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.     

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.

     

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 of Elusive Animals with DNA from Hunter-Collected Feces: Four Methods for Brown Bears

Abstract:  Noninvasive genetic methods can be used to estimate animal abundances and offer several advantages over conventional methods. Few attempts have been made, however, to evaluate the accuracy and precision of the estimates. We compared four methods of estimating population size based on fecal sampling. Two methods used rarefaction indices and two were based on capture-mark-recapture (CMR) estimators, one combining genetic and field data. Volunteer hunters and others collected 1904 fecal samples over 2 consecutive years in a large area containing a well-studied population of brown bears ( Ursus arctos ). On our 49,000-km² study area in south-central Sweden, population size estimates ranged from 378 to 572 bears in 2001 and 273 to 433 bears in 2002, depending on the method of estimation used. The estimates from the best model in the program MARK appeared to be the most accurate, based on the minimum population size estimate from radio-marked bears in a subsection of our sampling area. In addition, MARK models included heterogeneity and temporal variation in detection probabilities, which appeared to be present in our samples. All methods, though, incorrectly suggested a biased sex ratio, probably because of sex differences in detection probabilities and low overall detection probabilities. The population size of elusive animals can be estimated reliably over large areas with noninvasive genetic methods, but we stress the importance of an adequate and well-distributed sampling effort. In cases of biased sampling, calibration with independent estimates may be necessary. We recommend that this noninvasive genetic approach, using the MARK models, be used in the future in areas where sufficient numbers of volunteers can be mobilized.     

Comparison between Hansen-Hurwitz and Horvitz-Thompson estimators for adaptive cluster sampling

Thompson (1990) introduced the adaptive cluster sampling design. This sampling design has been shown to be a useful sampling method for parameter estimation of a clustered and scattered population (Roesch, 1993; Smith et al., 1995; Thompson and Seber, 1996). Two estimators, the modified Hansen-Hurwitz (HH) and Horvitz-Thompson (HT) estimators, are available to estimate the mean or total of a population. Empirical results from previous researches indicate that the modified HT estimator has smaller variance than the modified HH estimator. We analytically compare the properties of these two estimators. Some results are obtained in favor of the modified HT estimator so that practitioners are strongly recommended to use the HT estimator despite easiness of computations for the HH estimator.     

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.     

Will Observation Error and Biases Ruin the Use of Simple Extinction Models?

Abstract: Estimating the risk of extinction for populations of endangered species is an important component of conservation biology. These estimates must be made from data that contain both environmental noise in the year-to-year transitions in population size (so-called "process error"), random errors in sampling, and possible biases in sampling ( both forms of observation errors). To determine how much faith to place in estimated extinction rates, it is important to know how sensitive they are to observation error. We used three simple, commonly employed models of population dynamics to generate simulated population time series. We then combined random observation error or systematic biases with those data, fit models to the time series data, and observed how close the extinction dynamics of the fitted models compared with the dynamics of the underlying models. We found that systematic biases in sampling rarely affected estimates of extinction risk. We also found that even moderate levels of random observation error do not significantly affect extinction estimates except over a small range of process errors, corresponding to the region where extinction risk is most uncertain. With more substantial sampling error, estimates of extinction risk degraded rapidly. Field census techniques for a variety of taxa often involve observation errors within ±32% of actual population sizes. For typical time series used in conservation, therefore, we often may not need to be overly concerned about observation errors as an extra source of imperfection in our estimated extinction rates.     

