Evaluation of three methods to estimate density and detectability from roadside point counts

Roadside point counts are often used to estimate trends of bird populations. The use of aural counts of birds without adjustment for detection probability, however, can lead to incorrect population trend estimates. We compared precision of estimates of density and detectability of whistling northern bobwhites (Colinus virginianus) using distance sampling, independent double-observer, and removal methods from roadside surveys. Two observers independently recorded each whistling bird heard, distance from the observer, and time of first detection at 362 call-count stops in Ohio. We examined models that included covariates for year and observer effects for each method and distance from observer effects for the double-observer and removal methods using Akaike's Information Criterion (AIC). The best model of detectability from distance sampling included observer and year effects. The best models from the removal and double-observer techniques included observer and distance effects. All 3 methods provided precise estimates of detection probability (CV = 2.4–4.4%) with a range of detectability of 0.44–0.95 for a 6-min survey. Density estimates from double-observer surveys had the lowest coefficient of variation (2005 = 3.2%, 2006 = 1.7%), but the removal method also provided precise estimates of density (2005 CV = 3.4%, 2006 CV = 4.8%), and density estimates from distance sampling were less precise (2005 CV = 9.6%, 2006 CV = 7.9%). Assumptions of distance sampling were violated in our study because probability of detecting bobwhites near the observer was <1 or the roadside survey points were not randomly distributed with respect to the birds. Distances also were not consistently recorded by individual members of observer pairs. Although double-observer surveys provided more precise estimates, we recommend using the removal method to estimate detectability and abundance of bobwhites. The removal method provided precise estimates of density and detection probability and requires half the personnel time as double-observer surveys. Furthermore, the likelihood of meeting model assumptions is higher for the removal survey than with independent double-observers. © 2011 The Wildlife Society.     

Accommodating unmodeled heterogeneity in double-observer distance sampling surveys

Mark-recapture models applied to double-observer distance sampling data neglect the information on relative detectability of objects contained in the distribution of observed distances. A difference between the observed distribution and that predicted by the mark-recapture model is symptomatic of a failure of the assumption of zero correlation between detection probabilities implicit in the mark-recapture model. We develop a mark-recapture-based model that uses the observed distribution to relax this assumption to zero correlation at only one distance. We demonstrate its usefulness in coping with unmodeled heterogeneity using data from an aerial survey of crabeater seals in the Antarctic.     

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.     

Maximum likelihood analysis of logistic regression models with incomplete covariate data and auxiliary information

This article presents a new method for maximum likelihood estimation of logistic regression models with incomplete covariate data where auxiliary information is available. This auxiliary information is extraneous to the regression model of interest but predictive of the covariate with missing data. Ibrahim (1990, Journal of the American Statistical Association 85, 765-769) provides a general method for estimating generalized linear regression models with missing covariates using the EM algorithm that is easily implemented when there is no auxiliary data. Vach (1997, Statistics in Medicine 16, 57-72) describes how the method can be extended when the outcome and auxiliary data are conditionally independent given the covariates in the model. The method allows the incorporation of auxiliary data without making the conditional independence assumption. We suggest tests of conditional independence and compare the performance of several estimators in an example concerning mental health service utilization in children. Using an artificial dataset, we compare the performance of several estimators when auxiliary data are available.     

Detection Probabilities for Ground-Based Breeding Waterfowl Surveys

ABSTRACT Current methods for conducting ground-based surveys of breeding waterfowl pairs make the unlikely assumption that detection probabilities are constant and approach 100%. To test this assumption, we conducted independent double-observer pair surveys in North Dakota, USA, to evaluate sources of variation in detection probabilities for 8 common species of prairie-nesting ducks. An experienced observer had 0.911 detection probability averaged over all 8 species (range = 0.866-0.944) versus 0.790 (range = 0.537-0.890) for a novice observer. Detection probabilities also varied substantially among species, but patterns were not consistent between observers. Detection probabilities declined as number of ducks per wetland increased, presumably due to difficulty in identifying large numbers of flushing ducks. Other covariates affecting detection probabilities included size of social groups, precipitation, survey methodology (roadside vs. walk-up), cloud cover, time of day, and amount of wetland vegetation, but these covariates only affected detection probabilities by 2–5%. Our results demonstrated that the assumption of 100% detection probabilities for ground-based waterfowl counts was clearly false and surveys based on this erroneous assumption underestimated population size by 10–29%. We recommend that future investigators measure detection probabilities explicitly by using double-observer methodologies.     

