Analysing Designed Experiments in Distance Sampling
Stephen T. BUCKLAND, Robin E. RUSSELL, Brett G. DICKSON, Victoria A. SAAB, Donal
N. GORMAN and William M. BLOCK
Distance sampling is a survey technique for estimating the abundance or density
of wild animal populations. Detection probabilities of animals inherently differ by species,
age class, habitats, or sex. By incorporating the change in an observer’s ability to detect a
particular class of animals as a function of distance, distance sampling leads to density
estimates that are comparable across different species, ages, habitats, sexes, etc. Increasing
interest in evaluating the effects of management practices on animal populations in an
experimental context has led to a need for suitable methods of analysis of distance sampling
data.
We outline a two-stage approach for analysing distance sampling data from
designed experiments, in which a two-step bootstrap is used to quantify precision and
identify treatment effects. The approach is illustrated using data from a before-after
control-impact experiment designed to assess the effects of large-scale prescribed fire
treatments on bird densities in ponderosa pine forests of the southwestern United States.
Key Words: BACI design; Bootstrap; Distance sampling; Point transect sampling
_____________________________
Stephen T. Buckland is Professor of Statistics, University of St Andrews, Centre for
Research into Ecological and Environmental Modelling, The Observatory, Buchanan
Gardens, St. Andrews KY16 9LZ, Scotland (E-mail: steve@mcs.st-and.ac.uk). Robin E.
Russell is Post-doctoral Research Ecologist, US Forest Service, Rocky Mountain
Research Station, 1648 S. 7th Ave, Montana State University Campus, Bozeman, MT
59717, USA. Brett G. Dickson is David H. Smith Conservation Research Fellow, Center
for Environmental Sciences and Education, Northern Arizona University, Flagstaff, AZ
86001, USA. Victoria A. Saab is Research Wildlife Biologist, US Forest Service, Rocky
Mountain Research Station, 1648 S. 7th Ave, Montana State University Campus,
Bozeman, MT 59717, USA. Donal N. Gorman is a student in the School of Mathematics
and Statistics, University of St Andrews, funded for this project under the University of
St Andrews Undergraduate Research Internship Programme.
William M. Block is
Program Manager, US Forest Service, Rocky Mountain Research Station, Southwest
Forest Science Complex, 2500 South Pine Knoll Dr., Flagstaff, AZ 86001, USA.
1. INTRODUCTION
Distance sampling (Buckland et al. 2001, 2004) comprises a suite of methods for
estimating animal density. The most commonly-used methods are line and point transect
sampling.
Increasingly, the methods are being used in the context of designed
experiments, for example to assess the impact of forest restoration and fuel reduction
activities on bird communities (Dickson et al. in press; Russell et al. in press).
When the goal of a distance sampling study is to estimate overall abundance in a
region, there is no particular need to model heterogeneity in probability of detection of
animals, provided that the key assumption that animals on the line or at the point are
certain to be detected is met. This arises from the pooling robustness property of distance
sampling estimates, a proof of which is given by Buckland et al. (2004:389-392).
However, in the context of designed experiments, animal densities are typically needed
by plot, and plot-specific estimates may be biased in the presence of unmodelled
heterogeneity in the detection probabilities, unless a separate detection function (which
expresses the probability of detection as a function of distance from the line or point) is
estimated for each plot. Few studies are sufficiently large to allow this, so that the natural
way to analyse data from designed experiments is to use multiple-covariate distance
sampling (Marques and Buckland 2003, 2004; Marques et al. 2007). If detection is not
certain on the line or point, then mark-recapture distance sampling (Laake and Borchers
2004; Borchers et al. 2006) might be used. In either case, a single detection function
model is fitted to observations from all plots, so that plot-specific density estimates are
not independent. In this paper, we develop a strategy for analysing such data.
Commonly in designed distance sampling experiments, multiple species are
recorded. The analyses may have more power for detecting treatment effects if data for
species that are expected to show a similar response to treatment are pooled in a single
analysis. This can be achieved in the above context without having to assume equal
detectability across species by including species as a factor-type covariate in the multiplecovariate distance sampling analysis (Alldredge et al. 2007; Marques et al. 2007). We
illustrate our approach using an experiment to assess the effects of prescribed fire treatments
on bird densities in ponderosa pine- (Pinus ponderosa) dominated forests in the southwestern
United States using a before-after control-impact (BACI) study (Stewart-Oaten and Bence
2001).
We selected two species of warbler common in ponderosa pine forests of the
Southwest, Grace’s warbler (Dendroica graciae) and yellow-rumped warbler (Dendroica
coronata), to demonstrate our methodology. Both species are foliage insectivores and nest in
open cup nests located in the canopy. In their review of fire effects on avian communities in
the southwestern United States, Bock and Block (2005) report negative responses of both
species to wildfire. We expected declines in these species after prescribed fire due to
reduced opportunities for foraging and nesting.
