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The analysis of large heterogeneous data sets of avian point-count surveys compiled across studies is hindered by a lack of analytical approaches that can deal with detectability and variation in survey protocols.
We reformulated removal models of avian singing rates and distance sampling models of the effective detection radius (EDR) to control for the effects of survey protocol and temporal and environmental covariates on detection probabilities.
We estimated singing rates and EDR for 75 boreal forest songbird species and found that survey protocol, especially point-count radius, explained most of the variation in detectability. However, environmental and temporal covariates (date, time, vegetation) affected singing rates and EDR for 73% and 59% of species, respectively.
Unadjusted survey counts increased by an average of 201% from a 5-min, 50-m radius survey to a 10-min, 100-m radius survey (n = 75 species). This variability was decreased to 8·5% using detection probabilities estimated from a combination of removal and distance sampling models.
Our modelling approach reduced computation when fitting complex models to large data sets and can be used with a wide range of statistical techniques for inference and prediction of avian densities.
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Point-count surveys are the most widely used method for counting terrestrial birds (Rosenstock et al. 2002; Bart 2005). In recent years, considerable effort has been invested in compiling data from disparate point-count surveys across North America into continental-scale data sets that can address questions about avian ecology at spatial extents not possible with individual studies. Modelling spatial or temporal patterns in avian density and extrapolating densities to population sizes over large areas are goals of many of these programs (Bart 2005; Iliff et al. 2009; Cumming et al. 2010, 2013). Achieving an accurate density estimate is hindered by the lack of flexible analytical approaches that can deal with two classes of nuisance factors inherent in large compiled data bases of point-count surveys – avian detectability and variation in survey protocols (Nichols, Thomas & Conn 2009; Reidy, Thompson & Bailey 2011).
The first issue with estimating avian abundance from point-count data is that raw survey counts are often incomplete measures of abundance; they need to be adjusted for species-specific detection probabilities. Detection has two components: the probability that an individual bird present at the time of survey gave a visual or auditory cue and was therefore available for detection (p, availability), and the conditional probability that the available birds were detected (q, perceptibility, Nichols, Thomas & Conn 2009). The product pq is the overall detection probability, which links the raw survey counts with abundance. Joint estimators of p and q have been not been widely applied (Burnham et al. 2004; Farnsworth et al. 2005; Alldredge et al. 2007a), but products of marginal estimates of p and q have been used as detection probability (Handel et al. 2009).
Removal models (Farnsworth et al. 2002; Alldredge et al. 2007a) and distance sampling (Buckland et al. 2001) are commonly used to estimate p and q, respectively. These approaches require that observations are stratified into ≥2 time or distance intervals, respectively. Such stratification is a common but not universal practice in avian point-count surveys (Ralph, Droege & Sauer 1995). Thus, analysis of heterogeneous data sets such as considered here, marginal detection probabilities p and q might be estimated from subsets of the surveys where counts were appropriately stratified. The resultant estimates could then be applied to the entire data set when modelling density (Bart & Ernst 2002; Pollock et al. 2002).
The second issue in heterogeneous data sets is that survey protocols often vary between studies. Point-count survey effort often varies with respect to duration of counting time at a location (count duration; often 3, 5 or 10 min) and the point-count radius (often 50, 100 m or unlimited) within which birds are counted (Ralph, Droege & Sauer 1995). Raw counts of birds typically increase with both count duration and radius. For example, an increase in count duration from 3 to 10 min resulted in an average 65% increase in the mean counts of 54 species of boreal forest birds; increasing the count radius from 50 m to unlimited distance resulted in a 171% increase in mean counts of the same species (S. M. Matsuoka, unpublished data). Uncontrolled, these large differences in avian counts due to protocol could obscure the temporal and habitat-based trends in abundance that are typically the focus of investigations (Ralph, Droege & Sauer 1995).
In this study, we demonstrate a unified analytical approach for estimating breeding densities of birds, using point-count data compiled from a variety of disparate surveys conducted across the boreal region of North America (Cumming et al. 2010). We reformulate removal and distance sampling models into flexible estimators that can accommodate the effects of sampling protocol and temporal and environmental covariates on detection probabilities. We then apply these models to 75 species of boreal forest songbird species to evaluate the following: (1) the effects of sampling protocol and environmental covariates on p and q, (2) the relative influence of p and q on density estimates across species and (3) whether the combination of p and q produces estimates of avian density that are robust to variation in survey protocol.
