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

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

Animal abundance estimation in independent observer line transect surveys

The theory of conventional line transect surveys is based on an essential assumption that 100% detection of animals right on the transect lines can be achieved. When this assumption fails, independent observer line transect surveys are used. This paper proposes a general approach, based on a conditional likelihood, which can be carried out either parametrically or nonparametrically, to estimate the abundance of non-clustered biological populations using data collected from independent observer line transect surveys. A nonparametric estimator is specifically proposed which combines the conditional likelihood and the kernel smoothing method. It has the advantage that it allows the data themselves to dictate the form of the detection function, free of any subjective choice. The bias and the variance of the nonparametric estimator are given. Its asymptotic normality is established which enables construction of confidence intervals. A simulation study shows that the proposed estimator has good empirical performance, and the confidence intervals have good coverage accuracy.     

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.     

Propagation functions for monitoring distributional changes

In environmental assessment and monitoring, a primary objective of the investigator is to describe the changes occurring in the environmentally important variables over time. Propagation functions have been proposed to describe the distributional changes occurring in the variable of interest at two different times. McDonald et al. (1992, 1995) proposed an estimator of propagation function under the assumption of normality. We conduct a detailed sensitivity analysis of inference based on the normal model. It turns out that this model is appropriate only for small departures from normality whereas, for moderate to large departures, both estimation and testing of hypothesis break down. Non-parametric estimation of the propagation function based on kernel density estimation is also considered and the robustness of the choice of bandwidth for kernel density estimation is investigated. Bootstrapping is employed to obtain confidence intervals for the propagation function and also to determine the critical regions for testing the significance of distributional changes between two sampling epochs. Also studied briefly is the mathematical form and graphical shape of the propagation function for some parametric bivariate families of distributions. Finally, the proposed estimation techniques are illustrated on a data set of tree ring widths.     

Empirical likelihood confidence intervals for adaptive cluster sampling

Adaptive cluster sampling (ACS) is an efficient sampling design for estimating parameters of rare and clustered populations. It is widely used in ecological research. The modified Hansen-Hurwitz (HH) and Horvitz-Thompson (HT) estimators based on small samples under ACS have often highly skewed distributions. In such situations, confidence intervals based on traditional normal approximation can lead to unsatisfactory results, with poor coverage properties. Christman and Pontius (Biometrics 56:503–510, 2000) showed that bootstrap percentile methods are appropriate for constructing confidence intervals from the HH estimator. But Perez and Pontius (J Stat Comput Simul 76:755–764, 2006) showed that bootstrap confidence intervals from the HT estimator are even worse than the normal approximation confidence intervals. In this article, we consider two pseudo empirical likelihood functions under the ACS design. One leads to the HH estimator and the other leads to a HT type estimator known as the Hájek estimator. Based on these two empirical likelihood functions, we derive confidence intervals for the population mean. Using a simulation study, we show that the confidence intervals obtained from the first EL function perform as good as the bootstrap confidence intervals from the HH estimator but the confidence intervals obtained from the second EL function perform much better than the bootstrap confidence intervals from the HT estimator, in terms of coverage rate.     

Importance of Accounting for Detection Heterogeneity When Estimating Abundance: the Case of French Wolves

Abstract: Assessing conservation strategies requires reliable estimates of abundance. Because detecting all individuals is most often impossible in free‐ranging populations, estimation procedures have to account for a <1 detection probability. Capture–recapture methods allow biologists to cope with this issue of detectability. Nevertheless, capture–recapture models for open populations are built on the assumption that all individuals share the same detection probability, although detection heterogeneity among individuals has led to underestimating abundance of closed populations. We developed multievent capture–recapture models for an open population and proposed an associated estimator of population size that both account for individual detection heterogeneity (IDH). We considered a two‐class mixture model with weakly and highly detectable individuals to account for IDH. In a noninvasive capture–recapture study of wolves we based on genotypes identified in feces and hairs, we found a large underestimation of population size (27% on average) occurred when IDH was ignored.     

Variances and variance estimators of the improved ratio estimators under adaptive cluster sampling

Although not design-unbiased, the ratio estimator is recognized as more efficient when a certain degree of correlation exists between the variable of primary interest and the auxiliary variable. Meanwhile, the Rao–Blackwell method is another commonly used procedure to improve estimation efficiency. Various improved ratio estimators under adaptive cluster sampling (ACS) that make use of the auxiliary information together with the Rao–Blackwellized univariate estimators have been proposed in past research studies. In this article, the variances and the associated variance estimators of these improved ratio estimators are proposed for a thorough framework of statistical inference under ACS. Performance of the proposed variance estimators is evaluated in terms of the absolute relative percentage bias and the empirical mean-squared error. As expected, results show that both the absolute relative percentage bias and the empirical mean-squared error decrease as the initial sample size increases for all the variance estimators. To evaluate the confidence intervals based on these variance estimators and the finite-population Central Limit Theorem, the coverage rate and the interval width are used. These confidence intervals suffer a disadvantage similar to that of the conventional ratio estimator. Hence, alternative confidence intervals based on a certain type of adjusted variance estimators are constructed and assessed in this article.     

