Estimation of the CDF of a finite population in the presence of acalibration sample

We compare the performance of a number of estimators of the cumulative distribution function (CDF) for the following scenario: imperfect measurements are taken on an initial sample from afinite population and perfect measurements are obtained on a small calibration subset of the initial sample. The estimators we considered include two naive estimators using perfect and imperfect measurements; the ratio, difference and regression estimators for a two-phasesample; a minimum MSE estimator; Stefanski and Bay's SIMEX estimator (1996); and two proposed estimators. The proposed estimators take the form of a weighted average of perfect and imperfect measurements. They are constructed by minimizing variance among the class of weighted averages subject to an unbiasedness constraint. They differ in the manner of estimating the weight parameters. The first one uses direct sample estimates. The second one tunes the unknown parameters to an underlying normal distribution. We compare the root mean square error (RMSE) of the proposed estimator against other potential competitors through computer simulations. Our simulations show that our second estimator has the smallest RMSE among thenine compared and that the reduction in RMSE is substantial when the calibration sample is small and the error is medium or large.     

Estimation of rare and clustered population mean using stratified adaptive cluster sampling

For many clustered populations, the prior information on an initial stratification exists but the exact pattern of the population concentration may not be predicted. Under this situation, the stratified adaptive cluster sampling (SACS) may provide more efficient estimates than the other conventional sampling designs for the estimation of rare and clustered population parameters. For practical interest, we propose a generalized ratio estimator with the single auxiliary variable under the SACS design. The expressions of approximate bias and mean squared error (MSE) for the proposed estimator are derived. Numerical studies are carried out to compare the performances of the proposed generalized estimator over the usual mean and combined ratio estimators under the conventional stratified random sampling (StRS) using a real population of redwood trees in California and generating an artificial population by the Poisson cluster process. Simulation results show that the proposed class of estimators may provide more efficient results than the other estimators considered in this article for the estimation of highly clumped population.

     

Gray Whales and the Value of Monitoring Data in Implementing the U.S. Endangered Species Act

Abstract: Many scientists lament the absence of data for endangered species and argue that more funds should be spent acquiring basic information about population trends. Using 19 years of abundance estimates for the eastern North Pacific gray whale ( Eschrichtius robustus ), we sampled subsets of the original survey data to identify the number of years of data required to remove the population from the U.S. Endangered Species Act's (ESA) list of endangered and threatened wildlife. For any given duration of monitoring, we selected all possible combinations of consecutive counts. To incorporate variability in growth rates, we extracted a maximum likelihood estimator of growth rate and confidence interval about that growth rate on the assumption that the population changes can be approximated by a simple diffusion process with drift. We then applied a new approach to determine ESA status for each subset of survey data and found that a quantitative decision to delist is unambiguously supported by 11 years of data but is precariously uncertain with fewer than 10 years of data. The data needed to produce an unequivocal decision to delist gray whales cost the National Marine Fisheries Service an estimated U.S. $660,000, a surprisingly modest expense given the fact that delisting can greatly simplify regulatory constraints. This example highlights the value of population monitoring in administering the ESA and provides a compelling example of the utility of such information in identifying both imperiled and recovered species. The economic value of such data is that they provide the foundation for delisting, which could ultimately save much more money than the collection of the data would ever cost.     

Adaptive two-stage one-per-stratum sampling

We briefly describe adaptive cluster sampling designs in which the initial sample is taken according to a Markov chain one-per-stratum design (Breidt, 1995) and one or more secondary samples are taken within strata if units in the initial sample satisfy a given condition C. An empirical study of the behavior of the estimation procedure is conducted for three small artificial populations for which adaptive sampling is appropriate. The specific sampling strategy used in the empirical study was a single random-start systematic sample with predefined systematic samples within strata when the initially sampled unit in that stratum satisfies C. The bias of the Horvitz-Thompson estimator for this design is usually very small when adaptive sampling is conducted in a population for which it is suited. In addition, we compare the behavior of several alternative estimators of the standard error of the Horvitz-Thompson estimator of the population total. The best estimator of the standard error is population-dependent but it is not unreasonable to use the Horvitz-Thompson estimator of the variance. Unfortunately, the distribution of the estimator is highly skewed hence the usual approach of constructing confidence intervals assuming normality cannot be used here.     

