Imperfect detection during animal surveys biases estimates of abundance and can lead to improper conclusions regarding distribution and population trends. Farnsworth et al. (2005) developed a combined distance-sampling and time-removal model for point-transect surveys that addresses both availability (the probability that an animal is available for detection; e.g., that a bird sings) and perceptibility (the probability that an observer detects an animal, given that it is available for detection). We developed a hierarchical extension of the combined model that provides an integrated analysis framework for a collection of survey points at which both distance from the observer and time of initial detection are recorded. Implemented in a Bayesian framework, this extension facilitates evaluating covariates on abundance and detection probability, incorporating excess zero counts (i.e. zero-inflation), accounting for spatial autocorrelation, and estimating population density. Species-specific characteristics, such as behavioral displays and territorial dispersion, may lead to different patterns of availability and perceptibility, which may, in turn, influence the performance of such hierarchical models. Therefore, we first test our proposed model using simulated data under different scenarios of availability and perceptibility. We then illustrate its performance with empirical point-transect data for a songbird that consistently produces loud, frequent, primarily auditory signals, the Golden-crowned Sparrow (Zonotrichia atricapilla); and for 2 ptarmigan species (Lagopus spp.) that produce more intermittent, subtle, and primarily visual cues. Data were collected by multiple observers along point transects across a broad landscape in southwest Alaska, so we evaluated point-level covariates on perceptibility (observer and habitat), availability (date within season and time of day), and abundance (habitat, elevation, and slope), and included a nested point-within-transect and park-level effect. Our results suggest that this model can provide insight into the detection process during avian surveys and reduce bias in estimates of relative abundance but is best applied to surveys of species with greater availability (e.g., breeding songbirds).
Monitoring a species of interest to determine its population size, trend, or distribution requires robust, unbiased estimators of abundance or, at a minimum, a parameter directly proportional to abundance. Incomplete detection of animals during surveys can bias population estimates and lead to incorrect assessments of conservation status or of the effectiveness of management actions (Burnham 1981, Rosenstock et al. 2002, Thompson 2002, Norvell et al. 2003, Kéry and Schmid 2004). Numerous methods exist to adjust counts of animals for incomplete detection, including distance sampling (Buckland et al. 2001, Johnson et al. 2010), time removal (Farnsworth et al. 2002, Etterson et al. 2009), repeated counts (Royle and Nichols 2003, Royle 2004), double observer (Cook and Jacobson 1979, Nichols et al. 2000), mark–recapture (Laake et al. 2011), double sampling (Bart and Earnst 2002), and some combinations thereof (e.g., Farnsworth et al. 2005, Sólymos et al. 2013). The applicability of these methods varies widely with respect to the parameter of interest, behavior of the species, characteristics of the survey area, timing of the survey, and logistical, time, and resource constraints. For example, multiple-observer methods require additional manpower; repeated surveys within a season are expensive, logistically challenging, and affected by closure assumptions; and mark–recapture methods are not always possible or practical to implement.
Addressing imperfect detection is complicated by multiple causal mechanisms. Nichols et al. (2009) identified 2 parts of the detection process during bird surveys: coverage probability and detection probability. Coverage probability is the probability that the location of the bird is within a sampling unit at the time of the survey and is the product of the probability that (1) a bird has a home range that overlaps the sampling unit (ps) and (2) is present in the study area at the time of the survey (pp) (Nichols et al. 2009). These probabilities are a function of the timing and spatial coverage of a survey, and their computation is often based on knowledge of the study design. Detection probability also consists of 2 components: availability (pa), which is the probability that an animal is present in the survey area and signaling its presence to the observer (e.g., a bird is vocalizing or is in view); and perceptibility (pd), which is the probability that an animal available for detection (e.g., a bird sings) is detected by the observer (Marsh and Sinclair 1989). Availability and perceptibility are often strongly influenced by the animal's cue-production rate, the distance between the animal and the observer, and the observer's sensory abilities; these factors, in turn, may vary significantly with other influences such as time of season, time of day, habitat, and weather conditions (e.g., Farnsworth et al. 2002, Alldredge et al. 2007c, Nichols et al. 2009). Several methods or combinations of methods can estimate both components of detection probability; however, most require either multiple observers or repeat surveys at a point within a short period to avoid violating the assumption of population closure (e.g., time-of-detection, robust design, or multiple-observer capture–recapture; Bailey et al. 2004, Alldredge et al. 2007a, Nichols et al. 2009).