Abundance estimates to inform butterfly management: double-observer versus distance sampling

Abundance estimates are used to establish baselines, set recovery targets, and assess management actions, all of which are essential aspects of evidence-based natural resource management. For many rare butterflies, these estimates do not exist, and conservation decisions rely instead on expert opinion. Using Bartram’s scrub-hairstreak (Strymon acis bartrami, US Endangered) as a case study, we present a novel comparison of two methods that permit the incorporation of detection probabilities into abundance estimates, distance sampling and double-observer surveys. Additionally we provide a framework for establishing a systematic sampling scheme for monitoring very rare butterflies. We surveyed butterflies monthly in 2013, increasing intensity to weekly when butterflies were detected. We conducted 19 complete, island-wide surveys on Big Pine Key in the Florida Keys, detecting a total of 59 Bartram’s scrub-hairstreaks across all surveys. Peak daily abundances were similar as estimated with distance sampling, 156 butterflies (95 % CI 65–247), and double-observer, 169 butterflies (95 % CI 65–269). Selecting a method for estimating abundance of rare species involves evaluating trade-offs between methods. Distance sampling requires at least 40 detections, but only one observer, while double-observer requires only 10 detections, but two observers. Double-observer abundance estimates agreed with distance sampling estimates, which suggests that double-observer is a reasonable alternative method to use for estimating detection probability and abundance for rare species that cannot be surveyed with other, more commonly used methods.     

Statistical Efficiency in Distance Sampling

Distance sampling is a technique for estimating the abundance of animals or other objects in a region, allowing for imperfect detection. This paper evaluates the statistical efficiency of the method when its assumptions are met, both theoretically and by simulation. The theoretical component of the paper is a derivation of the asymptotic variance penalty for the distance sampling estimator arising from uncertainty about the unknown detection parameters. This asymptotic penalty factor is tabulated for several detection functions. It is typically at least 2 but can be much higher, particularly for steeply declining detection rates. The asymptotic result relies on a model which makes the strong assumption that objects are uniformly distributed across the region. The simulation study relaxes this assumption by incorporating over-dispersion when generating object locations. Distance sampling and strip transect estimators are calculated for simulated data, for a variety of overdispersion factors, detection functions, sample sizes and strip widths. The simulation results confirm the theoretical asymptotic penalty in the non-overdispersed case. For a more realistic overdispersion factor of 2, distance sampling estimation outperforms strip transect estimation when a half-normal distance function is correctly assumed, confirming previous literature. When the hazard rate model is correctly assumed, strip transect estimators have lower mean squared error than the usual distance sampling estimator when the strip width is close enough to its optimal value (± 75% when there are 100 detections; ± 50% when there are 200 detections). Whether the ecologist can set the strip width sufficiently accurately will depend on the circumstances of each particular study.     

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.     

Mixture Models for Distance Sampling Detection Functions

We present a new class of models for the detection function in distance sampling surveys of wildlife populations, based on finite mixtures of simple parametric key functions such as the half-normal. The models share many of the features of the widely-used “key function plus series adjustment” (K+A) formulation: they are flexible, produce plausible shapes with a small number of parameters, allow incorporation of covariates in addition to distance and can be fitted using maximum likelihood. One important advantage over the K+A approach is that the mixtures are automatically monotonic non-increasing and non-negative, so constrained optimization is not required to ensure distance sampling assumptions are honoured. We compare the mixture formulation to the K+A approach using simulations to evaluate its applicability in a wide set of challenging situations. We also re-analyze four previously problematic real-world case studies. We find mixtures outperform K+A methods in many cases, particularly spiked line transect data (i.e., where detectability drops rapidly at small distances) and larger sample sizes. We recommend that current standard model selection methods for distance sampling detection functions are extended to include mixture models in the candidate set.     

