Line Transect Sampling in Small and Large Regions

Melville and Welsh (2001, Biometrics 57, 1130-1137) consider an approach to line transect sampling using a separate calibration study to estimate the detection function g. They present a simulation study contrasting their results with poor results from a traditional estimator, labeled the "Buckland" estimator and referenced to Buckland et al. (1993, Distance Sampling: Estimating Abundance of Biological populations). The poor results from the "Buckland" estimator can be explained by the following observations: (i) the estimator is designated for untruncated distance data, but was applied by Melville and Welsh to truncated distance data; (ii) distance data were not pooled across transects, contrary to standard practice; and (iii) bias of the estimator was evaluated with respect to a fixed rather than a randomized grid of transect lines. We elaborate on the points above and show that the traditional methods perform to expectation when applied correctly. We also emphasize that the estimator labeled the "Buckland" estimator by Melville and Welsh is not an estimator recommended by Buckland et al. for practical survey applications.     

The Effect of Animal Movement on Line Transect Estimates of Abundance

Line transect sampling is a distance sampling method for estimating the abundance of wild animal populations. One key assumption of this method is that all animals are detected at their initial location. Animal movement independent of the transect and observer can thus cause substantial bias. We present an analytic expression for this bias when detection within the transect is certain (strip transect sampling) and use simulation to quantify bias when detection falls off with distance from the line (line transect sampling). We also explore the non-linear relationship between bias, detection, and animal movement by varying detectability and movement type. We consider animals that move in randomly orientated straight lines, which provides an upper bound on bias, and animals that are constrained to a home range of random radius. We find that bias is reduced when animal movement is constrained, and bias is considerably smaller in line transect sampling than strip transect sampling provided that mean animal speed is less than observer speed. By contrast, when mean animal speed exceeds observer speed the bias in line transect sampling becomes comparable with, and may exceed, that of strip transect sampling. Bias from independent animal movement is reduced by the observer searching further perpendicular to the transect, searching a shorter distance ahead and by ignoring animals that may overtake the observer from behind. However, when animals move in response to the observer, the standard practice of searching further ahead should continue as the bias from responsive movement is often greater than that from independent movement.     

A Model-Based Approach for Making Ecological Inference from Distance Sampling Data

Summary .  We consider a fully model-based approach for the analysis of distance sampling data. Distance sampling has been widely used to estimate abundance (or density) of animals or plants in a spatially explicit study area. There is, however, no readily available method of making statistical inference on the relationships between abundance and environmental covariates. Spatial Poisson process likelihoods can be used to simultaneously estimate detection and intensity parameters by modeling distance sampling data as a thinned spatial point process. A model-based spatial approach to distance sampling data has three main benefits: it allows complex and opportunistic transect designs to be employed, it allows estimation of abundance in small subregions, and it provides a framework to assess the effects of habitat or experimental manipulation on density. We demonstrate the model-based methodology with a small simulation study and analysis of the Dubbo weed data set. In addition, a simple ad hoc method for handling overdispersion is also proposed. The simulation study showed that the model-based approach compared favorably to conventional distance sampling methods for abundance estimation. In addition, the overdispersion correction performed adequately when the number of transects was high. Analysis of the Dubbo data set indicated a transect effect on abundance via Akaike's information criterion model selection. Further goodness-of-fit analysis, however, indicated some potential confounding of intensity with the detection function.     

Accounting for imperfect detection of groups and individuals when estimating abundance

If animals are independently detected during surveys, many methods exist for estimating animal abundance despite detection probabilities <1. Common estimators include double‐observer models, distance sampling models and combined double‐observer and distance sampling models (known as mark‐recapture‐distance‐sampling models; MRDS). When animals reside in groups, however, the assumption of independent detection is violated. In this case, the standard approach is to account for imperfect detection of groups, while assuming that individuals within groups are detected perfectly. However, this assumption is often unsupported. We introduce an abundance estimator for grouped animals when detection of groups is imperfect and group size may be under‐counted, but not over‐counted. The estimator combines an MRDS model with an N‐mixture model to account for imperfect detection of individuals. The new MRDS‐Nmix model requires the same data as an MRDS model (independent detection histories, an estimate of distance to transect, and an estimate of group size), plus a second estimate of group size provided by the second observer. We extend the model to situations in which detection of individuals within groups declines with distance. We simulated 12 data sets and used Bayesian methods to compare the performance of the new MRDS‐Nmix model to an MRDS model. Abundance estimates generated by the MRDS‐Nmix model exhibited minimal bias and nominal coverage levels. In contrast, MRDS abundance estimates were biased low and exhibited poor coverage. Many species of conservation interest reside in groups and could benefit from an estimator that better accounts for imperfect detection. Furthermore, the ability to relax the assumption of perfect detection of individuals within detected groups may allow surveyors to re‐allocate resources toward detection of new groups instead of extensive surveys of known groups. We believe the proposed estimator is feasible because the only additional field data required are a second estimate of group size.     