Avian survey data
We analysed a heterogeneous compilation of point-count data from surveys conducted throughout the boreal forest region of North America (Fig. 1) compiled by the Boreal Avian Modelling Project (bam, www.borealbirds.ca; Cumming et al. 2010). The bam data set version 3.0 (June 31, 2012) consists of 89,369 unique survey visits to 35,298 point-count locations between 1993 and 2011. These surveys included 47 combinations of 14 and 19 distinct protocols for time and distance intervals, respectively. All locations were georeferenced using global positioning systems (Fig. 1). The date and start time of each visit were recorded. We only analysed counts of vocalizing birds, which accounted for 98% of all detections in our data base.
QPAD approach to estimate density
For a point-count survey at a given location and time, the expected count of a given species may be written as: E[Y..] = N p(tJ) q(rK) = D A p(tJ) q(rK), where N is the species’ abundance, D is the point level density per unit area, A is the area sampled, p(tJ) is the probability of an individual bird singing at least once during the total cumulative time interval tJ, given presence and q(rK) is the probability that an individual bird within (the cumulative) point-count radius rK is detected, given singing. ‘QPAD’ refers to commutable components in this equation. Either A = π is known or the effective area sampled can be calculated in conjunction with q(rK). We also assume that the ‘population’ is closed during the sampling interval tJ within rK and that individuals are counted only once. We might have multiple consecutive time intervals within tJ, with start time t0 = 0 and end times tj, j = 1, 2,…, J, with J being the number of time intervals used. Similarly, we might have multiple consecutive distance bands with r0 = 0 and radii rk, k = 1, 2,…, K, with K being the number of distance bands. Values of tj, rk, J and K can vary between surveys. We use the ‘dot notation’ for raw counts to indicate sums over the time intervals j and distance bands k. We use conditional maximum likelihood estimators (Huggins 1991; Farnsworth et al. 2002, 2005; Sólymos, Lele & Bayne 2012) for the two components of detectability, when at least some of the surveys contain ≥2 time or distance intervals. The conditioning removes dependence of p(tJ) and q(rK) on the unknown density D.
The expected count of individuals first detected during time interval j can be written as: E[Yj.] = D A [p(tj) − p(tj–1)] q(rK), where p(t0) = 0. We assume that singing events by individuals follow a Poisson process and that the population (i.e. singing individuals, usually males) at a given location and time is homogeneous in its singing rate. It follows that probability of singing may be expressed by the singing rate (ϕ), which is the Poisson parameter. We count each individual only once, so individuals are ‘removed’ from the population after being detected. When singing rate is constant within a sampling event, the time to first singing event will follow an exponential distribution: f(t) = ϕ exp(−tϕ). Hence, the cumulative distribution of times to first detection in the time interval (0, tJ) is given by (Barker & Sauer 1995; Alldredge et al. 2007a):
We analysed the subset of BAM data with ≥2 time intervals (13,891 visits to 6018 point-count locations) using the r (R Core Team 2012) extension package ‘detect’ (Sólymos, Moreno & Lele 2013) for the conditional maximum likelihood estimation of singing rate parameters (see Appendix). Singing rates were estimated for each species detected during at least 75 visits using a loglinear model with and without visit-level covariates (see Appendix) for time since local sunrise (TSSR) and Julian day (JDAY). TSSR was estimated as the difference in hours between the survey start time and local sunrise on the date of the visit. Local sunrise was obtained from the National Oceanic and Atmospheric Administration's sunrise/sunset calculator (http://www.srrb.noaa.gov/highlights/sunrise/sunrise.html; Meeus 1991) as implemented in the R package ‘maptools’ (Lewin-Koh & Bivand 2012). JDAY had a mean of 165 and ranged 123–209 days; TSSR had a mean of 2·2 and ranged −1·3 to 9·1 h. We divided TSSR and JDAY by their maximum possible values of 24 and 365, respectively. For each species, we fitted removal models with the intercept only (i.e. assuming constant singing rate) and different combinations of JDAY, TSSR and their quadratic terms (Table 1). We evaluated the relative support for each model using Schwarz's Bayesian information criterion (BIC) and selected the model with lowest BIC.
