 2009. Population size was estimated to be 2267 (95% CI 1613-3007) and 1705 (1212-2369) during the fall and spring respectively, considerably lower than a previously published but statistically problematic estimate of 12 500. This large discrepancy emphasizes the importance of proper survey design and analysis for obtaining reliable information for management decisions. Jays were most abundant in low-elevation chaparral habitat; the detection function depended primarily on the percent cover of chaparral and forest within count circles. Vegetation change on the island has been dramatic in recent decades, due to release from herbivory following the eradication of feral sheep (Ovis aries) from the majority of the island in the mid-1980s. We applied best-fit fall and spring models of habitat-specific jay abundance to a vegetation map from 1985, and estimated the population size of A. insularis was 1400-1500 at that time. The 20-30% increase in the jay population suggests that the species has benefited from the recovery of native vegetation since sheep removal. Nevertheless, this jay's tiny range and small population size make it vulnerable to natural disasters and to habitat alteration related to climate change. Our results demonstrate that hierarchical distance-sampling models hold promise for estimating population size and spatial density variation at large scales. Our statistical methods have been incorporated into the R package unmarked to facilitate their use by animal ecologists, and we provide annotated code in the Supplement.

Using binomial distance-sampling models to estimate the effective detection radius of point-count surveys across boreal Canada

ABSTRACT. We used binomial distance-sampling models to estimate the effective detection radius (EDR) of point-count surveys across boreal Canada. We evaluated binomial models based on 0–50 m and >50 m distance categories for goodness-of-fit and sensitivities to variation in survey effort and habitats sampled. We also compared binomial EDRs to Partners in Flight's maximum detection distances (MDD) to determine differences in landbird population sizes derived from each. Binomial EDRs had a small positive bias (4%) averaged across 86 species and a large positive bias (30–82%) for two species when compared with EDRs estimated using multinomial distance sampling. Patterns in binomial EDRs were consistent with how bird songs attenuate in relation to their frequencies and transmission through different habitats. EDR varied 12% among habitats and increased 17% when birds were counted to an unlimited distance, compared with a limited distance of 100 m. The EDR did not vary with the duration of surveys, and densiti...

Using distance sampling and hierarchical models to improve estimates of Dall's sheep abundance

AbstractManagement of large mammal populations has often been based on aerial minimum count surveys that are uncorrected for incomplete detection and lack estimates of precision. These limitations can be particularly problematic for Dall's sheep (Ovis dalli dalli) due to the high cost of surveys and variation in detection probability across time and space. The limitations of these methods have been recognized for some time, but previously proposed alternatives for sheep surveys proved to be too costly and logistically unfeasible in most circumstances (Udevitz et al. 2006). We assessed the potential for a combination of distance sampling surveys and a hierarchical modeling approach to provide a more efficient means for estimating Dall's sheep abundance by conducting aerial contour transect surveys over all sheep habitat in Gates of the Arctic National Park and Preserve (GAAR), Alaska in 2009 and 2010. We estimated the population of Dall's sheep was 8,412 (95% CI: 6,517–11,090) and 10,072 (95% CI 8,081–12,520) in 2009 and 2010, respectively. Abundance within the Itkillik Preserve area within GAAR was 1,898 (95% CI: 1,421–2,578) and 1,854 (95% CI: 1,342–2,488) in 2009 and 2010, respectively. Estimates of lamb abundance in 2010 were more than double those of 2009 after correcting for detection bias related to group size, suggesting that the apparent estimate of lambs in the population may be biased in some years depending on the degree of aggregation. Overall, the contour transect surveys were feasible logistically, cost 70–80% less than minimum count surveys, and produced precise estimates of abundance, indicating that the application of these methods could be used effectively to increase the statistical rigor and spatial extent of Dall's sheep abundance surveys throughout Alaska. These methods could be used to improve the assessment of long‐term trends in populations and productivity and provide valuable information for harvest management at both local and landscape scales at reduced costs in comparison to traditional minimum count surveys. © 2011 The Wildlife Society.

A non-technical overview of spatially explicit capture–recapture models

Most capture–recapture studies are inherently spatial in nature, with capture probabilities depending on the location of traps relative to animals. The spatial component of the studies has until recently, however, not been incorporated in statistical capture–recapture models. This paper reviews capture–recapture models that do include an explicit spatial component. This is done in a non-technical way, omitting much of the algebraic detail and focussing on the model formulation rather than on the estimation methods (which include inverse prediction, maximum likelihood and Bayesian methods). One can view spatially explicit capture–recapture (SECR) models as an endpoint of a series of spatial sampling models, starting with circular plot survey models and moving through conventional distance sampling models, with and without measurement errors, through mark–recapture distance sampling (MRDS) models. This paper attempts a synthesis of these models in what I hope is a style accessible to non-specialists, placing SECR models in the context of other spatial sampling models.

Inference on population size in binomial detectability models

Many models for biological populations, including simple mark-recapture models and distance sampling models, involve a binomially distributed number, n, of observations x1, …, xn on members of a population of size N. Two popular estimators of (N, θ), where θ is a vector parameter, are the maximum likelihood estimator and the conditional maximum likelihood estimator based on the conditional distribution of x1, …, xn given n. We derive the large-N asymptotic distributions of and ⁠, and give formulae for the biases of and ⁠. We show that the difference is, remarkably, of order 1 and we give a simple formula for the leading part of this difference. Simulations indicate that in many cases this formula is very accurate and that confidence intervals based on the asymptotic distribution have excellent coverage. An extension to product-binomial models is given.