Separating Components of the Detection Process With Combined Methods: An Example With Northern Bobwhite

Abstract: There are various methods of estimating detection probabilities for avian point counts. Distance and multiple-observer methods require the sometimes unlikely assumption that all birds in the population are available (i.e., sing or are visible) during a count, but the time-of-detection method allows for the possibility that some birds are unavailable during the count. We combined the dependent double-observer method with the time-of-detection method and obtained field-based estimates of the components of detection probability for northern bobwhite (Colinus virginianus). Our approach was a special case of Pollock's robust capture-recapture design where the probability that a bird does not sing is analogous to the probability that an animal is a temporary emigrant. Top models indicated that observers' detection probabilities were similar (0.78–0.84) if bobwhite were available, but bobwhite only had an approximately 0.61 probability of being available during a 2.5-minute sampling interval. Additionally, observers' detection probabilities increased substantially after the initial encounter with an individual bobwhite (analogous to a trap-happy response on the part of the observer). A simulated data set revealed that the combined method was precise when availability and detection given availability were substantially lower. Combined methods approaches can provide critical information for researchers and land managers to make decisions regarding survey length and personnel requirements for point-count-based surveys.     

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.     

An open spatial capture–recapture model for estimating density,movement, and population dynamics from line‐transect surveys

The purpose of many wildlife population studies is to estimate density, movement, or demographic parameters. Linking these parameters to covariates, such as habitat features, provides additional ecological insight and can be used to make predictions for management purposes. Line‐transect surveys, combined with distance sampling methods, are often used to estimate density at discrete points in time, whereas capture–recapture methods are used to estimate movement and other demographic parameters. Recently, open population spatial capture–recapture models have been developed, which simultaneously estimate density and demographic parameters, but have been made available only for data collected from a fixed array of detectors and have not incorporated the effects of habitat covariates. We developed a spatial capture–recapture model that can be applied to line‐transect survey data by modeling detection probability in a manner analogous to distance sampling. We extend this model to a) estimate demographic parameters using an open population framework and b) model variation in density and space use as a function of habitat covariates. The model is illustrated using simulated data and aerial line‐transect survey data for North Atlantic right whales in the southeastern United States, which also demonstrates the ability to integrate data from multiple survey platforms and accommodate differences between strata or demographic groups. When individuals detected from line‐transect surveys can be uniquely identified, our model can be used to simultaneously make inference on factors that influence spatial and temporal variation in density, movement, and population dynamics.     

Rejection of Schmidt et al.'s estimators for bear population size

Aerial distance sampling of bears to estimate population size has been used throughout many parts of Alaska. The distance sampling models are complex since they need to account for undetected bears and differences in detection probabilities. This will require covariates and mark‐recapture data. The models proposed by Schmidt et al. do not use covariates or mark‐recapture data and are inappropriate for these surveys.     

Analysis of capture-recapture models with individual covariates using data augmentation

Summary .  I consider the analysis of capture–recapture models with individual covariates that influence detection probability. Bayesian analysis of the joint likelihood is carried out using a flexible data augmentation scheme that facilitates analysis by Markov chain Monte Carlo methods, and a simple and straightforward implementation in freely available software. This approach is applied to a study of meadow voles ( Microtus pennsylvanicus ) in which auxiliary data on a continuous covariate (body mass) are recorded, and it is thought that detection probability is related to body mass. In a second example, the model is applied to an aerial waterfowl survey in which a double-observer protocol is used. The fundamental unit of observation is the cluster of individual birds, and the size of the cluster (a discrete covariate) is used as a covariate on detection probability.     