Table 1. The frequency that models of detectability were best supported by the survey data for 75 species of songbirds, boreal North America, 1993–2011. Detectability estimated from a combination of singing rates from removal models and effective detection radius from distance sampling models
Removal models included a model with no covariate effects and models with different combinations of Julian day (JDAY), time since local sunrise (TSSR) and their quadratic terms.
Distance sampling models included a model with no effects, the proportion of tree cover measured at a 500-m2 resolution (TREE) and land cover class (LCC).
(0) No effects
(3) JDAY + JDAY2
(4) TSSR + TSSR2
(5) JDAY + TSSR
(6) JDAY + JDAY2 + TSSR
(7) JDAY + TSSR + TSSR2
(8) JDAY + JDAY2 + TSSR + TSSR2
Distance sampling model
We assume a half-normal detection function, so the probability of detecting an available individual at distance r from the observer is: g(r) = exp(−r2/τ2), where τ2 is the variance of the unfolded normal distribution describing the rate of distance decay and τ2/2 is the variance of the half-normal distribution. We also assume that r is measured without error, all available birds at r = 0 are detected, and that birds are detected at their initial location (Buckland et al. 2001). The parameter τ is also the effective detection radius (EDR) for unlimited distance counts and is defined as the distance at which as many of the available birds are detected beyond τ as remain undetected within τ (Buckland et al. 2001). The distribution of distances available for detection has the probability density function c(r) = π2r/πrK2, where rK is the maximum count radius, beyond which observations are not recorded. Therefore, the average probability that an individual randomly located within rK is detected may be adapted from Marques & Buckland (2004:44) as (terms are collapsed for notational simplicity):
The denominator is a normalizing constant based on perfect detection that allows probabilities to be calculated from the integral expression in the numerator, which refers to the volume under the probabilities of the circular half-Gaussian ‘pie’ shape cut at rK.
For distance models, we used the subset of BAM data with ≥2 consecutive distance intervals (60,959 visits to 21,497 point-count locations). We estimated parameters of the distance model for all species detected during at least 75 visits using the conditional maximum likelihood estimator described in the Appendix as implemented in the ‘detect’ r package (Sólymos, Moreno & Lele 2013). Perceptibility should be reduced in forest compared with open habitats because of the amount of foliage available to attenuate bird songs. Accordingly, we introduced a covariate for the proportion of tree cover at each location (TREE) by intersecting each point-count location with a 500-m resolution tree cover map (Hansen et al. 2003). TREE had a mean of 0·63 and ranged from 0 to 1. Attenuation of bird songs is also greater in deciduous than coniferous vegetation (Schieck 1997), so we also determined the land cover class at each survey point using the 250-m resolution Land Cover Map of Canada 2005 (Latifovic et al. 2008). We collapsed the 39 cover classes into five generalized land cover classes (LCC): open habitat (OH, <25% cover by trees), sparse canopy forests with a majority of conifers (SC) or deciduous trees (SD) (25–60% canopy cover), dense canopy forests with a majority of conifers (DC) or deciduous trees (DD) (>60% canopy cover). For each species, we fitted distance models with the intercept only (constant distance effect), the univariate effect of TREE (constrained to be negative) and the categorical effects of LCC, and then selected the model with lowest BIC.
Sources of variation in the detection process
We quantified the sensitivity of the detection probability (pq) and its separate subcomponents (p and q) to variation in covariates for bird species, sampling protocol (point-count duration and radius) and sampling effects (time, date, habitat). This involved (1) resampling covariate values from the BAM data set (n = 89,369), (2) using the best fit removal and distance sampling models (see methods above) to predicted p, q and pq for each resample, and (3) partitioning the variance in p, q and pq across resamples relative to each covariate.
For each resampling iteration, we randomly selected a value separately for each covariate from the data base (species, count radius, count duration, JDAY, TSSR, TREE, LCC) with replacement for a total of 106 resamples. We excluded unlimited radius from the resampled values of the point-count radius because we could not estimate without assumptions about the maximum detection distance (Matsuoka et al. 2012). We only included species with covariate effects in removal and/or distance sampling models. We used analysis of variance to partition the sum of squares (SSQ) in p, q or pq among the covariates (species, sampling protocol, covariate effects), with variance partitions expressed as a percentage of total SSQ.