Robust, comparable population metrics through collaborative photo-monitoring of whale sharks Rhincodon typus.

The formulation of conservation policy for species that are rare and migratory requires broad cooperation to ensure that adequate levels of standardized data collection are achieved and that the results of local analyses are comparable. Estimates of apparent survival rate, relative change in abundance, and proportions of newly marked and returning individuals can inform local management decisions while highlighting corresponding changes at other linked research stations. We have applied computer-assisted photo-identification and mark-recapture population modeling to whale sharks Rhincodon typus at Ningaloo Marine Park (NMP), Western Australia, to create a baseline trend for comparison with other regional aggregations of the species. We estimate several ecological parameters of interest, including an average apparent survival rate of 0.55 yr(-1) for sharks newly marked (new) and 0.83 yr(-1) for sharks captured in multiple seasons (philopatric). The average proportion of philopatric sharks is found to be 0.65 of the total population, and we derive an average population growth rate of 1.12 yr(-1) for them. Our analysis uncovered significant heterogeneity in capture and survival probabilities in this study population; our chosen model structures and data analysis account for these influences and demonstrate a good overall fit to the time-series data. The results show good correspondence between capture probability and an available measure of recapture effort, suggesting that unmodeled systematic effects contribute insignificantly to the model fits. We find no evidence of a decline in the whale shark population at NMP, and our results provide metrics of value to their future management. Overall, our study suggests an effective approach to analyzing and modeling mark-recapture data for a rare species using computer-assisted photo-identification and opportunistic data collection from ecotourism to ensure the quality and volume of data required for population analysis.     

Science or Slaughter: Need for Lethal Sampling of Sharks

Abstract: General consensus among scientists, commercial interests, and the public regarding the status of shark populations is leading to an increasing need for the scientific community to provide information to help guide effective management and conservation actions. Experience from other marine vertebrate taxa suggests that public, political, and media pressures will play an increasingly important part in setting research, management, and conservation priorities. We examined the potential implications of nonscientific influences on shark research. In particular, we considered whether lethal research sampling of sharks is justified. Although lethal sampling comes at a cost to a population, especially for threatened species, the conservation benefits from well‐designed studies provide essential data that cannot be collected currently in any other way. Methods that enable nonlethal collection of life‐history data on sharks are being developed (e.g., use of blood samples to detect maturity), but in the near future they will not provide widespread or significant benefits. Development of these techniques needs to continue, as does the way in which scientists coordinate their use of material collected during lethal sampling. For almost half of the known shark species there are insufficient data to determine their population status; thus, there is an ongoing need for further collection of scientific data to ensure all shark populations have a future. Shark populations will benefit most when decisions about the use of lethal sampling are made on the basis of scientific evidence that is free from individual, political, public, and media pressures.     

Estimating Population Size with Noninvasive Capture-Mark-Recapture Data

Abstract:  Estimating population size of elusive and rare species is challenging. The difficulties in catching such species has triggered the use of samples collected noninvasively, such as feces or hair, from which genetic analysis yields data similar to capture-mark-recapture (CMR) data. There are, however, two differences between classical CMR and noninvasive CMR. First, capture and recapture data are gathered over multiple sampling sessions in classical CMR, whereas in noninvasive CMR they can be obtained from a single sampling session. Second, because of genotyping errors and unlike classical CMR, there is no simple relationship between (genetic) marks and individuals in noninvasive CMR. We evaluated, through simulations, the reliability of population size estimates based on noninvasive CMR. For equal sampling efforts, we compared estimates of population size N obtained from accumulation curves, a maximum likelihood, and a Bayesian estimator. For a closed population and without sampling heterogeneity, estimates obtained from noninvasive CMR were as reliable as estimates from classical CMR. The sampling structure (single or multiple session) did not alter the results, the Bayesian estimator in the case of a single sampling session presented the best compromise between low mean squared error and a 95% confidence interval encompassing the parametric value of N in most simulations. Finally, when suitable field and lab protocols were used, genotyping errors did not substantially bias population size estimates (bias < 3.5% in all simulations). The ability to reliably estimate population size from noninvasive samples taken during a single session offers a new and useful technique for the management and conservation of elusive and rare species.     