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.     

Estimating the encounter rate variance in distance sampling

Summary .  The dominant source of variance in line transect sampling is usually the encounter rate variance. Systematic survey designs are often used to reduce the true variability among different realizations of the design, but estimating the variance is difficult and estimators typically approximate the variance by treating the design as a simple random sample of lines. We explore the properties of different encounter rate variance estimators under random and systematic designs. We show that a design-based variance estimator improves upon the model-based estimator of Buckland et al. (2001, Introduction to Distance Sampling. Oxford: Oxford University Press, p. 79) when transects are positioned at random. However, if populations exhibit strong spatial trends, both estimators can have substantial positive bias under systematic designs. We show that poststratification is effective in reducing this bias.     

A Field Evaluation of Distance Measurement Error in Auditory Avian Point Count Surveys

ABSTRACT Detection distance is an important and common auxiliary variable measured during avian point count surveys. Distance data are used to determine the area sampled and to model the detection process using distance sampling theory. In densely forested habitats, visual detections of birds are rare, and most estimates of detection distance are based on auditory cues. Distance sampling theory assumes detection distances are measured accurately, but empirical validation of this assumption for auditory detections is lacking. We used a song playback system to simulate avian point counts with known distances in a forested habitat to determine the error structure of distance estimates based on auditory detections. We conducted field evaluations with 6 experienced observers both before and after distance estimation training. We conducted additional studies to determine the effect of height and speaker orientation (toward or away from observers) on distance estimation error. Distance estimation errors for all evaluations were substantial, although training reduced errors and bias in distance estimates by approximately 15%. Measurement errors showed a nonlinear relationship to distance. Our results suggest observers were not able to differentiate distances beyond 65 m. The height from which we played songs had no effect on distance estimation errors in this habitat. The orientation of the song source did have a large effect on distance estimation errors; observers generally doubled their distance estimates for songs played away from them compared with distance estimates for songs played directly toward them. These findings, which we based on realistic field conditions, suggest measures of uncertainty in distance estimates to auditory detections are substantially higher than assumed by most researchers. This means aural point count estimates of avian abundance based on distance methods deserve careful scrutiny because they are likely biased.     

Incorporating covariates into standard line transect analyses

An implicit assumption of standard line transect methodology is that detection probabilities depend solely on the perpendicular distance of detected objects to the transect line. Heterogeneity in detection probabilities is commonly minimized using stratification, but this may be precluded by small sample sizes. We develop a general methodology which allows the effects of multiple covariates to be directly incorporated into the estimation procedure using a conditional likelihood approach. Small sample size properties of estimators are examined via simulations. As an example the method is applied to eastern tropical Pacific dolphin sightings data.     

Experimental Investigation of Observation Error in Anuran Call Surveys

ABSTRACT Occupancy models that account for imperfect detection are often used to monitor anuran and songbird species occurrence. However, presence—absence data arising from auditory detections may be more prone to observation error (e.g., false-positive detections) than are sampling approaches utilizing physical captures or sightings of individuals. We conducted realistic, replicated field experiments using a remote broadcasting system to simulate simple anuran call surveys and to investigate potential factors affecting observation error in these studies. Distance, time, ambient noise, and observer abilities were the most important factors explaining false-negative detections. Distance and observer ability were the best overall predictors of false-positive errors, but ambient noise and competing species also affected error rates for some species. False-positive errors made up 5% of all positive detections, with individual observers exhibiting false-positive rates between 0.5% and 14%. Previous research suggests false-positive errors of these magnitudes would induce substantial positive biases in standard estimators of species occurrence, and we recommend practices to mitigate for false positives when developing occupancy monitoring protocols that rely on auditory detections. These recommendations include additional observer training, limiting the number of target species, and establishing distance and ambient noise thresholds during surveys.     