The offset approach for density estimation
Following Hedley & Buckland (2004) and Buckland et al. (2009), estimates from removal and distance sampling models can be combined with Y to estimate density: for data with limited point-count radius, where is the correction factor, and , and , for i = 1, 2, …, n sampling events. For unlimited distance point counts, where Ai is unknown, we estimate effective detection radius. The effective detection radius (re) is equal to the half normal parameter (τi) for unlimited counts because re is the distance from observer where the number of individuals missed within () equals the number of individuals detected outside of it (). Consequently, the effective area sampled is , and we fix qi = 1. As a result, the correction factor for the unlimited radius counts is: .
The logarithm of the correction factors (log() can be applied as offsets in count regression models designed to estimate densities. This accounts for the effects of the nuisance parameters and simplifies the model structure. In the following illustrations, we assume that the marginal distributions of counts are (Yi..|λi) ∼ Poisson(λi), where the rate parameter is . We use the known area and the estimates of singing and perception probabilities as offsets on the log scale: . For unlimited radius point-count surveys, we alternatively use: . We can also model the log of population density as a function of covariates: , where β is a vector of parameters corresponding to columns in the covariate matrix X, including the intercept. We obtain the maximum likelihood estimate of the density parameters using the offsets in conventional statistical packages. This could be performed in a generalized linear model (GLM) for newly obtained count data by assuming that the effects of nuisance factors are the same as in the data set used to obtain the offsets.
The offset approach extends to related count models, such as the negative binomial, zero-inflated Poisson, zero-inflated negative binomial and Poisson–log-normal mixed model. The offsets can also be used in logistic regression with detection vs. non-detection data using the complementary log–log link function and in various machine learning methods, such as classification and regression trees (see Supporting Information).
The uncertainty in densities can be obtained using bootstrap (Hedley & Buckland 2004; Buckland et al. 2009) or Markov chain Monte Carlo (MCMC) techniques (Lele, Nadeem & Schmuland 2010; Sólymos 2010). In both cases, one may use a parametric bootstrap sample from the singing rate and distance model estimates to generate a series of offsets. In this way, the uncertainty in the offset term will be propagated into the variance of the parameter estimates for the density model (see Supporting Information for examples). We analysed survey data on Ovenbirds [Seiurus aurocapilla (Linnaeus, 1766)] from Alberta (n = 30,738 visits to 1710 locations) to demonstrate this approach. We used the ‘lme4’ r package (Bates, Maechler & Bolker 2013) and fit a Poisson–log-normal, mixed-effects model to the survey data to estimate Ovenbird densities in each of five land cover classes (LCC), with point-count location included as a random intercept term in the model. We fitted the models separately with each of four offsets [log(A), log(Ap), log(Aq) and log(Apq)] to compare the ranked order of habitat densities estimated from each offset. We used a nonparametric bootstrap approach as described in the Supporting Information to propagate the errors in the offsets into the mixed modelling estimation. We therefore assess uncertainties in the expected densities that are due to uncertainties in the offsets versus the fixed effects parameters. The singing rate model included the effect of JDAY and the EDR model included the effect of LCC. These were best supported by BIC among the candidate models we evaluated (Table 1). Unlimited distance point counts were excluded because A is not defined. As a result, the log(A) offset had no offset related uncertainty.
Robustness of density estimates to survey protocol
We tested the robustness of our density estimates to varying sampling protocol using a subset of the BAM data (n = 10,414 visits to 3605 point-count locations) where counts were recorded for all four combinations of 0–5 and 5–10 min intervals and 0–50 and 50–100 m distance bands. We estimated mean densities ( ) using a Poisson GLM model for each of the 75 species based on the counts from each of the four combinations of cumulative duration and distance intervals (5 min and 50 m, 10 min and 50 m, 5 min and 100 m, 10 min and 100 m), and each of the four types of offsets [log(A), log(Ap), log(Aq), log(Apq)].