2. METHODS
In the following, we assume that the units for analysis are plots (e.g., a prescribed
fire treatment), where each plot has a number of line or point transects randomly placed
within the plot. Typically, lines or points will be systematically placed through the plot,
but assumed random. See Fewster et al. (in press) for a detailed discussion of this.
Commonly, as in our example below, the same plot may be surveyed in more than one
time period, say year. In that case, the units for analysis will be a ‘plot-year’. For
simplicity, we simply refer to these units as ‘plots’.
The basic data in distance sampling experiments are, for each plot, a count of
number of animals or animal clusters detected, together with distances of detections from
the line or point for estimating detectability. If all animals have the same probability of
detection, the counts are estimates of relative abundance in their own right, so that we can
simply analyse them, for example using Poisson regression (Brand and George, 2001); if
effort varies between plots, the log link function of this model enables this to be
accommodated using an appropriate offset term. We consider here how that strategy
might be extended to the more plausible case of variable detectability.
2.1 LINE TRANSECT SAMPLING
Suppose we have K plots, laid out according to some design, and consider first
line transect sampling. We will suppose that animals occur in clusters; if they do not,
then cluster size is set to unity in the following. We can estimate density in plot k as
1
Dˆ k =
2l k
nk
∑s
i =1
i
fˆy|z (0 | z i )
(2.1)
where lk = total effort conducted on plot k (total line length if each line is covered once)
nk = number of animal clusters detected on plot k
si = number of animals in ith detected cluster on plot k
fˆy | z ( y | z i ) = estimated pdf of distance y from the line for ith detected cluster on
plot k, for which covariates z i were recorded
fˆy |z (0 | z i ) = fˆy | z ( y | z i ) evaluated at y = 0 .
The above result is obtained by converting the expression for M̂ from section 4.2
of Marques and Buckland (2003) into a density, by dividing through by the covered area,
2wlk , where w is the truncation distance for y (i.e. it is the half-width of the covered
strip).
We can express (2.1) as
H
Dˆ k = nk k
2lk
where H k =
1
nk
nk
∑ s fˆ
i =1
i
y|z
(0 | z i ) is the sample mean of the product si fˆy | z (0 | z i ) , where the
mean is taken over detections on plot k. (If nk = 0 , we might estimate H k by the mean
of this product across all detections from other plots.)
If the H k were known constants, it would be natural to model the counts using a
generalized linear model (GLM) with a Poisson or negative binomial error distribution, a
log link function and an offset of − log e (H k / 2lk ) :
⎛ p
⎞
E (nk ) = exp⎜⎜ ∑ x jk β j − log e (H k / 2lk )⎟⎟
⎝ j =1
⎠
where the x’s are determined by the design matrix, and the β are the corresponding
parameters. We will consider how to accommodate uncertainty in estimation of H k
below.
2.2 POINT TRANSECT SAMPLING
For point transect sampling, (2.1) is replaced by
1
Dˆ k =
2πek
nk
∑ s hˆ
i =1
i y|z
(0 | z i )
(2.2)
where ek = total effort conducted on plot k (number of points on plot multiplied by
(mean) number of visits per point)
nk = number of animal clusters detected on plot k
si = number of animals in ith detected cluster on plot k
hˆy |z ( y | z i ) = fˆy′| z ( y | z i ) = slope of estimated pdf of distance y from the point for ith
detected cluster on plot k, for which covariates z i were recorded
hˆy |z (0 | z i ) = hˆy | z ( y | z i ) evaluated at y = 0 .
Equation (2.2) is obtained by dividing Equation (3.44) of Marques and Buckland
(2004) by the covered area, ekπw2 . We can now write
H
Dˆ k = nk k
2πek
1
where H k =
nk
nk
∑ s hˆ
i =1
i y|z
(0 | z i ) , and then proceed as for line transect sampling.
2.3 QUANTIFYING PRECISION
A two-stage bootstrap allows uncertainty arising from estimating H k to be
accounted for. For a given bootstrap resample, we first generate a resample of the
distances of detections from the line or point. If there is an adequate number of replicate
lines or points on each plot (say at least ten), this can be achieved by resampling lines or
points with replacement for each plot. Note that if a line or point is selected for a
resample, data (including any relevant covariates) from all visits to that line or point are
included. If for example the same lines or points are visited in more than one year, this
strategy ensures that we do not assume that the data recorded from the line or point in
different years are independent. For each resample, we re-estimate the H k and hence the
offset. We also find the counts for each resample on each plot (or each plot-year if for
example a plot is surveyed each year), and these counts are modelled using a Poisson or
negative binomial GLM. At each stage, we could condition on the model selected for the
original data, or reassess which model was best in each resample, as judged for example
by Akaike’s information criterion (AIC), to allow for model uncertainty.
If there is insufficient replication in the design, so that there are too few lines or
points per plot for the above strategy, the naïve bootstrap might be used, where distances
yi are resampled along with the associated covariates z i . Another option is to condition
on z i , and generate a parametric bootstrap resample from the fitted model for f y | z ( y | z ) .