Distance sampling and time removal are methods that require only a single survey and observer. For distance sampling, an observer measures the distance to each detected animal. This method is used extensively in line-transect and point sampling because of its relative efficiency in terms of cost and time (Buckland et al. 2001, Rosenstock et al. 2002, Norvell et al. 2003, Sillett et al. 2012). Conventional distance sampling will, however, produce negatively biased estimates of density if the critical assumption of perfect detection at zero distance is violated (Buckland et al. 2001). Such conditions may arise if an animal that is present at the survey point does not give a cue during the survey period (availability bias) or if the observer does not detect a cue that was given (perception bias) (Laake and Borchers 2004).
To account for availability in avian point-count surveys, Farnsworth et al. (2002) described a time-removal model (also called “time-to-detection model”) that treats subsets of the survey period as independent replicates (“occasions”) in which birds are “captured” (i.e. counted) and mentally removed from the population (i.e. not counted again) during later subperiods. As animals within a closed population are counted and removed from further consideration, the number of individuals available for detection will decrease through time. Hence, following the logic of a removal model to estimate population size (Zippin 1958), Farnsworth et al.'s (2002) model estimates the proportion of the population never detected during a survey and, by subtraction, the proportion of the population available for detection (pa).
Together, distance-sampling and time-removal methods can quantify both important components of detection. Farnsworth et al.'s (2002) method relies on birds perceived within a time interval to estimate availability and, thus, estimates the product of pa and pd (Marsh and Sinclair 1989, Nichols et al. 2009); including distance sampling allows separation of each detection component. Additionally, combining these methods allows explicit modeling of heterogeneity in detection probability. Some heterogeneity arises from individual variation in the frequency, intensity, or duration of signals to the observer, which for songbirds is often strongly related to distance from the observer, date within season, and time of day. Such heterogeneity is often ignored when modeling detection probability (Farnsworth et al. 2002, 2005), but ignoring it can positively bias estimates of detection probability (and negatively bias density estimates) because the group of detected individuals contains a greater proportion of the population that is consistently easy to detect during repeated surveys (e.g., birds closer to the observer or singing more frequently) than is included in the group of undetected individuals.
Conventional distance sampling is robust to some individual heterogeneity in pd as long as other modeling assumptions are met (Buckland et al. 2001). Perhaps the greatest benefit of combining distance sampling with a method to address availability, such as Farnsworth et al.'s (2002) time-removal model, is that it allows one to relax the assumption that detection probability on the line or at the point is perfect (i.e. p(0) = 1; Buckland et al. 2001:30). Pollock and Kendall (1987) recommended mark–recapture methods to estimate detection probability as an alternative to distance sampling because of this bias. However, heterogeneity tends to increase at greater distances from the observer (Laake et al. 2011) and may render capture–recapture-based multiple-observer methods of all but the smallest fixed-radius areas subject to such bias unless detection probability is also modeled as a function of distance.
Combining distance-sampling with time-removal methods can address this issue, along with heterogeneity, using less survey effort than multiple observers or repeated surveys. An observer need only record the time interval and estimated distance to each animal when it is first detected during a single survey. Such efficient survey methodology is particularly attractive for large-scale wildlife surveys, especially in areas where temporally repeated sampling is logistically difficult and expensive.