Pooling robustness in distance sampling: Avoiding bias when there is unmodelled heterogeneity

The pooling robustness property of distance sampling results in unbiased abundance estimation even when sources of variation in detection probability are not modeled. However, this property cannot be relied upon to produce unbiased subpopulation abundance estimates when using a single pooled detection function that ignores subpopulations. We investigate by simulation the effect of differences in subpopulation detectability upon bias in subpopulation abundance estimates. We contrast subpopulation abundance estimates using a pooled detection function with estimates derived using a detection function model employing a subpopulation covariate. Using point transect survey data from a multispecies songbird study, species-specific abundance estimates are compared using pooled detection functions with and without a small number of adjustment terms, and a detection function with species as a covariate. With simulation, we demonstrate the bias of subpopulation abundance estimates when a pooled detection function is employed. The magnitude of the bias is positively related to the magnitude of disparity between the subpopulation detection functions. However, the abundance estimate for the entire population remains unbiased except when there is extreme heterogeneity in detection functions. Inclusion of a detection function model with a subpopulation covariate essentially removes the bias of the subpopulation abundance estimates. The analysis of the songbird point count surveys shows some bias in species-specific abundance estimates when a pooled detection function is used. Pooling robustness is a unique property of distance sampling, producing unbiased abundance estimates at the level of the study area even in the presence of large differences in detectability between subpopulations. In situations where subpopulation abundance estimates are required for data-poor subpopulations and where the subpopulations can be identified, we recommend the use of subpopulation as a covariate to reduce bias induced in subpopulation abundance estimates.     

Accounting for preferential sampling in species distribution models

Species distribution models (SDMs) are now being widely used in ecology for management and conservation purposes across terrestrial, freshwater, and marine realms. The increasing interest in SDMs has drawn the attention of ecologists to spatial models and, in particular, to geostatistical models, which are used to associate observations of species occurrence or abundance with environmental covariates in a finite number of locations in order to predict where (and how much of) a species is likely to be present in unsampled locations. Standard geostatistical methodology assumes that the choice of sampling locations is independent of the values of the variable of interest. However, in natural environments, due to practical limitations related to time and financial constraints, this theoretical assumption is often violated. In fact, data commonly derive from opportunistic sampling (e.g., whale or bird watching), in which observers tend to look for a specific species in areas where they expect to find it. These are examples of what is referred to as preferential sampling, which can lead to biased predictions of the distribution of the species. The aim of this study is to discuss a SDM that addresses this problem and that it is more computationally efficient than existing MCMC methods. From a statistical point of view, we interpret the data as a marked point pattern, where the sampling locations form a point pattern and the measurements taken in those locations (i.e., species abundance or occurrence) are the associated marks. Inference and prediction of species distribution is performed using a Bayesian approach, and integrated nested Laplace approximation (INLA) methodology and software are used for model fitting to minimize the computational burden. We show that abundance is highly overestimated at low abundance locations when preferential sampling effects not accounted for, in both a simulated example and a practical application using fishery data. This highlights that ecologists should be aware of the potential bias resulting from preferential sampling and account for it in a model when a survey is based on non‐randomized and/or non‐systematic sampling.     

Improvements on GPS Location Cluster Analysis for the Prediction of Large Carnivore Feeding Activities: Ground-Truth Detection Probability and Inclusion of Activity Sensor Measures

Animal space use studies using GPS collar technology are increasingly incorporating behavior based analysis of spatio-temporal data in order to expand inferences of resource use. GPS location cluster analysis is one such technique applied to large carnivores to identify the timing and location of feeding events. For logistical and financial reasons, researchers often implement predictive models for identifying these events. We present two separate improvements for predictive models that future practitioners can implement. Thus far, feeding prediction models have incorporated a small range of covariates, usually limited to spatio-temporal characteristics of the GPS data. Using GPS collared cougar (Puma concolor) we include activity sensor data as an additional covariate to increase prediction performance of feeding presence/absence. Integral to the predictive modeling of feeding events is a ground-truthing component, in which GPS location clusters are visited by human observers to confirm the presence or absence of feeding remains. Failing to account for sources of ground-truthing false-absences can bias the number of predicted feeding events to be low. Thus we account for some ground-truthing error sources directly in the model with covariates and when applying model predictions. Accounting for these errors resulted in a 10% increase in the number of clusters predicted to be feeding events. Using a double-observer design, we show that the ground-truthing false-absence rate is relatively low (4%) using a search delay of 2–60 days. Overall, we provide two separate improvements to the GPS cluster analysis techniques that can be expanded upon and implemented in future studies interested in identifying feeding behaviors of large carnivores.     