We compared mean density estimates when not accounting for differences in sampling protocol (no offset) and when accounting for methodology differences via QPAD offsets from the best supported removal and distance models to calculate the offset components. We calculated deviation in adjusted mean densities by dividing the estimates from each combination with the estimates based on the 10-min and 100-m radius count with the offset incorporating all terms [log(Apq)]. We expected that mean density estimates should not show systematic deviation (average ratio close to one) w.r.t. survey duration and radius if our QPAD method was robust to sampling protocol. We also expected that density estimates based on raw counts (no offset) and incomplete offsets [log(A), log(Ap), log(Aq)] should show systematic deviation w.r.t. survey duration and radius.
Estimated singing rates and effective detection radii
Estimates of constant singing rate () from removal models varied from 0·15 (±0·02 SE of species-specific estimate) songs/min for the blue jay [Cyanocitta cristata (Linnaeus, 1758)] to 0·63 (±0·02 SE) songs/min for clay-coloured sparrow [Spizella pallida (Swainson, 1832)] and averaged 0·32 (±0·09 SD) songs/min across 75 species (Fig. 2a). Adding covariates improved models of singing rate for 73% of 75 species (Table 1). Covariate models received the most support when they included the univariate linear effect of JDAY (36 species) or JDAY and TSSR (seven species). Quadratic effects were supported more frequently for TSSR than for JDAY (Table 1); quadratic effects rarely resulted in unimodal responses over the range of the data (Fig. 3). Among species with covariate effects, estimates of singing rate consistently declined with increases in JDAY. The shape of the relationship with TSSR was more variable among species (Fig. 3).
Estimates for the constant distance parameter (EDR, ) for unlimited radius surveys varied from 39·2 (±0·9 SE of species-specific estimate) m for the Cape May Warbler [Dendroica tigrina (Gmelin, 1789)] to 171·4 (±4·4 SE) m for the American Crow (Corvus brachyrhynchos Brehm, 1822) and averaged 74·0 ± (24·2 SD) m across the 75 species (Fig. 2b). Models with covariate effects were better supported by the data than the constant distance model for 59% of 75 species (Table 1). Distance sampling models with TREE and LCC were best supported for 29 and 15 species, respectively. Covariate effects on EDR varied by species. EDR decreased with increasing forest cover, and tended to be higher in coniferous than in deciduous forests (Fig. 4).
Sources of variation in the detection process
The marginal probabilities of availability () and perceptibility () based on constant singing rate () and constant effective detection radii ( ) were sensitive to variation in survey protocol. Estimates of ( averaged across the 75 species increased by 56% from 3-min (mean 0·61 ± 0·10 SD) to 10-min (mean 0·95 ± 0·05 SD) duration surveys. Estimates of averaged across species increased by 75% from surveys of 100 m (mean 0·44 ± 0·18 SD) to 50 m in radius (mean 0·77 ± 0·11 SD). Sampling protocol had a large effect on the product of the marginal probabilities, with increasing by 170% from a 3-min, 100-m radius survey (mean 0·27 ± 0·11 SD) to a 10-min, 50-m radius survey (mean 0·73 ± 0·11 SD; Fig. 2c). Short-duration, small-radius surveys (3 min, 50 m) appear to be dominated by bias in p, while long-duration, large-radius surveys (10 min, 100 m) surveys appear to be dominated by bias in q.
Among the 55 species with covariate effects on singing rate, the variation in among the removal models was mostly attributable to species (40%), count duration (21%), and JDAY (13%), and <1% to TSSR. The variation in among distance sampling models for the 44 species with covariate effects was mostly attributable to species (65%) and to count radius (23%). TREE and LCC together contributed only 4%. The variation in detection probability across 39 species with covariate effects in both removal and distance sampling models was attributable primarily to species (66%), secondarily to count radius (15%), and much less to count duration (4%), JDAY (3%), TREE and LCC (1%), and TSSR (<0·01%).
Robustness of density estimates to sampling protocol
The mean density estimates from Poisson GLM based on the complete offset [log(Apq)] did not show systematic deviation from the reference density estimates (10 min and 100 m) w.r.t. survey duration or radius across the 75 species (Fig. 5). Mean density estimates based on raw counts (no offset) showed the most severe deviation reflecting the relationship between mean counts and sampling protocol. This was followed by the large negative deviation based on the log(A) and log(Ap) types of offsets where 50-m radius counts led to smaller deviation due to high probability of detection close to the observer. The deviation was smaller for the log(Aq) offset because probability of singing contributed less to the overall offset. This was well reflected by the density estimates from 10-min counts where deviations were smaller compared with 5-min counts (Fig. 5). Unadjusted survey counts increased by an average of 201% from a 5-min, 50-m radius survey to a 10-min, 100-m radius survey. This variability was decreased to 8.5% using detection probabilities estimated from a combination of removal and distance sampling models.