A Note on Interval Estimation of the Simple Difference in Data with Correlated Matched Pairs

When we employ cluster sampling to collect data with matched pairs, the assumption of independence between all matched pairs is not likely true. This paper notes that applying interval estimators, that do not account for the intraclass correlation between matched pairs, to estimate the simple difference between two proportions of response can be quite misleading, especially when both the number of matched pairs per cluster and the intraclass correlation between matched pairs within clusters are large. This paper develops two asymptotic interval estimators of the simple difference, that accommodate the data of cluster sampling with correlated matched pairs. This paper further applies Monte Carlo simulation to compare the finite sample performance of these estimators and demonstrates that the interval estimator, derived from a quadratic equation proposed here, can actually perform quite well in a variety of situations.     

On the importance of estimating detection probabilities from at‐sea surveys of flying seabirds

The primary and accepted method used to estimate seabird densities at sea from ships is the strip transect method, designed to correct for the effect of random directional bird movement relative to that of the ship. However, this method relies on the critical assumption that all of the birds within the survey strip are detected. We used the distance sampling method from line‐transects to estimate detection probability of a number of species of flying seabirds, and to test whether distance from the ship and bird body size affected detectability. Detection probability decreased from 0.987 (SE=0.029) to 0.269 (SE=0.035) with increasing strip half‐width from 100 to 1400 m. Detection probability also varied between size‐groups of species with strip half‐width. For all size‐groups, this probability was close to 1 for strip half‐width of 100 m, but was 0.869 (SE=0.115), 0.725 (SE=0.096) and 0.693 (SE=0.091) for strip half‐width of 300 m, a typical strip width used in seabird surveys, for respectively large, medium and small size flying seabirds. For larger strip half‐width, detection probability was higher for large sized species, intermediate for medium sized species and lower for smaller sized species. For strip half‐width larger than 100 m we suggest that more attention should be paid to testing the assumption of perfect detectability, because abundance estimates may be underestimated when this assumption is violated. Finally, the effect of the speed of travel of flying seabird on the detection probability was estimated in a simulation study, which suggests that detection probability was underestimated with increasing flying speed.     

Estimating overdispersion in sparse multinomial data

Multinomial data arise in many areas of the life sciences, such as mark-recapture studies and phylogenetics, and will often by overdispersed, with the variance being higher than predicted by a multinomial model. The quasi-likelihood approach to modeling this overdispersion involves the assumption that the variance is proportional to that specified by the multinomial model. As this approach does not require specification of the full distribution of the response variable, it can be more robust than fitting a Dirichlet-multinomial model or adding a random effect to the linear predictor. Estimation of the amount of overdispersion is often based on Pearson's statistic X² or the deviance D. For many types of study, such as mark-recapture, the data will be sparse. The estimator based on X² can then be highly variable, and that based on D can have a large negative bias. We derive a new estimator, which has a smaller asymptotic variance than that based on X², the difference being most marked for sparse data. We illustrate the numerical difference between the three estimators using a mark-recapture study of swifts and compare their performance via a simulation study. The new estimator has the lowest root mean squared error across a range of scenarios, especially when the data are very sparse.     

An Evaluation of Survey Methods for Estimating Northern Bobwhite Abundance in Southern Texas

Abstract: Distance sampling has been identified as a reliable and well-suited method for estimating northern bobwhite (Colinus virginianus) density. However, distance sampling using walked transects requires intense sampling to obtain precise estimates, thus making the technique impractical for large acreages. Researchers have addressed this limitation by either resorting to the use of indices (e.g., morning covey-call surveys) or incorporating the use of aerial surveys with distance sampling. Both approaches remain relatively untested. Our objectives were to 1) compare density estimates among morning covey-call surveys, helicopter transects, and walked transects; 2) test a critical assumption of distance sampling pertinent to helicopter surveys (i.e., all objects on line are detected); and 3) evaluate the underlying premise of morning covey-call surveys (i.e., that the no. of calling coveys correlates with bobwhite density). Our study was conducted on 3 study sites in Brooks County, Texas, USA, during October to December, 2001 to 2005. Comparisons between walked transects and morning covey-call surveys involved the entire 5-year data set, whereas helicopter transects involved only the latter 2 years. Density estimates obtained from helicopter transects were similar to walked transect estimates for both years. We documented a detection probability on the helicopter transect line of 70 ± 10.2% (% ± SE; n = 20 coveys). Morning covey-call surveys yielded similar density estimates to walked transect estimates during only 2 of 5 years, when walked transect estimates were the least accurate and precise. We detected a positive relationship (R² = 0.51; 95% CI for slope: 29.5–53.1; n = 63 observations) between covey density and number of coveys heard calling. We conclude that helicopter transects appear to be a viable alternative to walked transects for estimating density of bobwhites. Morning covey-call surveys appear to be a poor method to estimate absolute abundance and to depict general population trajectories.     