Assessing plausible rates of population growth in humpback whales from life-history data

The rate of growth of any population is a quantity of interest in conservation and management and is constrained by biological factors. In this study, recent data on life-history parameters influencing rates of population growth in humpback whales, including survival, age at first parturition and calving rate are reviewed. Monte Carlo simulations are used to compute a distribution of rates of increase (ROIs) taking into account uncertainty in biological parameter estimates. Two approaches for computing juvenile survival are proposed, which taken into account along with other life-history data, resulted in the following estimates of the rate of population growth: Approach A: mean of 7.3%/year (95% CI = 3.5–10.5%/year) and Approach B: mean of 8.6%/year (95% CI = 5.0–11.4%/year). It is proposed that the upper 99% quantile of the resulting distribution of the ROI for Approach B (11.8%/year) be established as the maximum plausible ROI for humpback whales and be used in population assessment of the species. Possible sources of positive and negative biases in the present estimates are presented and include measurement error in estimation of life-history parameters, changes in the environment within the period these quantities are measured, density dependence or other natural factors. However, it is difficult to evaluate potential biases without additional data. The methods presented in this study can be applied to other species for which life-history parameters are available and are useful in assessing plausibility in the estimation of population growth rates from time series of abundance estimates.     

Estimation of population parameters from aerial counts of North American mallards: a cautionary tale.

The accuracy of population estimates strongly interferes with our ability to obtain unbiased estimates of population parameters based on analyses of time series of population fluctuations. Here we use long-term data on fluctuations in the size of Mallard populations collected as part of the May Breeding Waterfowl Survey covering a large section of North America. We assume a log-linear model of density dependence and use a hierarchical Bayesian state-space approach in which all parameters are assumed to be realizations from a common underlying distribution. Thus, parameters for different populations are not allowed to vary independently of each other. We then simulated independent time series of aerial counts, using the estimated parameters and adding various levels of observation error. These simulations showed that the estimates of stochastic population growth rate and strength of density dependence were biased even when moderate sampling errors were present. In contrast, the estimates of the environmental stochasticity and the carrying capacity were unbiased even for short time series and large observation error. Our results underline the importance of reducing the magnitude of sampling error in the design of large-scale monitoring programs of population fluctuations.     

Risk-Based Viable Population Monitoring

Abstract:  We describe risk-based viable population monitoring, in which the monitoring indicator is a yearly prediction of the probability that, within a given timeframe, the population abundance will decline below a prespecified level. Common abundance-based monitoring strategies usually have low power to detect declines in threatened and endangered species and are largely reactive to declines. Comparisons of the population's estimated risk of decline over time will help determine status in a more defensible manner than current monitoring methods. Monitoring risk is a more proactive approach; critical changes in the population's status are more likely to be demonstrated before a devastating decline than with abundance-based monitoring methods. In this framework, recovery is defined not as a single evaluation of long-term viability but as maintaining low risk of decline for the next several generations. Effects of errors in risk prediction techniques are mitigated through shorter prediction intervals, setting threshold abundances near current abundance, and explicitly incorporating uncertainty in risk estimates. Viable population monitoring also intrinsically adjusts monitoring effort relative to the population's true status and exhibits considerable robustness to model misspecification. We present simulations showing that risk predictions made with a simple exponential growth model can be effective monitoring indicators for population dynamics ranging from random walk to density dependence with stable, decreasing, or increasing equilibrium. In analyses of time-series data for five species, risk-based monitoring warned of future declines and demonstrated secure status more effectively than statistical tests for trend.     

Using the bootstrap and fast Fourier transform to estimate confidence intervals of 2D kernel densities

Kernel density estimators are often used to estimate the utilization distributions (UDs) of animals. Kernel UD estimates have a strong theoretical basis and perform well, but are usually reported without estimates of error or uncertainty. It is intuitively and theoretically appealing to estimate the sampling error in kernel UD estimates using bootstrapping. However, standard equations for kernel density estimates are complicated and computationally expensive. Bootstrapping requires computing hundreds or thousands of probability densities and is impractical when the number of observations, or the area of interest is large. We used the fast Fourier transform (FFT) and discrete convolution theorem to create a bootstrapping algorithm fast enough to run on commonly available desktop or laptop computers. Application of the FFT method to a large (n>20,000) set of radio telemetry data would provide a 99.6% reduction in computation time (i.e., 1.6 as opposed to 444 hours) for 1000 bootstrap UD estimates. Bootstrap error contours were computed using data from a radio-collared polar bear (Ursus maritimus) in the Beaufort Sea north of Alaska.     

