Model Description

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 (max_d) 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 (y_k) and the period (j_i) and distance class (b_i) 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) p_a and p_d 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, p_p = 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 (y_k). Then a second model component is described for y_k, conditional on the population size at each sample point (N_k), 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 N_k 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 y_k), point-specific counts (i.e. y_k conditional on N_k), and population size (i.e. N_k).

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 (p_a), derived from detections during J time intervals, and perceptibility (p_d), derived from detections within B distance classes. Cell probabilities π can be expressed as a categorical distribution for individual observations such that dclass_i is the distance class and tinterval_i is the time interval of detection of individual i. Thus, the observation model, specified conditional (^c) on y_k, has the following 2 components:

for i = 1, 2, … , y_k.

For p_a, 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 p_ak 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 = a_k(1 – a_k)^j−1, where a_k 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(a_k) = α + β_xx_k. We derived point-specific p_ak by summing the time-interval-specific probability of availability π_ajk 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 M_c 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. p_a = 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 a_k 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 p_a 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 a_k, and resulting estimates of p_a 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 p_a 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 p_a using only our covariate model approach, which uses all detections to inform p_a 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 as

.

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 y_k as the product of 2 components. First, we estimated the number of individuals in the local population that were available for sampling (navail_k) as a random variable with sample size N_k and probability of availability p_ak where navail_k ∼ Binomial(N_k, p_ak). Then, the observed (i.e. detected) number of individuals per point y_k is a binomial random variable with sample size navail_k and probability of detection p_dk: y_k ∼ Binomial(navail_k, p_dk).

Level 3: Population Size

Because abundance may vary among points in relation to measurable attributes, we modeled the population size for each point N_k as a Poisson distribution with mean expected value λ, N_k ∼ Poisson(λ_k) (Royle et al. 2004). Point-level covariates x affecting abundance can be incorporated into the expected value where log(λ_k) = α + β_xx_k. Density D_k is then N_k-adjusted for the area surveyed A. For example, birds detected within a 300-m radius of the observer would have

Simulated Dataset

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 p_d < 1 and p_a < 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 α_t as a random effect on the expected count to account for non-independence among points within a transect where log(λ_k) = α_t + β_xx_k.

We modeled a population of a simulated grassland bird species as a function of moderately correlated (r_max = 0.5) covariates (i.e. habitat). Abundance was positively or negatively associated with the proportion of grass (Grass, β_Grass = 1.0), agriculture (Ag, β_Ag = −0.5), forest (Trees, β_Trees = −0.05), and wetland (Wet, β_Wet = 0.5; Wet², β_Wet2 = −0.5) at the survey point. Furthermore, p_d declined with the proportion of trees at a point (β.d_Trees = −0.3) and p_a declined with date within season (Date, β.a_Date = −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 max_d = 300 m, p_a = 0.9, p_d = 0.4, and λ ∼ 9. We further examined model performance when availability and perceptibility were low (p_a = 0.4, p_d = 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).

Field Study

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 p_d 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 p_a 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 λ_k. 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.

Simulated Dataset

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 (p_a ∼ 0.9) and perceptibility was low (p_d ∼ 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 p_d 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 p_a was similar to the baseline values but needed ∼10× greater run time to reach convergence.

TABLE 2.

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 p_a ∼ 0.9, p_d ∼ 0.4, λ ∼ 9, and y ∼ 300 (95% CI: 254–333); and low availability, where p_a ∼ 0.4, p_d ∼ 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.

FIGURE 1.

Smoothed frequency distributions of parameter estimates for (A) abundance, (B) perceptibility, and (C) availability under 2 scenarios of mean availability of individuals for detection during point-transect surveys: high (p_a ∼ 0.9, solid lines) and low (p_a ∼ 0.4, dashed lines). Parameter estimates from 500 realized datasets were centered by subtracting estimates from true simulated values (truth — realized) and scaled by dividing by the mean true value of each simulation. Scaling the estimates provides the relative uncertainty around each parameter estimate in relation to true values. For example, a value of 0.2 is equivalent to stating that the estimated value was ∼20% greater than the true value. We evaluated the following 2 scenarios at 100 points, adjusting abundance to obtain similar numbers of detections under the 2 levels of availability: high availability (λ ∼ 5, y ∼ 300, p_a ∼ 0.9, p_d ∼ 0.4, N ∼ 900) and low availability (λ ∼ 20, y ∼ 300, p_a ∼ 0.4, p_d ∼ 0.4, N ∼ 1900) , where λ = expected abundance per point, y = number of individuals detected, p_a = availability, p_d = perceptibility, and N = total population size.