Locally efficient estimation of the quality-adjusted lifetime distribution with right-censored data and covariates

Zhao and Tsiatis (1997) consider the problem of estimation of the distribution of the quality-adjusted lifetime when the chronological survival time is subject to right censoring. The quality-adjusted lifetime is typically defined as a weighted sum of the times spent in certain states up until death or some other failure time. They propose an estimator and establish the relevant asymptotics under the assumption of independent censoring. In this paper we extend the data structure with a covariate process observed until the end of follow-up and identify the optimal estimation problem. Because of the curse of dimensionality, no globally efficient nonparametric estimators, which have a good practical performance at moderate sample sizes, exist. Given a correctly specified model for the hazard of censoring conditional on the observed quality-of-life and covariate processes, we propose a closed-form one-step estimator of the distribution of the quality-adjusted lifetime whose asymptotic variance attains the efficiency bound if we can correctly specify a lower-dimensional working model for the conditional distribution of quality-adjusted lifetime given the observed quality-of-life and covariate processes. The estimator remains consistent and asymptotically normal even if this latter submodel is misspecified. The practical performance of the estimators is illustrated with a simulation study. We also extend our proposed one-step estimator to the case where treatment assignment is confounded by observed risk factors so that this estimator can be used to test a treatment effect in an observational study.     

Double-Observer Line Transect Methods: Levels of Independence

Summary .  Double-observer line transect methods are becoming increasingly widespread, especially for the estimation of marine mammal abundance from aerial and shipboard surveys when detection of animals on the line is uncertain. The resulting data supplement conventional distance sampling data with two-sample mark–recapture data. Like conventional mark–recapture data, these have inherent problems for estimating abundance in the presence of heterogeneity. Unlike conventional mark–recapture methods, line transect methods use knowledge of the distribution of a covariate, which affects detection probability (namely, distance from the transect line) in inference. This knowledge can be used to diagnose unmodeled heterogeneity in the mark–recapture component of the data. By modeling the covariance in detection probabilities with distance, we show how the estimation problem can be formulated in terms of different levels of independence. At one extreme, full independence is assumed, as in the Petersen estimator (which does not use distance data); at the other extreme, independence only occurs in the limit as detection probability tends to one. Between the two extremes, there is a range of models, including those currently in common use, which have intermediate levels of independence. We show how this framework can be used to provide more reliable analysis of double-observer line transect data. We test the methods by simulation, and by analysis of a dataset for which true abundance is known. We illustrate the approach through analysis of minke whale sightings data from the North Sea and adjacent waters.     

A Random Field Approach to Transect Counts of Wildlife Populations

Line transect counting of a wildlife population is considered a sampling from a planar marked point process, where the marks describe the detectability of the animals. Sampling properties of transect counts and a new density estimator are derived from a counting process, which is a shot-noise field induced by the marked point process. A general formula for the sampling variance of a transect is derived and applied to compare five common types of transects. Some stereological connections of transect sampling and density estimators are shown.     

Semi‐parametric analysis of overdispersed count and metric data with varying follow‐up times: Asymptotic theory and small sample approximations

Count data are common endpoints in clinical trials, for example magnetic resonance imaging lesion counts in multiple sclerosis. They often exhibit high levels of overdispersion, that is variances are larger than the means. Inference is regularly based on negative binomial regression along with maximum‐likelihood estimators. Although this approach can account for heterogeneity it postulates a common overdispersion parameter across groups. Such parametric assumptions are usually difficult to verify, especially in small trials. Therefore, novel procedures that are based on asymptotic results for newly developed rate and variance estimators are proposed in a general framework. Moreover, in case of small samples the procedures are carried out using permutation techniques. Here, the usual assumption of exchangeability under the null hypothesis is not met due to varying follow‐up times and unequal overdispersion parameters. This problem is solved by the use of studentized permutations leading to valid inference methods for situations with (i) varying follow‐up times, (ii) different overdispersion parameters, and (iii) small sample sizes.     