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.     

A framework for adapting survey design through time for wildlife population assessment

Sampling strategies for monitoring the status and trends in wildlife populations are often determined before the first survey is undertaken. However, there may be little information about the distribution of the population and so the sample design may be inefficient. Through time, as data are collected, more information about the distribution of animals in the survey region is obtained but it can be difficult to incorporate this information in the survey design. This paper introduces a framework for monitoring motile wildlife populations within which the design of future surveys can be adapted using data from past surveys whilst ensuring consistency in design-based estimates of status and trends through time. In each survey, part of the sample is selected from the previous survey sample using simple random sampling. The rest is selected with inclusion probability proportional to predicted abundance. Abundance is predicted using a model constructed from previous survey data and covariates for the whole survey region. Unbiased design-based estimators of status and trends and their variances are derived from two-phase sampling theory. Simulations over the short and long-term indicate that in general more precise estimates of status and trends are obtained using this mixed strategy than a strategy in which all of the sample is retained or all selected with probability proportional to predicted abundance. Furthermore the mixed strategy is robust to poor predictions of abundance. Estimates of status are more precise than those obtained from a rotating panel design.     

Relative Contributions of Sampling Error in Initial Population Size and Vital Rates to Outcomes of Population Viability Analysis

Abstract:  We evaluated the relative contributions of sampling error (randomly chosen standard errors applied as 0–30% of parameter estimates) in initial population size and vital rates (survival and reproduction) to the outcome of a simulated population viability analysis for grizzly bears (  Ursus arctos ). Error in initial population size accounted for the largest source of variation (model II analysis of variance, F _25,5= 10.8, p = 0.00001) in simulation outcomes, explaining 60.5% of the variance. In contrast, error in vital rates contributed little to simulation outcomes ( F _25,5= 0.61, p = 0.70), accounting for only 2.4% of model variation. Reduced global variation in vital rates, as a result of independent random sampling of annual deviates for each parameter, likely contributed to the results. Errors in estimates of initial population size, if ignored in PVA, have the potential to leave managers with estimates of population persistence that are of little value for making management decisions.     

Statistical inference using stratified judgment post-stratified samples from finite populations

This paper develops statistical inference for population mean and total using stratified judgment post-stratified (SJPS) samples. The SJPS design selects a judgment post-stratified sample from each stratum. Hence, in addition to stratum structure, it induces additional ranking structure within stratum samples. SJPS is constructed from a finite population using either a with or without replacement sampling design. Inference is constructed under both randomization theory and a super population model. In both approaches, the paper shows that the estimators of population mean and total are unbiased. The paper also constructs unbiased estimators for the variance (mean square prediction error) of the sample mean (predictor of population mean), and develops confidence and prediction intervals for the population mean. The empirical evidence shows that the proposed estimators perform better than their competitors in the literature.     

Sampling variability and estimates of density dependence: a composite-likelihood approach

It is well known that sampling variability, if not properly taken into account, affects various ecologically important analyses. Statistical inference for stochastic population dynamics models is difficult when, in addition to the process error, there is also sampling error. The standard maximum-likelihood approach suffers from large computational burden. In this paper, I discuss an application of the composite-likelihood method for estimation of the parameters of the Gompertz model in the presence of sampling variability. The main advantage of the method of composite likelihood is that it reduces the computational burden substantially with little loss of statistical efficiency. Missing observations are a common problem with many ecological time series. The method of composite likelihood can accommodate missing observations in a straightforward fashion. Environmental conditions also affect the parameters of stochastic population dynamics models. This method is shown to handle such nonstationary population dynamics processes as well. Many ecological time series are short, and statistical inferences based on such short time series tend to be less precise. However, spatial replications of short time series provide an opportunity to increase the effective sample size. Application of likelihood-based methods for spatial time-series data for population dynamics models is computationally prohibitive. The method of composite likelihood is shown to have significantly less computational burden, making it possible to analyze large spatial time-series data. After discussing the methodology in general terms, I illustrate its use by analyzing a time series of counts of American Redstart (Setophaga ruticilla) from the Breeding Bird Survey data, San Joaquin kit fox (Vulpes macrotis mutica) population abundance data, and spatial time series of Bull trout (Salvelinus confluentus) redds count data.     