Field Results

For this analysis we included 536 male Golden-crowned Sparrows detected within a maximum truncation distance of 280 m and 162 ptarmigan (n_willow = 92, n_rock = 70) during 1,021 point-transect surveys in southwest Alaska. After detections were binned into 4 distance classes, detection functions appeared to be generally appropriate, with little clustering or spikes at intermediate distances. Markov chains in models reached convergence (R_max < 1.1), and Bayesian P values ranged from 0.33 for ptarmigan availability to 0.67 for sparrow perceptibility, which suggests that the model adequately fit the data. Mean availability for detection during the 5-min surveys (p_a) was 0.95 (95% CI: 0.93–0.97) for sparrows and 0.64 (95% CI: 0.42–0.85) for ptarmigan. Cumulative perceptibility (p_d, i.e. the probability of detecting an individual within the maximum truncation distance, given that it was available to be detected) was 0.39 (95% CI: 0.33–0.46) for sparrows within 280 m and 0.33 (95% CI: 0.24–0.41) for ptarmigan within 450 m; p_d within 100 m of the observer was 0.87 for sparrows and 0.93 for ptarmigan. Perceptibility was negatively influenced for sparrows by the amount of dense cover at the survey point (Figure 2B), but there was no evidence of cover effects for ptarmigan, and no suggestion of wind-speed effect or substantial difference between the 2 observer groups for either ptarmigan species (Figures 2A and 3A; Table 3). Cumulative perceptibility at mean values for wind speed and dense cover were slightly greater for observer group 1 than for group 2 for both sparrows (group 1 p_d = 0.40, 95% CI: 0.35–0.47; group 2 p_d = 0.36, 95% CI: 0.31–0.41) and ptarmigan (group 1 p_d = 0.34, 95% CI: 0.26–0.45; group 2 p_d = 0.31, 95% CI: 0.24–0.39), but CIs overlapped broadly (Table 3). Availability (p_a) of sparrows for detection decreased as the season progressed (Figure 2C), and availability of ptarmigan decreased with time of day (Figure 3B and Table 3).

FIGURE 2.

Modeled relationships between covariates and (A–C) components of detection probability and (D–E) predicted abundance per point for Golden-crowned Sparrows observed during 1,021 point-transect surveys in 3 national parks in southwest Alaska. (A) Fitted detection functions for observer group 1 (solid line) and observer group 2 (dashed line) in relation to frequencies of observations in each distance bin scaled by detection distance. (B) Perceptibility, given availability for detection (p_d), declined with proportion of dense habitat cover (closed tall shrub and forest) at a point and averaged 0.39 (95% credible interval [CI]: 0.33–0.46) within a 280-m radius of the observer. (C) Availability for detection, given presence at a point, (p_a), averaged 0.95 (95% CI: 0.93–0.97) during the 5-min surveys and declined with season date (Julian date within year; x̄ = 144; May 14). Predicted abundance per point within a 280-m radius (∼25 ha) averaged 2.9 adults (95% CI: 1.0–7.5) and (D) increased with the proportion of shrub > 20 cm tall and (E) deciduous forest within a 50-m or 150-m buffer around each point and (F) decreased with elevation in Katmai and Lake Clark. We provide predicted values per point (points) and loess-smoothed trends (solid lines) with 95% CIs (dashed lines).

FIGURE 3.

Modeled relationships between covariates and components of detection probability for Willow and Rock ptarmigan (combined) observed during 1,021 point-transect surveys in 3 national parks in southwest Alaska. (A) Fitted detection functions for observer group 1 (solid line) and observer group 2 (dashed line) in relation to frequencies of observations in each distance bin scaled by detection distance. (B) Availability of ptarmigan for detection, given presence at a point, (p_a), averaged 0.64 (95% credible interval [CI]: 0.43–0.85) during the 5-min surveys and declined with time of day of the survey (x̄ = 11:15 Alaska Daylight Time). Mean perceptibility of ptarmigan, given availability for detection, (p_d), was 0.34 (95% CI: 0.24–0.39) within a 450-m radius of the observer; CIs of all modeled covariates for perceptibility contained zero.

TABLE 3.