Estimating the abundance of rare and elusive carnivores from photographic-sampling data when the population size is very small

Conservation and management agencies require accurate and precise estimates of abundance when considering the status of a species and the need for directed actions. Due to the proliferation of remote sampling cameras, there has been an increase in capture–recapture studies that estimate the abundance of rare and/or elusive species using closed capture–recapture estimators (C–R). However, data from these studies often do not meet necessary statistical assumptions. Common attributes of these data are (1) infrequent detections, (2) a small number of individuals detected, (3) long survey durations, and (4) variability in detection among individuals. We believe there is a need for guidance when analyzing this type of sparse data. We highlight statistical limitations of closed C–R estimators when data are sparse and suggest an alternative approach over the conventional use of the Jackknife estimator. Our approach aims to maximize the probability individuals are detected at least once over the entire sampling period, thus making the modeling of variability in the detection process irrelevant, estimating abundance accurately and precisely. We use simulations to demonstrate when using the unconditional-likelihood M ₀ (constant detection probability) closed C–R estimator with profile-likelihood confidence intervals provides reliable results even when detection varies by individual. If each individual in the population is detected on average of at least 2.5 times, abundance estimates are accurate and precise. When studies sample the same species at multiple areas or at the same area over time, we suggest sharing detection information across datasets to increase precision when estimating abundance. The approach suggested here should be useful for monitoring small populations of species that are difficult to detect.     

Interval Estimation of Simple Difference under Independent Negative Binomial Sampling

When the sample size is not large or when the underlying disease is rare, to assure collection of an appropriate number of cases and to control the relative error of estimation, one may employ inverse sampling, in which one continues sampling subjects until one obtains exactly the desired number of cases. This paper focuses discussion on interval estimation of the simple difference between two proportions under independent inverse sampling. This paper develops three asymptotic interval estimators on the basis of the maximum likelihood estimator (MLE), the uniformly minimum variance unbiased estimator (UMVUE), and the asymptotic likelihood ratio test (ALRT). To compare the performance of these three estimators, this paper calculates the coverage probability and the expected length of the resulting confidence intervals on the basis of the exact distribution. This paper finds that when the underlying proportions of cases in both two comparison populations are small or moderate (≤0.20), all three asymptotic interval estimators developed here perform reasonably well even for the pre-determined number of cases as small as 5. When the pre-determined number of cases is moderate or large (≥50), all three estimators are essentially equivalent in all the situations considered here. Because application of the two interval estimators derived from the MLE and the UMVUE does not involve any numerical iterative procedure needed in the ALRT, for simplicity we may use these two estimators without losing efficiency.     

Feral donkey Equus asinus populations on the Karpaz peninsula, Cyprus

Numerous researchers have documented the adverse effects of feral donkeys Equus asinus introduced to semi-arid ecosystems. With the release of feral donkeys and potential increasing populations in natural habitats in northern Cyprus, there is concern for negative impacts on vegetation and native species. In the north of the island, there has been only one published study of feral donkey populations, and population estimators were relatively subjective. We estimated feral donkey populations on the Karpaz Peninsula using line transect surveys and quantitative distance sampling estimators. We stratified the sampling by using 11 sample units within the study area. We evaluated potential biases associated with habitat, topography, and perpendicular distance from the transect line and found that these variables did not bias donkey detections during our surveys. Using program DISTANCE, we found that a hazard rate cosine model was the best model that described our distance data based on model selection criterion (Akaikes Information Criteria adjusted for small sample bias). Estimated effective strip width was 280.19 m and detection probability was 0.47 with this model. Estimated donkey density was 6.7 donkeys/km², and estimated total abundance was 800 donkeys for the entire 132.5 km² study area. Of 95 donkey groups detected: 16% were detected in agricultural habitats with flat topography, 9% were detected in agricultural habitats with sloped topography, 24% were detected in shrub/forest habitats with flat topography, and 51% were detected in shrub/forest habitats with sloped topography. Of 102 behavioral observations recorded (multiple behaviors were detected in groups), frequencies of behaviors were 1% bedded, 70% standing, 22% grazing, 6% moving, and 2% other. Our estimated donkey population density in the Karpaz Peninsula was >2 times densities reported in arid regions of the United States and Australia, but slightly lower than earlier density estimates reported for the Karpaz region. These estimates of feral donkey populations in the Karpaz Peninsula provide a quantitative baseline from which to make population management decisions.