Uncertainties in Ovenbird density estimates
Ovenbird densities in Alberta were highest in deciduous forests and lowest in sparse coniferous and open habitats (Fig. 6). The log(A) and log(Ap) offset failed to differentiate between dense and sparse deciduous habitats based on point predictions. The log(Aq) and log(Apq) offsets showed differences between these habitat types based on point predictions; however, 95% confidence intervals did not reveal significant differences. The difference in offsets, including the component q, was due to differences in EDR across habitat types. EDR was 79 m in dense deciduous forests, while it was larger (83–87 m) in other land cover types, causing a 10% difference between estimated densities in dense and sparse deciduous forests. Densities based on log(A), log(Ap) and log(Aq) offsets were 45–60%, 50–66% and 89–92% of the densities based on the log(Apq) offset, respectively. Contribution of the offsets related uncertainty to the total 95% confidence interval was smaller for the log(Ap) offset (2–4%) than for the log(Aq) and log(Apq) offsets (8–15% of the total confidence interval; Fig. 6).
The analysis of large compilations of heterogeneous point-count data has been hampered by the lack of tractable analytical approaches for dealing with nuisance factors that bias the raw counts and affect the probability that individuals present at the survey will actually be detected. Therefore, we reformulated removal and distance sampling models (Buckland et al. 2001; Farnsworth et al. 2002) to provide a unified, computationally efficient and flexible approach for estimating avian densities from point-count surveys that vary in survey duration, survey radius and the length and number of time or distance subintervals within each. Our approach, that we have called QPAD, can also incorporate environmental covariates directly into the components of singing rate and distance effects, further reducing bias in estimates of abundance.
Behaviour of the marginal estimators
Density estimates can vary widely with survey protocol when distance and removal models are used separately (Reidy, Thompson & Bailey 2011; Matsuoka et al. 2012). Our approach appears to solve this problem. We suggest that this is important because assumptions are relaxed when the models are used in combination (Farnsworth et al. 2005). For example, distance sampling assumes that availability (p), and therefore perceptibility (q), is equal to one at the point-count centre (Buckland et al. 2001:19). However, we found that p increased with survey duration for all 75 species evaluated, but still averaged only 0·94 (range: 0·68–0·99) across species for our maximum survey duration of 10 min. The assumption that p = 1 was therefore violated, particularly for short-duration surveys. This violation is reflected in density estimates from distance sampling, which increase with survey duration: 19% from 3-min to 5-min surveys and 25% from 5-min to 10-min surveys (Matsuoka et al. 2012). Removal model estimates of p remedied this problem by relaxing the distance sampling assumption that q = 1 at the point (Farnsworth et al. 2005) and therefore led to density estimates with minimal availability bias. Removal models produce density estimates that vary widely with changes in the survey radius (Reidy, Thompson & Bailey 2011) and will only yield unbiased estimates of density when q = 1 (Efford & Dawson 2009). Reducing the survey radius to 50 m can reduce the magnitude of violations to q = 1 (Reidy, Thompson & Bailey 2011), but often result in sparse counts, which can lead to inaccurate density estimates. Our distance sampling estimates of q often declined sharply with distance, but still averaged only 0·77 (range: 0·49–0·96) across 75 species when the survey radius was 50 m. Because the assumption that q = 1 is rarely met, distance sampling estimates of q are needed to obtain density estimates from removal models (Efford & Dawson 2009).