Shifting elasmobranch community assemblage at Cocos Island—an isolated marine protected area

Fishing pressure has increased the extinction risk of many elasmobranch (shark and ray) species. Although many countries have established no‐take marine reserves, a paucity of monitoring data means it is still unclear if reserves are effectively protecting these species. We examined data collected by a small group of divers over the past 21 years at one of the world's oldest marine protected areas (MPAs), Cocos Island National Park, Costa Rica. We used mixed effects models to determine trends in relative abundance, or probability of occurrence, of 12 monitored elasmobranch species while accounting for variation among observers and from abiotic factors. Eight of 12 species declined significantly over the past 2 decades. We documented decreases in relative abundance for 6 species, including the iconic scalloped hammerhead shark (Sphyrna lewini) (?45%), whitetip reef shark (Triaenodon obesus) (?77%), mobula ray (Mobula spp.) (?78%), and manta ray (Manta birostris) (?89%), and decreases in the probability of occurrence for 2 other species. Several of these species have small home ranges and should be better protected by an MPA, which underscores the notion that declines of marine megafauna will continue unabated in MPAs unless there is adequate enforcement effort to control fishing. In addition, probability of occurrence at Cocos Island of tiger (Galeocerdo cuvier), Galapagos (Carcharhinus galapagensis), blacktip (Carcharhinus limbatus), and whale (Rhincodon typus) sharks increased significantly. The effectiveness of MPAs cannot be evaluated by examining single species because population responses can vary depending on life history traits and vulnerability to fishing pressure.     

A spatially explicit estimate of the prewhaling abundance of the endangered North Atlantic right whale

The North Atlantic right whale (NARW) (Eubalaena glacialis) is one of the world's most threatened whales. It came close to extinction after nearly a millennium of exploitation and currently persists as a population of only approximately 500 individuals. Setting appropriate conservation targets for this species requires an understanding of its historical population size, as a baseline for measuring levels of depletion and progress toward recovery. This is made difficult by the scarcity of records over this species’ long whaling history. We sought to estimate the preexploitation population size of the North Atlantic right whale and understand how this species was distributed across its range. We used a spatially explicit data set on historical catches of North Pacific right whales (NPRWs) (Eubalaena japonica) to model the relationship between right whale relative density and the environment during the summer feeding season. Assuming the 2 right whale species select similar environments, we projected this model to the North Atlantic to predict how the relative abundance of NARWs varied across their range. We calibrated these relative abundances with estimates of the NPRW total prewhaling population size to obtain high and low estimates for the overall NARW population size prior to exploitation. The model predicted 9,075–21,328 right whales in the North Atlantic. The current NARW population is thus <6% of the historical North Atlantic carrying capacity and has enormous potential for recovery. According to the model, in June–September NARWs concentrated in 2 main feeding areas: east of the Grand Banks of Newfoundland and in the Norwegian Sea. These 2 areas may become important in the future as feeding grounds and may already be used more regularly by this endangered species than is thought.     

Effects of uncertainty and variability on population declines and IUCN Red List classifications

The International Union for Conservation of Nature (IUCN) Red List Categories and Criteria is a quantitative framework for classifying species according to extinction risk. Population models may be used to estimate extinction risk or population declines. Uncertainty and variability arise in threat classifications through measurement and process error in empirical data and uncertainty in the models used to estimate extinction risk and population declines. Furthermore, species traits are known to affect extinction risk. We investigated the effects of measurement and process error, model type, population growth rate, and age at first reproduction on the reliability of risk classifications based on projected population declines on IUCN Red List classifications. We used an age‐structured population model to simulate true population trajectories with different growth rates, reproductive ages and levels of variation, and subjected them to measurement error. We evaluated the ability of scalar and matrix models parameterized with these simulated time series to accurately capture the IUCN Red List classification generated with true population declines. Under all levels of measurement error tested and low process error, classifications were reasonably accurate; scalar and matrix models yielded roughly the same rate of misclassifications, but the distribution of errors differed; matrix models led to greater overestimation of extinction risk than underestimations; process error tended to contribute to misclassifications to a greater extent than measurement error; and more misclassifications occurred for fast, rather than slow, life histories. These results indicate that classifications of highly threatened taxa (i.e., taxa with low growth rates) under criterion A are more likely to be reliable than for less threatened taxa when assessed with population models. Greater scrutiny needs to be placed on data used to parameterize population models for species with high growth rates, particularly when available evidence indicates a potential transition to higher risk categories.     