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.     

Modelling temporal and spatial variability in tag reporting-rates for Newfoundland cod (<Emphasis Type="Italic">Gadus morhua</Emphasis>)

Mark–recapture experiments can be used to estimate the exploitation rate of a fishery; however, the estimate is influenced by the tag reporting-rate by the fishers. We present two methods to estimate the reporting rates in high/low reward ($100 and $10 CAD respectively) long-term cod tagging experiments. We fit two binomial logistic mixed-effect models, one with temporal auto-correlation in the reporting-rate year-effects and one with independent year-effects. We estimate reporting-rates separately for recreational and commercial fishers, and test for spatial variation using fixed-effects for spatial regions. Due to the complexity of the fishery, our models account for factors such as recapture-fishery type, fish-size and time-at-liberty. Our results indicate that the recreational fishers reporting-rate was constant at 0.51 across all regions and years. The commercial fishery showed more spatial and temporal variation, with the reporting-rates estimates lying between 0.67 and 0.87 for the independent year-effect model, and between 0.57 and 0.84 for the random walk model. Furthermore, we assessed the model performance as well as the coverage probability of nominal 95 % confidence intervals using simulations. We found that the models performed adequately; however, the nominal 95 % confidence intervals tended to be too narrow.     

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

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

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.     

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

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

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.     

Species abundance estimation using point-to-plant sampling in a design-based setting

A new species abundance estimator is proposed when point-to-plant sampling is adopted in a design-based framework. The method is based on the relationship between each species abundance and the probability density function of the relative squared point-to-plant distance. Using this result, a kernel estimator for species abundance is provided and the nearest neighbor method is suggested for bandwidth selection. The proposed estimator requires no assumptions about the species point patterns nor corrections for sampling near the edges of the study region. Moreover, the estimator shows suitable statistical properties as well as good practical performance as is shown in a simulation study.     

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

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

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.     

Bayesian estimation of age-specific bird nest survival rates with categorical covariates

The populations of many North American landbirds are showing signs of declining. Gathering information on breeding productivity allows critical assessment of population performance and helps identify good habitat-management practices. He (Biometrics (2003) 59 962–973) proposed a Bayesian model to estimate the age-specific nest survival rates. The model allows irregular visiting schedule under the assumption that the observed nests have homogeneous nest survival. Because nest survival studies are often conducted in different sites and time periods, it is not realistic to assume homogeneous nest survival. In this paper, we extend He’s model by incorporating these factors as categorical covariates. The simulation results show that the Bayesian hierarchical model can produce satisfactory estimates on nest survival and capture different factor effects. Finally the model is applied to a Missouri red-winged blackbird data set.     

确定湖泊参照状态过程中频率分析的精度估计

确定湖泊参照状态是建立湖泊水质基准的关键步骤之一。以频率分析为基础的方法,如湖泊群体分布法、频率分析法、三分法等广泛地应用于参照状态的研究中;但是,由于湖泊观测数据具有关联性以及难以确定概率分布,这些研究都未给出参照状态估计的置信区间。滑块自助法无需确定观测数据的理论概率分布,同时能很好地克服数据关联性引起的问题,给出这些方法得到的参照状态的置信区间。以太湖为例,分析了确定频率分析过程中,正态分布法和普通自助法的缺陷;结果说明这一方法适合于确定湖泊参照状态的精度。     

Removal process estimation of population size for a population with a known sex ratio

A removal model for estimating population size which uses the known population sex ratio is studied. A maximum likelihood estimate and an optimal martingale estimate of the population size are proposed. Their standard errors and large sample properties are obtained. Simulation studies are reported, and the performance of the proposed estimators are compared with the standard maximum likelihood estimator which ignores the sex ratio information. An example on a capture study of deer mice is given.     

The Reliability of Using Population Viability Analysis for Risk Classification of Species

I examine whether or not it is appropriate to use extinction probabilities generated by population viability analyses, based on best estimates for model parameters, as criteria for listing species in Red Data Book categories as recently proposed by the World Conservation Union. Such extinction probabilities are influenced by how accurately model parameters are estimated and by how accurately the models depict actual population dynamics. I evaluate the effect of uncertainty in parameter estimation through simulations. Simulations based on Steller sea lions were used to evaluate bias and precision in estimates of probability of extinction and to consider the performance of two proposed classification schemes. Extinction time estimates were biased (because of violation of the assumption of stable age distribution) and underestimated the variability of probability of extinction for a given time (primarily because of uncertainty in parameter estimation). Bias and precision in extinction probabilities are important when these probabilities are used to compare the risk of extinction between species. Suggestions are given for population viability analysis techniques that incorporate parameter uncertainty. I conclude that testing classification schemes with simulations using quantitative performance objectives should precede adoption of quantitative listing criteria.