Farnsworth et al. (2005) first described a combined distance-sampling and time-removal model for auditory point-count surveys that assumes the following: (1) A bird that is present during a survey may be detected only if it vocalizes; and (2) detection probability, given availability, declines as a function of distance from the observer. Thus, overall detection probability p is the product of each component of detection, p = pa × pd (Farnsworth et al. 2005). They implemented the model in Program SURVIV (White 1992), but their model did not include ways to incorporate covariates on either component of detection or an integrated way to estimate abundance. Availability and perceptibility can be highly influenced by weather, observer ability, habitat, species, seasonal date, and time of day, among other factors (Alldredge et al. 2007c, Fitzpatrick et al. 2009). Although estimates of density across broad regions and survey periods may be robust to unmodeled sources of heterogeneity in detection (i.e. “pooling robust”; Buckland et al. 2001), not incorporating sources of variability at the survey-point level may lead to biased estimates of detection probability and density at more refined spatial and temporal scales (Thompson 2002). Wildlife are not distributed evenly across the landscape, and habitat and environmental factors may strongly influence the abundance of a species at a given survey point. Elucidating patterns and factors that influence detection and abundance is often of primary interest to ecologists and can provide important information to inform monitoring and management efforts, beyond how many animals are present in an area.
Royle et al. (2004) developed a hierarchical distance-sampling model for spatially replicated counts with a distance-sampling protocol to estimate and model abundance adjusted for pd (also see Hedley and Buckland 2004, Johnson et al. 2010, Sillett et al. 2012, Oedekoven et al. 2013, Schmidt and Rattenbury 2013). Etterson et al. (2009) created a similar hierarchical N-mixture model using Farnsworth et al.'s (2002) time-removal method to estimate abundance adjusted for pa. In these models, covariate effects on both abundance and detection can be modeled directly as a generalized linear model. Especially when implemented in a Bayesian framework, straightforward extensions include fixed and random effects, spatial autocorrelation, excess zeros, and replicated counts through time (Thogmartin et al. 2004, Martin et al. 2005, Etterson et al. 2009, Sauer and Link 2011, Kéry and Schaub 2012). However, because models of both Royle et al. (2004) and Etterson et al. (2009) incorporate only 1 component of detection probability (i.e. pd or pa, respectively), the modeled detection probability is really a conflated estimate of pa × pd and the resulting abundance estimates are subject to unmeasured amounts of negative bias.
Here, we extend and integrate Farnsworth et al.'s (2005) combined time-removal and distance-sampling model for point-count surveys and Royle et al.'s (2004) hierarchical distance-sampling model into a Bayesian hierarchical N-mixture framework with separate processes for abundance (i.e. the total number of birds present in the surveyed area during the survey period, N), perceptibility (pd), and availability (pa). Our new model (1) accommodates covariates on all processes, (2) allows estimates of detection-adjusted population density, (3) allows modeling of spatial autocorrelation, and (4) accounts for extra dispersion in the form of zero inflation (Table 1). To illustrate our model, we use simulated data and field examples from avian point-transect data collected across a broad landscape in Alaska. Furthermore, we implement the model in a Bayesian framework using the accessible JAGS program (Plummer 2003), allowing flexible model development by nonspecialists.
Summary of the capabilities that we incorporated into our distance-sampling and time-removal N-mixture model compared with those of similar existing models.
We created this model to accommodate single-visit point-transect data replicated at k = 1, 2, … , K points in an area of interest. At each survey point during a prescribed period (e.g., 5 min), a single observer records distance from the central observation point to each bird detected and the survey time elapsed to its initial detection (i.e. time-to-detection), at which time the bird is “removed” from further counting. For each bird detected i, radial distance is recorded into discrete classes b = 1, 2, … , B out to a maximum distance (maxd) and time-to-detection is assigned to a time interval j = 1, 2, … , J. The observed data, then, are the counts of individuals at each point (yk) and the period (ji) and distance class (bi) for individuals i = 1, 2, … , y, where y is the total number of birds detected across all spatial sample units. Although the model is formulated using discrete time and distance intervals for ease of computation, truly continuous observations can be accommodated to greater levels of precision simply by using a large number of fine intervals.
The model requires the following primary assumptions: (1) Points are placed randomly with respect to bird distribution; (2) birds are detected at their initial location prior to any movement; (3) birds are identified without error (e.g., with respect to species and without double counting at a point); (4) distances are measured accurately or observations are recorded in the appropriate distance classes; (5) pa and pd are independent; (6) the population is closed during surveys to births, deaths, immigration, and emigration; and (7) the entire population is present during surveys (i.e. probability of presence, pp = 1; Nichols et al. 2009).