Hazard rate models with covariates.

Many problems, particularly in medical research, concern the relationship between certain covariates and the time to occurrence of an event. The hazard or failure rate function provides a conceptually simple representation of time to occurrence data that readily adapts to include such generalizations as competing risks and covariates that vary with time. Two partially parametric models for the hazard function are considered. These are the proportional hazards model of Cox (1972) and the class of log-linear or accelerated failure time models. A synthesis of the literature on estimation from these models under prospective sampling indicates that, although important advances have occurred during the past decade, further effort is warranted on such topics as distribution theory, tests of fit, robustness, and the full utilization of a methodology that permits non-standard features. It is further argued that a good deal of fruitful research could be done on applying the same models under a variety of other sampling schemes. A discussion of estimation from case-control studies illustrates this point.     

Regression analysis of longitudinal binary data with time-dependent environmental covariates: bias and efficiency

Generalized estimating equations (Liang and Zeger, 1986) is a widely used, moment-based procedure to estimate marginal regression parameters. However, a subtle and often overlooked point is that valid inference requires the mean for the response at time t to be expressed properly as a function of the complete past, present, and future values of any time-varying covariate. For example, with environmental exposures it may be necessary to express the response as a function of multiple lagged values of the covariate series. Despite the fact that multiple lagged covariates may be predictive of outcomes, researchers often focus interest on parameters in a 'cross-sectional' model, where the response is expressed as a function of a single lag in the covariate series. Cross-sectional models yield parameters with simple interpretations and avoid issues of collinearity associated with multiple lagged values of a covariate. Pepe and Anderson (1994), showed that parameter estimates for time-varying covariates may be biased unless the mean, given all past, present, and future covariate values, is equal to the cross-sectional mean or unless independence estimating equations are used. Although working independence avoids potential bias, many authors have shown that a poor choice for the response correlation model can lead to highly inefficient parameter estimates. The purpose of this paper is to study the bias-efficiency trade-off associated with working correlation choices for application with binary response data. We investigate data characteristics or design features (e.g. cluster size, overall response association, functional form of the response association, covariate distribution, and others) that influence the small and large sample characteristics of parameter estimates obtained from several different weighting schemes or equivalently 'working' covariance models. We find that the impact of covariance model choice depends highly on the specific structure of the data features, and that key aspects should be examined before choosing a weighting scheme.     

Improving inference for aerial surveys of bears: The importance of assumptions and the cost of unnecessary complexity

Obtaining useful estimates of wildlife abundance or density requires thoughtful attention to potential sources of bias and precision, and it is widely understood that addressing incomplete detection is critical to appropriate inference. When the underlying assumptions of sampling approaches are violated, both increased bias and reduced precision of the population estimator may result. Bear (Ursus spp.) populations can be difficult to sample and are often monitored using mark‐recapture distance sampling (MRDS) methods, although obtaining adequate sample sizes can be cost prohibitive. With the goal of improving inference, we examined the underlying methodological assumptions and estimator efficiency of three datasets collected under an MRDS protocol designed specifically for bears. We analyzed these data using MRDS, conventional distance sampling (CDS), and open‐distance sampling approaches to evaluate the apparent bias‐precision tradeoff relative to the assumptions inherent under each approach. We also evaluated the incorporation of informative priors on detection parameters within a Bayesian context. We found that the CDS estimator had low apparent bias and was more efficient than the more complex MRDS estimator. When combined with informative priors on the detection process, precision was increased by >50% compared to the MRDS approach with little apparent bias. In addition, open‐distance sampling models revealed a serious violation of the assumption that all bears were available to be sampled. Inference is directly related to the underlying assumptions of the survey design and the analytical tools employed. We show that for aerial surveys of bears, avoidance of unnecessary model complexity, use of prior information, and the application of open population models can be used to greatly improve estimator performance and simplify field protocols. Although we focused on distance sampling‐based aerial surveys for bears, the general concepts we addressed apply to a variety of wildlife survey contexts.