Mean coefficients and derived parameters with 95% credible intervals (CIs) from posterior distributions of Bayesian hierarchical models that estimated breeding densities of Golden-crowned Sparrows and Rock and Willow ptarmigan in 3 national parks in southwest Alaska. Observers counted birds during 1,021 point-transect surveys from 2004 to 2008. We divided observers into 2 groups (Obs1 and Obs2) by ability and provide scale parameters on detection (σ) for each group. We report perceptibility (p_d) and availability (p_a) of birds detected as well as a zero-inflation parameter (z), and Bayesian P values for the 2 components of detection probability (i.e. measure of goodness-of-fit). Bold denotes covariates with 95% CIs that do not contain zero (i.e. a measure of significance). See text for description of covariates.

For Golden-crowned Sparrows, abundance was positively associated with shrub habitat >20 cm tall and with deciduous forest and showed a quadratic relationship with elevation in Katmai and Lake Clark (Figure 2D–2F and Table 3). Abundance of both species of ptarmigan was negatively associated with mixed and coniferous forest types (Figure 4A–4D and Table 3). Both ptarmigan species showed a quadratic relationship with elevation in Katmai and Lake Clark, but this relationship held only for Rock Ptarmigan in Aniakchak. Predicted abundance peaked at lower elevations in both regions for Willow Ptarmigan than for Rock Ptarmigan (Figure 4E–4H and Table 3). The estimated zero-inflation factor z was 1.00 (95% CI: 0.99–1.00) for sparrows and 0.99 (95% CI: 0.81–1.00) for ptarmigan, which indicated that there was not an excess of true zero counts (Martin et al. 2005). Mean predicted density of individuals (ha⁻¹) across points from 2004 to 2008 was 0.117 (95% CI: 0.042–0.306) for Golden-crowned Sparrow, 0.007 (95% CI: 0.001–0.031) for Rock Ptarmigan, and 0.006 (95% CI: 0.001–0.026) for Willow Ptarmigan. For Golden-crowned Sparrows detected within max_d = 280 m, mean detection distance increased by ∼40 m from the first to the third period (F = 4.50, df = 2 and 534, P < 0.001), violating the assumption of independence between p_d and p_a. When we reran the models with max_d = 200 m, within which detection distance was independent of time (F = 1.06, df = 2 and 443, P = 0.29), we obtained a similar but less precise estimate of density (mean = 0.121, 95% CI: 0.023–0.324).

FIGURE 4.

Modeled relationships between covariates and predicted abundance per point for Willow Ptarmigan (WIPT) and Rock Ptarmigan (ROPT) observed during 1,021 point-transect surveys in 3 national parks in southwest Alaska. Mean abundance per 64-ha point (450-m radius) was 0.35 adults (95% credible interval [CI]: 0.07–1.58) for Willow Ptarmigan and 0.44 adults (95% CI: 0.09–1.92) for Rock Ptarmigan. Across all parks, abundance of both species declined with the proportion of (A–B) mixed forest and (C–D) coniferous forest within a 50-m or 150-m buffer around each point. Abundance of the 2 species differed in relation to elevation in (E–F) Katmai and Lake Clark versus (G–H) Aniakchak, but in both areas Rock Ptarmigan occurred at higher elevations than Willow Ptarmigan. We provide predicted values per point (points) and loess-smoothed trends (solid lines) with 95% CIs (dashed lines).

Heterogeneity in Detection Probability

Importantly, our analytical approach allows one to model heterogeneity in both components of detection probability. Latent heterogeneity in capture probability has long been recognized as an important factor to consider when estimating the size of a closed population so as to minimize bias (e.g., Burnham and Overton 1978, Norris and Pollock 1996). For breeding songbirds, models that incorporate heterogeneity in availability for detection have been invariably supported over those that do not (Farnsworth et al. 2002, Alldredge et al. 2007a, Reidy et al. 2011). Our empirical example confirmed these findings, in that we found significant support for a declining detection probability as the season progressed (most likely due to less frequent singing). The ability to model point-level covariates directly, such as date and time of day of the survey, facilitates less biased estimates of abundance at the point level.

Our simulations and ptarmigan examples suggest that although density estimates are likely unbiased, precision is greatly reduced when mean availability for detection is low (p_a ∼ 0.4) compared to high (p_a ∼ 0.9), primarily because of the greater uncertainty in estimating the lower p_a even with similar numbers of detections. Low availability resulted in imprecise estimates, even with a robust sample size in the simulated dataset (i.e. n ∼ 300). Thus, we caution against designing studies to sample during periods when detection rates are expected to be low (e.g., low rates of singing for auditory surveys). For multispecies surveys for songbirds, for example, one should focus efforts during morning periods and examine pilot data to determine the tradeoffs for different species from sampling during different seasonal periods. Furthermore, simulation studies similar to ours could be used to determine the sampling effort needed (e.g., spatial replicates or total observations) to obtain desired precision for species with inherently low availability (e.g., those detected primarily through visual observations).