This model uses Royle's (2004) N-mixture model with a novel observation-level formulation of Farnsworth et al.'s (2005) joint distance-sampling and time-removal model to estimate detection probability (Farnsworth et al. 2002). The observation-level formulation is convenient to implement in popular Bayesian analysis software (e.g., JAGS; Plummer 2003) and also allows for some flexibility to incorporate other effects at the individual level. The model is expressed in terms of the “conditional likelihood,” in which the observation model is expressed as conditional on the observed count of individuals at each sample point (yk). Then a second model component is described for yk, conditional on the population size at each sample point (Nk), which is assumed to be a random variable itself so that one can model variation among sample points in the population. Analyzing the joint likelihood (e.g., Royle et al. 2004) cannot be done in JAGS easily because the multinomial parameter Nk cannot be an unobserved random variable. However, it is straightforward to implement the model on the basis of factorization of the joint model into 3 hierarchical levels: individual-level data (i.e. observations conditional on yk), point-specific counts (i.e. yk conditional on Nk), and population size (i.e. Nk).
Level 1: Individual-level Data
We assumed independence of time-of-removal and distance so that the overall probability of detection can be expressed as the product of the probabilities of availability (pa), derived from detections during J time intervals, and perceptibility (pd), derived from detections within B distance classes. Cell probabilities π can be expressed as a categorical distribution for individual observations such that dclassi is the distance class and tintervali is the time interval of detection of individual i. Thus, the observation model, specified conditional (c) on yk, has the following 2 components:
for i = 1, 2, … , yk.
For pa, we generally followed Farnsworth et al. (2002) in constructing a time-removal model to estimate availability based on initial detections of birds within each of 3 equal time intervals, but we added a covariate model to address individual heterogeneity. We expected point-level covariates (e.g., date within season and time of day each point was surveyed) to provide useful information on the detection process for all individuals in the population. Specifically, individuals are detected with probability during each time interval and the conditional cell probabilities are defined by
where is the probability of availability in time interval j at point k and pak is the probability of an individual being available during at least 1 time interval at point k. The time-interval-specific probability of availability is calculated as = ak(1 – ak)j−1, where ak is an individual's probability of detection during 1 time interval at point k. We accounted for heterogeneity by modeling point-level covariates x affecting the probability of availability as a logistic regression logit(ak) = α + βxxk. We derived point-specific pak by summing the time-interval-specific probability of availability πajk across j (j = 1 to J), where
We also explored 2 alternative models to account for heterogeneity in availability: a simple 2-point mixture model and a combined mixture–covariate model. (1) In the simple mixture model, we considered the case (general model Mc described in Farnsworth et al. 2002) in which the population as a whole is modeled simply as a combination of 2 groups, the first of which comprises birds that are readily available for detection (e.g., dominant males that sing often or for longer duration), all of which are assumed to be detected during the first time interval (i.e. pa = 1). Individuals in the second group, comprising an expected proportion c of the population, are less available (e.g., submissive males that sing less frequently) and are detected with probability a during each time interval (where a = 1 − q; cf. Farnsworth et al. 2002). Multinomial probabilities are specified as in our covariate model above, except that they incorporate c and are not indexed by point k: for j = 1, = 1 – c(1 – a) and for j = 2 to J, = ca(1 – a)j−1. (2) In the combined model, we modeled the population as a combination of 2 groups as in the preceding simple mixture model, but then further modeled availability ak of the second (less available) group as a logistic regression function of point-level covariates, as in our covariate model. We found that both of these approaches were inferior to our covariate model during simulations and analysis of field data. In the simple mixture model, estimates of pa were similar to those from the covariate model but much less precise, with concomitant effects on precision of density estimates. In the combined model, we found problems of nonidentifiability in attempting to estimate both c and ak, and resulting estimates of pa and density were again highly imprecise. In both the simple mixture and combined models, estimates of c and a rely on the detections of individuals in group 2, thereby leading to poor estimability if pa is high or c is low, or both. Furthermore, heterogeneity in detection probabilities has been demonstrated to be problematic in analysis of capture–recapture data if there is uncertainty in the underlying distributions (e.g., beta vs. logit normal distribution), even when sample sizes are large (Link 2004). We therefore recommend and present results for pa using only our covariate model approach, which uses all detections to inform pa and includes only biologically meaningful covariates.