Bayes Estimation of a Distribution Function Using Ranked Set Samples

Aranked set sample (RSS), if not balanced, is simply a sample of independent order statistics gener- ated from the same underlying distribution F. Kvam and Samaniego (1994) derived maximum likelihood estimates of F for a general RSS. In many applications, including some in the environ- mental sciences, prior information about F is available to supplement the data-based inference. In such cases, Bayes estimators should be considered for improved estimation. Bayes estimation (using the squared error loss function) of the unknown distribution function F is investigated with such samples. Additionally, the Bayes generalized maximum likelihood estimator (GMLE) is derived. An iterative scheme based on the EM Algorithm is used to produce the GMLE of F. For the case of squared error loss, simple solutions are uncommon, and a procedure to find the solution to the Bayes estimate using the Gibbs sampler is illustrated. The methods are illustrated with data from the Natural Environmental Research Council of Great Britain (1975), representing water discharge of floods on the Nidd River in Yorkshire, England     

A Bayesian-weighted approach to predicting the number of newly discovered rare species

In natural ecological communities, most species are rare and thus susceptible to extinction. Consequently, the prediction and identification of rare species are of enormous value for conservation purposes. How many newly found species will be rare in the next field survey? We took a Bayesian viewpoint and used observed species abundance information in an ecological sample to develop an accurate way to estimate the number of new rare species (e.g., singletons, doubletons, and tripletons) in an additional unknown sample. A similar method has been developed for incidence-based data sets. Five seminumerical tests (3 abundance cases and 2 incidence cases) showed that our proposed Bayesian-weight estimator accurately predicted the number of new rare species with low relative bias and low relative root mean squared error and, accordingly, high accuracy. Finally, we applied the proposed estimator to 6 conservation-directed empirical data sets (3 abundance cases and 3 incidence cases) and found the prediction of new rare species was quite accurate; the 95% CI covered the true observed value very well in most cases. Our estimator performed similarly to or better than an unweighted estimator derived from Chao et al. and performed consistently better than the naïve unweighted estimator. We recommend our Bayesian-weight estimator for conservation applications, although the unweighted estimator of Chao et al. may be better under some circumstances. We provide an R package RSE (r are s pecies e stimation) at https://github.com/ecomol/RSE for implementation of the estimators.     

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

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

Robust optimal observation of a metapopulation

We consider a two-parameter continuous-time Markovian model for patch occupancy in metapopulation, and address the question of when to observe the population in order to obtain the most accurate and precise estimator of the parameters. Using the likelihood obtained from a diffusion approximation we derive a robust procedure for determining the optimal observation schedule, which is considerably simpler than would otherwise be possible. We investigate the performance of two optimality criteria, ED-optimality and a particular form of maximin-optimality. Both allow one to incorporate prior belief about the parameter values before any data is collected. Kernel density estimates of the maximum likelihood estimator are compared under the various combinations of optimality criteria and prior distributions. Our methods are illustrated with reference to a model for the spread of crown of thorns starfish (Acanthaster planci) among the 55 islands comprising the Ryukyu group in Japan.     

Population Viability Analysis with Species Occurrence Data from Museum Collections

Abstract: The most comprehensive data on many species come from scientific collections. Thus, we developed a method of population viability analysis (PVA) in which this type of occurrence data can be used. In contrast to classical PVA, our approach accounts for the inherent observation error in occurrence data and allows the estimation of the population parameters needed for viability analysis. We tested the sensitivity of the approach to spatial resolution of the data, length of the time series, sampling effort, and detection probability with simulated data and conducted PVAs for common, rare, and threatened species. We compared the results of these PVAs with results of standard method PVAs in which observation error is ignored. Our method provided realistic estimates of population growth terms and quasi‐extinction risk in cases in which the standard method without observation error could not. For low values of any of the sampling variables we tested, precision decreased, and in some cases biased estimates resulted. The results of our PVAs with the example species were consistent with information in the literature on these species. Our approach may facilitate PVA for a wide range of species of conservation concern for which demographic data are lacking but occurrence data are readily available.     