Farnsworth et al. (2005) stressed that an estimate of area sampled during a point count is necessary for accurate estimates of density or abundance. Unless either perceptibility or the effective area sampled is known to be constant, a measure of the area sampled (or perceptibility within the survey area) is necessary for unbiased inferences about relative abundance of different species within an area, of a species across habitats, or of a species across time (Buckland et al. 2001:299). Our model accounts for heterogeneity in perceptibility by estimating distance to each observation—arguably the largest source of individual heterogeneity in perceptibility (Burnham et al. 2004, Simons et al. 2009)—and modeling the rate at which perceptibility declines explicitly as a function of distance. Our estimates of perceptibility for the Golden-crowned Sparrow, which gives a loud, clearly whistled song (Norment et al. 1998), indicate that densities in our study area would have been underestimated by 55% within a radius of 280 m and 13% within 100 m if we had not corrected estimates for distance from the observer.

For typical songbird point counts, ignoring distance-related heterogeneity or accounting for it only as a covariate in time-removal or capture–recapture (multiple-observer) models will generally result in estimates of density with a large negative bias, even if counts are restricted to a small (e.g., 50-m) radius (Kissling and Garton 2006, Efford and Dawson 2009, Sólymos et al. 2013). Such bias arises because unmodeled heterogeneity increases with distance, because only the more detectable birds are detected as distance increases, which inflates estimates of detection probability associated with the perception process (Laake et al. 2011). Furthermore, an implicit assumption of multiple-observer methods is that the same group of birds is available for detection by all observers, which will not hold true if one observer can detect birds at greater distances than other observers (Nichols et al. 2000). Restricting counts to a smaller fixed radius may satisfy this assumption (Nichols et al. 2000) but will sacrifice observations, result in a loss of precision in estimates, and still not resolve the problem of variation in effective area sampled due to spatial heterogeneity in perceptibility.

Model Assumptions

Although incorporating the time-removal component in our model allowed us to relax the assumption in conventional distance sampling that the probability of detection at the point is perfect, care should be taken to meet the other 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) p_a and p_d are independent; (6) the population is closed during surveys to births, deaths, immigration, and emigration; and (7) the entire breeding population is present at the study areas. Assumption 1 can be met through proper study design, such as the stratified random design of our field example, which will ensure that birds are distributed independently of the point locations. Assumptions 2–3 can be problematic if count duration is too long, such that undetected movement occurs (Rosenstock et al. 2002). Simons et al. (2009) suggested that among trained observers, double counting may be a more significant source of bias than misidentification of some species. Point-transect sampling is considered a “snapshot” method in which the observer estimates the number of birds present at a single instant, and even random movement can cause a positive bias in density estimates (Buckland 2006). Duration of the count should thus be long enough to estimate availability but short enough to minimize undetected movement. Thus, surveys should be timed to coincide with periods of optimal cue production (e.g., diurnally and seasonally). Additionally, movement toward or away from observers is likely more prevalent at the beginning of surveys, and Ralph et al. (1995) recommended an acclimation period before beginning point counts. Surveys in open habitats, such as in our study area, are less likely to violate the assumption of undetected movement than those in closed habitats. If the observer is being swamped during multispecies surveys by large numbers of individuals, species can be subdivided into groups and recorded in a series of consecutive surveys to ensure accurate counts (Buckland 2006).

Meeting assumption 4 of accurate estimation of distances is less problematic in open habitats, including our field example, than in forested habitats, where distances to auditory detections can be difficult to determine correctly (e.g., Alldredge et al. 2007b, Simons et al. 2009). In open habitats, observers can use laser rangefinders and visual cues, such as location of shrub clusters, to determine the accurate locations of singing but visually hidden passerines. Experimental studies of broadcast vocalizations in forested habitat with dense understory revealed that errors in distance estimation were particularly pronounced at distances of ∼75 m and when songs were played away from instead of toward observers; training in distance estimation reduced but did not eliminate errors (Alldredge et al. 2007b). When analyzing songbird point-transect counts from a variety of habitats in interior Alaska, Hoekman and Lindberg (2012) found problematic deficits or surpluses of detections near the observer and peaks in detections at intermediate distances (30–60 m), particularly for thrushes (Turdidae); the authors attributed these patterns as most likely due to errors in distance estimation, undetected movement, or both. Pooling observations into broad distance classes, especially near the observer, reduced poor fit of observations to the detection function for some species but resulted in high uncertainty in density estimates (Hoekman and Lindberg 2012).