For modeling probability of perception, we followed a similar approach. The conditional multinomial cell probabilities for distance are constructed asBuckland et al. 2001). The scale parameter σk represents the rate of decay of g(r)bk as a function of distance for each point. Point-level covariates x on pd can be modeled as a log-linear function of σk where log(σk) = log(σ0) + βxxk. We derived point-specific pdk by summing the multinomial cell probability πdbk, the probability that an individual was detected at point k in distance bin b, across b (b = 1 to B), where
Level 2: Point-specific Counts
Because we wanted to examine goodness-of-fit of the availability component of the model independently from that of the perceptibility part of the model, we expressed the model for the point-level counts yk as the product of 2 components. First, we estimated the number of individuals in the local population that were available for sampling (navailk) as a random variable with sample size Nk and probability of availability pak where navailk ∼ Binomial(Nk, pak). Then, the observed (i.e. detected) number of individuals per point yk is a binomial random variable with sample size navailk and probability of detection pdk: yk ∼ Binomial(navailk, pdk).
Level 3: Population Size
Because abundance may vary among points in relation to measurable attributes, we modeled the population size for each point Nk as a Poisson distribution with mean expected value λ, Nk ∼ Poisson(λk) (Royle et al. 2004). Point-level covariates x affecting abundance can be incorporated into the expected value where log(λk) = α + βxxk. Density Dk is then Nk-adjusted for the area surveyed A. For example, birds detected within a 300-m radius of the observer would have
We investigated model performance by simulating data to evaluate model assumptions and examined the range of parameter values under which our model provides acceptable estimates. We simulated overdispersed point-transect data with pd < 1 and pa < 1, both of which were influenced by several covariates, and with spatial autocorrelation among points within a transect. Ignoring spatial autocorrelation can lead to underestimating standard errors and overfitting models (Legendre 1993). Thus, we included transect-specific intercepts αt as a random effect on the expected count to account for non-independence among points within a transect where log(λk) = αt + βxxk.
We modeled a population of a simulated grassland bird species as a function of moderately correlated (rmax = 0.5) covariates (i.e. habitat). Abundance was positively or negatively associated with the proportion of grass (Grass, βGrass = 1.0), agriculture (Ag, βAg = −0.5), forest (Trees, βTrees = −0.05), and wetland (Wet, βWet = 0.5; Wet2, βWet2 = −0.5) at the survey point. Furthermore, pd declined with the proportion of trees at a point (β.dTrees = −0.3) and pa declined with date within season (Date, β.aDate = −0.3). We simulated data at 100 points surveyed along 10 transects of 10 points each using baseline values of 5 distance classes, 3 equal time intervals, maximum truncation distance maxd = 300 m, pa = 0.9, pd = 0.4, and λ ∼ 9. We further examined model performance when availability and perceptibility were low (pa = 0.4, pd = 0.4, and λ ∼20). We created 500 dataset realizations for each scenario to assess bias and coverage of the 95% credible intervals (CIs) for each parameter of interest. We estimated scaled relative bias as the deviation of each realized parameter estimate minus the true value scaled as a proportion of the true value, where
We defined “coverage” as the percentage of realizations with 95% CIs that included the true value for each parameter. R code to simulate the dataset and create the JAGS model used in this example are provided in Supplemental Material Appendices A (AUK-14-11_Supplemental Material Appendix A.doc) and B (Amundson_et_al_SUPP_2_updated_1_JUL_14.doc).