Estimating Population Size in a Continuous-time Removal Experiment with a Known Sub-population Size Ratio

We study a continuous-time removal model for estimating the population size for a population in which a sub-population size ratio is known. The maximum likelihood estimate and the optimal martingale estimate of the population size are obtained; these are shown to be equivalent. A comparison between the proposed estimator and the maximum likelihood estimate which ignores the information on the known size ratio is made, using a simulation study. The asymptotic variances of these two estimators are also obtained, and a comparison between them is made. The sensitivity of mis-specification of the known size ratio is examined. We also apply the corresponding discrete-time model to the proposed continuous-time setting, and study the efficiency of the corresponding discrete-time type estimator relative to the proposed estimator.     

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

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

Long‐Range Movement of Humpback Whales and Their Overlap with Anthropogenic Activity in the South Atlantic Ocean

Humpback whales (Megaptera novaeangliae) are managed by the International Whaling Commission as 7 primary populations that breed in the tropics and migrate to 6 feeding areas around the Antarctic. There is little information on individual movements within breeding areas or migratory connections to feeding grounds. We sought to better understand humpback whale habitat use and movements at breeding areas off West Africa, and during the annual migration to Antarctic feeding areas. We also assessed potential overlap between whale habitat and anthropogenic activities. We used Argos satellite‐monitored radio tags to collect data on 13 animals off Gabon, a primary humpback whale breeding area. We quantified habitat use for 3 cohorts of whales and used a state‐space model to determine transitions in the movement behavior of individuals. We developed a spatial metric of overlap between whale habitat and models of cumulative human activities, including oil platforms, toxicants, and shipping. We detected strong heterogeneity in movement behavior over time that is consistent with previous genetic evidence of multiple populations in the region. Breeding areas for humpback whales in the eastern Atlantic were extensive and extended north of Gabon late in the breeding season. We also observed, for the first time, direct migration between West Africa and sub‐Antarctic feeding areas. Potential overlap of whale habitat with human activities was the highest in exclusive economic zones close to shore, particularly in areas used by both individual whales and the hydrocarbon industry. Whales potentially overlapped with different activities during each stage of their migration, which makes it difficult to implement mitigation measures over their entire range. Our results and existing population‐level data may inform delimitation of populations and actions to mitigate potential threats to whales as part of local, regional, and international management of highly migratory marine species. Cuantificación de los Movimientos de Gran Amplitud y el Traslape Potencial con Actividad Antropogénica y las Ballenas Jorobadas en el Océano Atlántico Sur     

A proposal for a two-step sampling design to oversample units responding to prescribed characteristics

We introduce a novel method to extract a sample from a finite population where units with desired characteristics are over-represented. The approach is both sequential and adaptive and allows, via suitable compositions of predictive and objective functions, to target specific subsets of the population. We consider the problem of estimation and conjecture the validity of a modified Horvitz–Thompson estimator capable to account for the imbalance induced by the targeting procedure. After discussing how to apply the method to the sampling of geographically distributed units, we investigate its potential via simulations.     

Diversity,abundance, and distribution of reef sharks on outer-shelf reefs of the Great Barrier Reef,Australia

Quantifying the distribution and habitat use of sharks is critical for understanding their ecological role and for establishing appropriate conservation and management regimes. On coral reefs, particularly the Great Barrier Reef (GBR), little is known regarding the distribution of sharks across major reef habitat types. In this study, we surveyed shark populations across outer-shelf reefs of the GBR in order to determine the diversity, abundance, and distribution of reef sharks across three major coral reef habitats: (1) the reef slope, (2) the back reef and (3) the reef flat. Model selection revealed that habitat was the principal factor influencing shark distribution and abundance. Specifically, overall shark abundance and diversity were significantly higher on the reef slope (and to a lesser degree, the back reef) than the reef flat. This confirms that shark populations are not homogeneously distributed across coral reefs. Thus, the results presented herein have important implications for shark population assessments. In addition, our results highlight the potential importance of the reef slope, with high levels of live coral cover and structural complexity, for sustaining reef shark populations. As this habitat is highly susceptible to disturbance events, this study provides a useful context for predicting and understanding how environmental degradation may influence reef shark populations in the future.