In our field example, we selected cutoff points to reduce heaping within distance bins, minimize bias near the observer, and facilitate smooth detection functions (Buckland et al. 2001), and we recommend that studies implementing our model also take similar steps to maximize fit of the data to the detection function. We further stress the importance of minimizing errors in distance estimation by using laser rangefinders and training programs for the array of species and habitats encountered within each study area (Rosenstock et al. 2002). Although we did not measure such errors in our field study, we recognize the importance of doing so (Alldredge et al. 2007b, Simons et al. 2009), particularly if detections are almost exclusively auditory, with no assistance from visual cues. Maximum-likelihood methods have been developed to use such auxiliary data (paired true and estimated distances) to correct for bias in density estimates that may result from additive or multiplicative errors in distance estimation (Borchers et al. 2010); we suggest that our model can be usefully extended to do so in a Bayesian framework.

Assumption 5, which requires that the 2 components of detection probability be independent, could be violated if both were influenced by the same factor or if both were functions of distance from the observer or time from the start of the survey period. For example, adverse weather may reduce not only the birds' singing rates but also the perceptibility of their songs to the observer. Concentrating initial efforts on detecting individuals nearest to the observer in an attempt to ensure that p(0) ∼ 1 may result in distant birds being more likely to be detected during the later time intervals. Similarly, availability and perceptibility may differ within a species depending on the type of cue (visual or auditory) given, which may induce heterogeneity in detection probability that is difficult to model jointly (cf. Marques et al. 2007). For example, for more visible species that provide infrequent auditory cues, individuals close to the observer may be more likely to be detected during initial periods, whereas those farther away, obscured by vegetation, may be more likely than observed individuals to be detected during later periods. For secretive species, individuals close to the observer may alter their singing rate, thereby affecting availability in relation to distance. Other studies of multiple species across broad regions (Handel et al. 2009, Matsuoka et al. 2012, Sólymos et al. 2013) have found that confounding of estimates of p_a and p_d from removal models and distance sampling, respectively, was minimal and that marginal estimation of these parameters was therefore justified. Results from our field example corroborate these findings, in that density estimates for Golden-crowned Sparrow did not differ when data were restricted to a significantly smaller radius within which distance and time intervals were independent. This assumption should be tested for each species and, if necessary, observations should be truncated to an appropriate radius, detection probability should be modeled by type of cue (Marques et al. 2007), surveys should be short in duration and include an acclimation period to minimize effects on availability of individuals close to the observer (Ralph et al. 1995), or the 2 components should be estimated jointly (Burnham et al. 2004).

Single surveys of short duration, such as those employed in our field study, will satisfy the sixth assumption of population closure. Models that employ repeated visits to points across a season for estimating abundance (e.g., Kéry et al. 2005, Royle et al. 2007, Schmidt et al. 2013) will generally violate this assumption because temporary immigration or emigration can occur between visits. Thus, the closure assumption will apply to the territory rather than the bird, and such models will estimate the “superpopulation” of home ranges that overlap the sampling area, rather than the breeding densities estimated by our model. The superpopulation of home ranges exposed to sampling will depend on how mobile the birds are and may vary with density, such that comparisons of estimates should be evaluated carefully (Nichols et al. 2009). Furthermore, if floaters (i.e. nonbreeding individuals) occur in the population, they may move longer distances and be present at multiple survey points across repeated surveys, which could lead to overestimating population size from double counting individuals and underestimating detection probability (Oppel et al. 2014). For most breeding migratory songbirds, the proportion of the population present in an area increases in the spring as birds begin establishing territories on the breeding grounds, approaches an asymptote of 1 during incubation and brood rearing, and then decreases as failed and successful breeders depart for wintering grounds. If surveys are not conducted during peak breeding, our seventh and final assumption that the entire population is present in an area during surveys could be violated, thus biasing relative abundance estimates low. Multispecies surveys are vulnerable to violating this assumption because timing of breeding varies among species of sympatrically nesting birds. Birds breeding at higher latitudes and elevations typically have a more abbreviated nesting season (e.g., Slagsvold 1977, Spottiswoode and Møller 2004, Bears et al. 2009), which generally results in greater breeding synchrony both within and among species (Best 1981). Although breeding seasons of many species within a community may overlap considerably, those of others may be almost mutually exclusive (Slagsvold 1977, Best 1981). If the entire population of a given species is not present at the time of a multispecies survey, we suggest limiting inference from this model to those that are fully engaged in breeding activities during the time of the survey or exploring alternative models that allow explicit estimation of p_p (e.g., repeat surveys; Nichols et al. 2009).