We further evaluated the model using empirical survey data collected in southwest Alaska. From mid-May to mid-June, 2004–2008, 5 observers conducted unlimited-radius point counts of birds primarily in upland habitats (>100 m elevation) at 1,021 points along 169 transects in 63 randomly selected sample plots within 3 national parks: Aniakchak National Monument and Preserve, Katmai National Park and Preserve, and Lake Clark National Park and Preserve (Ruthrauff et al. 2007, Ruthrauff and Tibbitts 2009). Points were spaced ∼500 m apart along transects with a random start and oriented across habitat and elevational gradients. Habitats in the parks are largely unfragmented except by natural disturbance (e.g., volcanic eruptions and wildfires). Observers recorded exact radial distance (with laser rangefinder) and exact time to initial detection for each bird during 5-min point-transect surveys and recorded data on habitat and physiographic features at each point. Observers detected >100 species of landbirds and shorebirds, but we illustrate the model with 2 examples: Golden-crowned Sparrow (Zonotrichia atricapilla) and 2 ptarmigan species, Willow Ptarmigan (Lagopus lagopus) and Rock Ptarmigan (L. muta). Because the numbers of observations were low and the detections were primarily visual for both species of ptarmigan, we estimated both availability and perceptibility jointly, but examined species-specific habitat associations for abundance.
After an exploratory analysis of the raw distance data (Buckland et al. 2001), we truncated ∼10% of the farthest observations, which were those beyond 280 m for Golden-crowned Sparrows and beyond 450 m for ptarmigan. We then created 4 unequal distance bins that had approximately equal numbers of observations and adequately fit a half-normal density function. We divided the survey into 3 equal (100-s) periods for estimating availability. We used an analysis of variance of mean detection distance by time interval to examine the assumption of independence between distance and time intervals within the selected truncation radius. After finding evidence of increasing mean detection distance in later time intervals only for Golden-crowned Sparrow, we reran the models with data truncated at a smaller radius (200 m) within which the independence assumption was satisfied. We compared resulting density estimates to assess the effect of violating this assumption for this species.
We modeled 8 coarse-scale habitat categories, elevation, and slope as covariates on abundance. We derived elevation and slope from the National Elevation Dataset (Gesch 2007) and summarized mean values within a 150-m radius of each point. We examined elevation effects for Aniakchak National Monument and Preserve separately from the other 2 parks combined because the broad array of habitats sampled across Lake Clark and Katmai (approximately 100–1,600 m) were found to be very compressed elevationally (to ∼600 m) in Aniakchak, most likely because of the more severe weather conditions (wind, snow, and cold) typical of the Aniakchak area (Ruthrauff and Tibbitts 2009). Phenology of vegetation was relatively delayed the year Aniakchak was sampled, and the highest-elevation points were extensively snow covered (Ruthrauff and Tibbitts 2009). Across all parks, habitat categories included the following: shrub <20 cm tall and mesic herbaceous cover (Dshrubherb); shrub >20 cm tall (Shrub); bare ground and perennial ice and snow (Baresnow); open water (Water); wetlands and wet sedge (Wetland); coniferous forest consisting of white spruce (Picea glauca), Sitka spruce (P. sitchensis), or black spruce (P. mariana) (Spruce); mixed deciduous–coniferous forest (Mixed); and deciduous forest (Dec). Habitat was characterized to a 150-m radius at most points (n = 779), except those in closed forest or tall shrub habitat with limited visibility, where habitat was characterized within 50 m (n = 242). We assumed that habitat composition recorded by observers applied to larger spatial scales because habitat composition was strongly correlated at multiple scales; correlation between habitat composition at 150 m and 800 m was 0.81.
We modeled pd as a function of the following covariates: wind speed (mph), 2 observer groups (i.e. multiple observers were grouped by hearing ability), and the proportion of dense habitat cover (i.e. closed tall shrub and forest cover) within either a 50- or 150-m radius of the point. We modeled pa as a function of Julian date within season and time of day. For Golden-crowned Sparrow, we restricted analysis to detections of singing males; but for ptarmigan, we analyzed all visual and auditory detections of adults, including males, females, and those of unknown sex. Thus, to estimate total breeding density of Golden-crowned Sparrows, we assumed a 50:50 sex ratio and multiplied estimated density of males by 2. We fit transect-specific intercepts as described in the simulated data example. We assumed that either (1) abundances at higher levels of spatial nesting (e.g., transects within plots and plots within parks) were independent because of the large distances between transects, plots, and parks; or (2) the spatial autocorrelation at larger scales was related to features of the landscape, which could be alleviated with the inclusion of relevant spatially structured environmental covariates like the ones we included in our analysis (Wintle and Bardos 2006).