Our density estimator assumes that p and q result from independent processes that can be estimated separately and then combined to adjust the survey counts for detection bias (Diefenbach et al. 2007; Handel et al. 2009). If such confounding between p and q were a serious problem, then the independent estimates of p and q would be sensitive to changes in survey radius and duration, respectively. Farnsworth et al. (2002) found that removal model estimates of p varied little between 50-m and unlimited radius surveys for four temperate forest bird species and ratios of first detections in two time intervals remained stable relative to survey radii out to a distance of ≥100 m in 82% of 28 boreal bird species (Handel et al. 2009). Also, distance sampling estimates of the effective detection radius (τ) varied only 1·6% between 3- and 5-min surveys across 55 species and 1·8% between 5- and 10-min counts across 81 boreal bird species (Matsuoka et al. 2012). Thus, confounding of removal model estimates of p and distance sampling estimates of q appears to be minimal, and the marginal estimation of p and q seems justified. The latter example (Matsuoka et al. 2012) also suggests that undetected movement of birds during a 10-min point-count survey may not be a serious problem. This is because τ should decrease with count duration because probabilities of movement will increase with duration, and birds are more likely to be detected if moving towards than away from the point.
Sources of variation in the detection process
Our assumption that p and q result from separate processes allowed us to more easily incorporate environmental covariate effects into our removal models of singing rate and distance sampling models of the effective detection radius. Previous efforts to jointly model p and q have not included such effects (Burnham et al. 2004; Farnsworth et al. 2005). We found that differences between species and protocols greatly affected variation in detectability accounting for 66% and 19% of the total variation in detectability, respectively. Environmental covariate effects were quite prevalent, but their relative contribution to the overall variation was small (4%). Time-of-day or time-of-year effects on singing rates were detected among 73% of species, and effects of tree cover or land cover type on the effective detection radii were detected among 59% of species. Environmental covariate effects can be important to control for when comparing densities among habitats or years. For example, we demonstrated how accounting for relatively small differences in EDR among land cover types affected whether we could detect habitat differences in Ovenbird density. This emphasizes the importance of controlling for both survey protocol and environmental covariate effects on detection probabilities when combining data or comparing densities among studies that vary in sampling (Matsuoka et al. 2012).
We demonstrated how the constant and covariate-related QPAD model estimates developed in this study can be used in statistical models of density for inference and prediction. We also showed that ignoring either the singing rate or the distance component in modelling can lead to large differences in density estimates depending on sampling protocol and covariate effects. Localized predictions of absolute density are often important, for example in quantifying incidental take by human development, or determining conservation offsets. In such situations, our estimates and the offset approach can be applied to data collected under any combination of point-count duration and radius in a variety of count modelling frameworks as first approximation solutions to most nuisance factors (see examples in Supporting Information), thus greatly simplifying the model structure and reducing the computational burden when fitting complex models to large data sets. Our proposed QPAD approach also allows for the inclusion of unlimited distance counts by estimating the effective area sampled when A is unknown. Further, the probabilities from our removal- and distance model estimates provide important insights for designing timing and protocol in field surveys.
The Ovenbird example suggests that patterns of relative density remain largely unchanged irrespective of the use of an offset, thus reducing the need for corrected estimates when absolute measures of population size are not required. This may be true for this particular data set because the pattern in abundance among habitats was stronger than the ‘noise’ introduced by protocol variability. But the importance of controlling for protocol effects will vary with circumstance. For example, spatial or temporal bias in the distribution of survey protocols would confound relative density patterns, and could lead to erroneous study conclusions. As well, measures of absolute rather than relative density can be important when analysing community composition, as community patterns may be distorted when detectability varies largely among species.
We benefitted from large sample sizes of detections to estimate singing rates, effective detection radii and corresponding values of p and q. Our estimates might therefore be considered robust to heterogeneity in detection rates relative to observers, weather and other spatial and temporal factors because we pooled over such a large data set (Buckland et al. 2001; Marques et al. 2007). However, we recommend that researchers conducting point-count surveys incorporate multiple time and distance intervals as advocated by standardized survey protocols (e.g. Ralph, Droege & Sauer 1995) so that p and q can be directly estimated and applied to their own studies. This would also allow their data to be analysed as part of larger compilations of survey data as considered here (Farnsworth et al. 2005). Although unbiased relative to survey protocol, we do caution that density estimators using our methods might still be biased for species that move at high rates (Scott & Ramsey 1981; Burnham et al. 2004:352–356), move in response to observers (Buckland et al. 2001), or that sing in irregular bouts or at low or variable rates (Alldredge et al. 2007b). These are general issues with distance sampling and removal models – field studies are much needed to evaluate the actual prevalence of these violations and the extent that they truly bias density estimators from these and also other estimators of avian abundance (Johnson 2008; Nichols, Thomas & Conn 2009; Reidy, Thompson & Bailey 2011).