Practical Application

Methods to adjust raw counts for incomplete detection of individuals rely on assumptions and, thus, still result in indices of population size, though the adjusted estimates should approximate the true population size of interest better than the raw counts would alone (Johnson 2008). Such methods to estimate detection probability may increase costs of monitoring programs (e.g., by requiring repeated surveys or multiple observers) or have unrealistic assumptions, and complex models such as the one presented here may be prohibitive for managers to implement without consultation with a statistician or quantitative ecologist. The benefit, then, of using our model may depend on study objectives and available funding.

Studies focused on examining geographic or temporal variation in population size, mapping species distributions, or exploring mechanisms that underlie imperfect detection of animals will benefit from modeling detection probability using spatially or temporally varying covariates. For example, our model for Golden-crowned Sparrows in southwest Alaska indicated that abundance was positively associated with tall shrubs and deciduous forest at relatively low elevations in upland areas, which is consistent with their predominant distribution at or above tree line in forest–tundra ecotones elsewhere in the state (Kessel and Gibson 1978, Kessel 1989, Petersen et al. 1991). If we had not accounted for either component of detection probability, our model suggests that densities would have been underestimated by an average of 63% and that predicted distributions would have been biased in relation to habitat type because perceptibility decreased with increasing dense cover, with implications for species-specific habitat management. Similarly, not accounting for significant temporal effects on singing rates would have resulted in spatially biased estimates of density because sampling order was not random with respect to time (i.e. survey order was geographically clustered for efficiency in time and cost). Furthermore, the model presented here can provide unbiased, reasonable estimates of model components for observation (i.e. detection probability) and process (i.e. abundance) using data collected by a single observer during a single survey, thus reducing survey cost and effort and better approximating a “snapshot” of abundance in space and time. The combined time-of-detection, distance-sampling method also shares these characteristics but requires tracking individual birds throughout the survey period (Alldredge et al. 2007a), which may not be feasible in areas with high bird densities or for multispecies surveys.

Studies focused on uncommon species or that violate model assumptions may not be appropriate for use with our model. Among some other species in our study, models that had a marginal number of detections (i.e. <75; Buckland et al. 2001) converged and produced reasonable estimates of detection probability and density but, upon closer examination, revealed potential problems such as non-normal posterior distributions and possibly spurious covariate relationships. Additionally, like all model-based estimates, unbiased inference is dependent on adherence to model assumptions, which may be violated under some sampling designs or for certain species. In these cases, unadjusted indices of abundance may produce similar conclusions and management decisions if objectives require only estimates pooled across space and time and population trends are strong (Thompson and La Sorte 2008), or if variance in detection probability is substantially smaller than the variance in population size to be detected and detection probability and population size are independent (Johnson 2008).

Our model can improve understanding of avian populations by (1) reducing bias associated with imperfect detection probability and spatial variability that may lead to inaccurate inferences regarding population size or distribution and (2) increasing our understanding of the processes associated with abundance and detection probability through examination of relevant covariates. If assumptions can be met and sampling design is appropriate, our model can be used to produce predictive landscape-level maps of distribution and abundance that can be useful for both management and conservation. Thus, broad-scale programs to model abundance of birds should not only use rigorous study design and implementation to minimize bias in population-level estimates, but also include methods to estimate survey-specific detection probabilities if objectives include point- or time-specific inferences.

In sum, we propose a novel extension to 2 existing models to estimate detection probability and abundance from spatially replicated surveys. Minimal additional effort by an observer during a single survey can provide a wealth of additional information regarding the detection process and mechanisms underlying species abundance. Our model can assist land managers, hindered by logistical and financial constraints on research and monitoring, in obtaining unbiased estimates of population density while minimizing costs. Furthermore, implementation of our model in a Bayesian framework greatly increases flexibility over maximum-likelihood methods (e.g., by allowing random effects at multiple hierarchical levels) while accounting for important but often overlooked sources of heterogeneity in detection probability.