We accounted for an overabundance of zero counts during surveys, which can bias model fit when birds are not observed at a large proportion of survey points (Martin et al. 2005, Joseph et al. 2009). To do so, we included a zero-inflation term z multiplied by the Poisson mean λk. The population size per point is then a Poisson distribution with mean λ′, where λ′ is the product of the expected count λ and a Bernoulli draw z of the zero-inflation parameter ψ: z∼Bernoulli(ψ).
Model Implementation and Goodness-of-Fit
For simulated and field data, we conducted a Bayesian analysis in JAGS version 3.2.0 (Plummer 2003), in which we called JAGS remotely from R version 2.15.2 (R Core Team 2012). We standardized (i.e. x̄ = 0, SD = 1) all covariates to facilitate convergence. We assigned random effects including nested point-within-transect intercepts and observer groups on perceptibility as normal distributions with mean μ and precision τ (i.e. τ = 1/variance). For fixed effects, including the hyperparameter μ, we specified vague normal prior distributions with mean 0 and variance 100 for coefficients; for variances, we chose uniform priors ranging from 0 to 1,000 at the σ scale (Rota et al. 2011, Kéry and Schaub 2012). We conducted 100,000–250,000 iterations from 3 Markov chains, thinned by 1 in 50, and discarded the first 50,000–100,000 draws as burn-in. We assessed model convergence using the Gelman-Rubin potential scale reduction parameter, R̂, where R̂ = 1 at convergence (Gelman and Rubin 1992). We accepted coefficient estimates with R̂ between 1.0 and 1.1. Finally, for the availability and perceptibility components of the models, we used Bayesian P values generated from the posterior predictive distributions to assess goodness-of-fit (Gelman et al. 1996), where a P value close to 0.5 indicates a fitting model but a value close to 0 or 1 suggests doubtful fit (Kéry 2010:108). For each model parameter, we present the point estimate with 95% CIs.
Point estimates of model parameters based on posterior summaries resembled true parameter values for all model components, and coverage ranged from 0.87 to 0.98 for the default model in which availability was high (pa ∼ 0.9) and perceptibility was low (pd ∼ 0.4) and from 0.82 to 0.98 when both availability and perceptibility were low (∼0.4). Both scenarios had similar numbers of simulated detections (∼300; Table 2). Although coverage was similar for the 2 simulated datasets, the precision of the abundance estimate was lower when availability was reduced, despite similar numbers of observations (Table 2 and Figure 1). To compare results from binned (B = 5) versus essentially unbinned distance data, we analyzed the same simulated dataset (high availability) with 300 one-meter distance classes. Exploratory analysis of unbinned data resulted in similar posterior distributions of mean pd per points but was computationally less efficient, with ∼50× greater run times needed to reach convergence (i.e. ∼7 days vs. ∼3.5 hr on a computer with a 3.4-GHz processor and 16 GB memory). Similarly, when we divided the data into 10 vs. 3 periods, the resulting mean pa was similar to the baseline values but needed ∼10× greater run time to reach convergence.
Results from 500 realized datasets with 95% Bayesian credible intervals (CIs) in relation to the true value of covariate coefficients and model parameters and the proportion of simulations with 95% CIs that included the true value (coverage). We simulated data at 100 points located in 10 transects with 10 points each, using baseline values of 5 distance classes and 3 time intervals. We created 2 scenarios: high availability, where pa ∼ 0.9, pd ∼ 0.4, λ ∼ 9, and y ∼ 300 (95% CI: 254–333); and low availability, where pa ∼ 0.4, pd ∼ 0.4, λ ∼ 20, and y ∼ 300 (95% CI: 250–333). Model fit for each component of detection probability was assessed with Bayesian P values. See text for description of covariates.