Design based estimation for ranked set sampling in finite populations

In this paper, we consider design-based estimation using ranked set sampling (RSS) in finite populations. We first derive the first and second-order inclusion probabilities for an RSS design and present two Horvitz–Thompson type estimators using these inclusion probabilities. We also develop an alternate Hansen–Hurwitz type estimator and investigate its properties. In particular, we show that this alternate estimator always outperforms the usual Hansen–Hurwitz type estimator in the simple random sampling with replacement design with comparable sample size. We also develop formulae for ratio estimator for all three developed estimators. The theoretical results are augmented by numerical and simulation studies as well as a case study using a well known data set. These show that RSS design can yield a substantial improvement in efficiency over the usual simple random sampling design in finite populations.     

Estimating trend precision and power to detect trends across grouped count data

Ecologists commonly use grouped or clustered count data to estimate temporal trends in counts, abundance indices, or abundance. For example, the U.S. Breeding Bird Survey data represent multiple counts of birds from within each of multiple, spatially defined routes. Despite a reliance on grouped counts, analytical methods for prospectively estimating precision of trend estimates or statistical power to detect trends that explicitly acknowledge the characteristics of grouped count data are undescribed. These characteristics include the fact that the sampling variance is an increasing function of the mean, and that sampling and group-level variance estimates are generally estimated on different scales (the sampling and log scales, respectively). We address these issues for repeated sampling of a single population using an analytical approach that has the flavor of a generalized linear mixed model, specifically that of a negative binomial-distributed count variable with random group effects. The count mean, including grand intercept, trend, and random group effects, is modeled linearly on the log scale, while sampling variance of the mean is estimated on the log scale via the delta method. Results compared favorably with those derived using Monte Carlo simulations. For example, at trend = 5% per temporal unit, differences in standard errors and in power were modest relative to those estimated by simulation (< or = /11/% and < or = /16/%, respectively), with relative differences among power estimates decreasing to < or = /7/% when power estimated by simulations was > or = 0.50. Similar findings were obtained using data from nine surveys of fingernail clams in the Mississippi River. The proposed method is suggested (1) where simulations are not practical and relative precision or power is desired, or (2) when multiple precision or power calculations are required and where the accuracy of a fraction of those calculations will be confirmed using simulations.     

Reliability of Absolute and Relative Predictions of Population Persistence Based on Time Series

Abstract:  Conventional population viability analysis (PVA) is often impractical because data are scarce for many threatened species. For this reason, simple count-based models are being advocated. The simplest of these models requires nothing more than a time series of abundance estimates, from which variance and autocorrelation in growth rate are estimated and predictions of population persistence are generated. What remains unclear, however, is how many years of data are needed to generate reliable estimates of these parameters and hence reliable predictions of persistence. By analyzing published and simulated time series, we show that several decades of data are needed. Predictions based on short time series were very unreliable mainly because limited data yielded biased, unreliable estimates of variance in growth rate, especially when growth rate was strongly autocorrelated. More optimistically, our results suggest that count-based PVA is sometimes useful for relative risk assessment (i.e., for ranking populations by extinction risk), even when time series are only a decade long. However, some conditions consistently lead to backward rankings. We explored the limited conditions under which simple count-based PVA may be useful for relative risk assessment.     

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.