We thank the Handling Editor and two anonymous reviewers for helpful comments on an earlier version of the manuscript. This publication is a contribution of the Boreal Avian Modelling (BAM) Project, an international research collaboration for the ecology, management and conservation of boreal birds. We acknowledge the BAM funding and data partners, and Technical Committee members who made this project possible (http://www.borealbirds.ca/index.php/acknowledgements). This project was funded by Environment Canada, Natural Sciences and Engineering Research Council of Canada, U.S. Fish and Wildlife Service's Neotropical Migratory Bird Conservation Act, and Alberta Biodiversity Monitoring Institute.
Conditional maximum likelihood estimation
Removal sampling model parameters
We would like to estimate the unknown parameter ϕ. For this, we find the probability πj that an individual y is member of the set of individuals in Yj. given that it is the member of the set of individuals in the total count (Y.. = ΣjYj.). By this conditioning, D A q(rK) is removed from the equation:
which depends only on ϕ. The conditional multinomial density function for the removal model is:
For a given observation, the value of Eq. 2 depends solely on the total count, the distribution of the count within the intervals j = 1,..,J, and the rate parameter ϕ. It is not necessary for the intervals to be the same for any two observations, provided that ϕ is either constant or depends on covariates. For i = 1, 2, …, n sampling events, we can use covariates that affect singing rate (e.g. time of year, time of day): log(ϕi) = , where γ is a vector of parameters corresponding to columns in the covariate matrix Z, including the intercept. We specify a loglinear model because support for ϕ is non-negative. We define a sampling event as a single visit to a single location. Multiple visits to the same location are possible, but in our framework, these visits (unique sampling events) might differ in terms of sampling design. For n sampling events, we obtain conditional maximum likelihood estimates of the singing rate parameters by maximizing the conditional likelihood function:
This formulation allows us to use the part of the data with at least two consecutive time intervals and nonzero total counts to estimate singing rate, as the multinomial density function is identical to one when there is only one time interval or when the total count is zero.
Distance sampling model parameters
The expected count for any distance band k can be written as:
where the term s(rk) − s(rk–1) is the part of the integral within distance band k, compared with the total volume defined by the whole point-count area of radius rK under prefect detection. As we are comparing against the whole point-count area, A refers to the entire area sampled rather than just the area within a band. We would like to estimate the unknown parameter τ. For this, we find the probability that an individual y is member of the set of individuals in Y.k given that it is the member of the set of individuals in the total count (Y.. = ΣkY.k). By this conditioning, D A p(tJ) is removed from the equation:
This expression allows us to use point counts with unlimited radius, rK= ∞, because the expression does not include A, which is unknown for unlimited distance counts. In the unlimited distance case, π.k = u(rk) − u(rk −1), because u(rK) = u(∞) = 1. Removing area by conditioning is important for dealing with the unlimited distance cases, which make up 24% of the BAM multiple distance data set. For example, the integrated likelihood (N-mixture) approach (Royle 2004; Royle, Dawson & Bates 2004; Sólymos, Lele & Bayne 2012) cannot directly deal with this situation, because A is part of the Poisson probability density (abundance model), while q is part of the binomial/multinomial probability density (detectability model). Note that the expression in Eq. 5 depends only on τ.
The conditional multinomial density function for the distance model is:
For i = 1, 2, …, n unique sampling events, we can use covariates that influence the effective detection radius (e.g. observer, habitat type): , where θ is a vector of parameters corresponding to columns in the covariate matrix Q, including the intercept. We use a loglinear model to ensure non-negativity for τ. For n sampling events, we obtain conditional maximum likelihood estimate of the effective detection radius parameters by maximizing the conditional likelihood function:
As with the removal model for duration effects, this formulation allows us to use data with at least two distance intervals and nonzero total counts to estimate the effective detection radius, because the multinomial density function equals one when there is only one distance interval or the total count is zero. We used 100 m as the unit in the distance model so that density is expressed as the number of males/ha.
The joint estimation of model parameters ϕ and τ follows from Eqs. 1 and 5: The marginal removal and distance sampling models are orthogonal to each other, and the joint estimator does not improve statistical efficiency and consistency (see Supporting Information for numerical proof).