We then estimate the H k for the resample. Neither of these options generates variation
in the count, so we cannot now proceed to the second stage as before. Instead, for each
resample, we could condition on the estimates of H k obtained from stage one, and
generate a bootstrap resample corresponding to the GLM. This could be a parametric
bootstrap resample, obtained by generating bootstrap counts from the fitted model, or the
probability integral transform resampling method might be used (Bravington 1994;
Borchers et al. 1997).
The latter method has the advantage that overdispersion is
preserved in the resamples when a quasi-Poisson GLM is used.
3. EXAMPLE: PONDEROSA PINE PRESCRIBED FIRE STUDY
3.1 STUDY DESIGN
Four National Forests in the southwestern United States were selected as study sites:
the Kaibab, Coconino, and Apache-Sitgreaves National Forests in Arizona, and the Gila
National Forest in New Mexico. On each site, a 247–405-ha prescribed fire treatment plot
was paired with one (Coconino and Gila) or two (Apache-Sitgreaves and Kaibab) control
plots of similar extent and vegetation composition. In our analyses, for sites with two
control plots, we combine the data on the control plots as if they had been a single
contiguous plot. Although overstory vegetation on each site was dominated by ponderosa
pine, other common tree species included Gambel oak (Quercus gambelii) and alligator
juniper (Juniperus deppeana) at lower elevations and Douglas fir (Pseudotsuga
menziesii) at higher elevations in New Mexico (Dickson 2006, Saab et al. 2006).
The four sites were surveyed for birds annually from 2003 to 2005. Prescribed fire
treatments were implemented by US Forest Service District personnel and completed in
the fall of 2003 (Apache-Sitgreaves and Coconino) and spring of 2004 (Gila and Kaibab).
Each treatment was characterized by fire behaviors of low to moderate intensity (see also
Dickson 2006). For each pair of plots, one was burned and the other was not. This was
therefore a BACI design, and we are interested in the treatment by year interaction, as under
the alternative hypothesis that burning affects bird densities, we expect any change from preburning to post-burning to differ between burn and control plots. Point count sampling was
conducted at 25-50 points per plot, where points were located at least 250m apart and
150m from the edge of the study plot. Point counts began just after the dawn chorus and
were completed within five hours. Each point was surveyed either three or four times
during the breeding season between late May and early July. At each point, observers
recorded all birds detected for five minutes and used a laser rangefinder to estimate the
distance between the observer and the detected bird as 0-10 m, 10-25 m, 25-50 m, 50-75
m, 75-100 m and > 100 m (Birds and Burns Network 2003; Dickson 2006).
Several covariates that might affect detectability were recorded: site, plot, cluster
size (number of birds in a detected group), visit number, observer, species, and year.
Additionally, information regarding the vegetation surrounding the point count was
collected including large (>23 cm dbh) tree and snag densities, and fire effects data to
compute a composite burn severity index (Key and Benson 2005; see Saab et al. (2006)
for detailed vegetation sampling description).
3.2 ESTIMATING THE DETECTION FUNCTION
We analyse here data on two warblers: Grace’s warbler and the yellow-rumped
warbler. As both species were expected to show a similar response to burning, we pooled
the data from the two species to increase power. (A more comprehensive analysis would
also analyse the two species separately.) A truncation distance of w = 100 m was found
to be adequate (Figure 1), leaving n = 2300 detections. (Choice of truncation distance is
discussed by Buckland et al., 2001:15-17.) The multiple covariate distance sampling
(MCDS) engine of Distance (Thomas et al. 2006) was used, and AIC used to identify a
suitable detection function model (Table 1). The hazard-rate model with year as a factor
and tree and snag density as covariates was found to fit the data well (Figure 1). Note
that the covariate species was not selected; the two warbler species gave nearly identical
estimated detection functions. Estimated detection probabilities were lower at points
with high tree density, and slightly lower at points with high snag density.
3.3 MODELLING THE PLOT COUNTS
The glm command of the statistical package R (R Development Core Team 2007)
was used to fit a Poisson model to the warbler plot counts (Table 2). The counts analysed
were the sum of counts across the two species, the points on the plot, and the visits made
within a year, giving rise to 24 counts (four sites by two plots per site by three years). To
estimate the offset, the hazard-rate detection function model with year as a factor and tree
and snag density as continuous covariates was used. The number of visits to each point
(i.e. the effort) was also included in the offset. AIC selected the model with all twofactor interactions but no three-factor interaction (Table 2).
We also tried adding
covariates (averaged across the points within the plot for each year) – tree density, snag
density and burn index – to this model, but none led to a reduction in the AIC score, and
so they were not included. (Thus covariates were included in the model for detectability,
and hence in the offset of the count model, but were not included in the linear predictor of
the count model.) When a model with the same structure but with a negative binomial
error distribution was fitted (using the glm.nb command from the MASS package in R,
Venables and Ripley 2002), the limiting form corresponding to Poisson regression was
obtained, so results were unaffected. Similarly, as the residual deviance under this model
(6.04, Table 2) is almost equal to the degrees of freedom (6), inference is unaffected if a
quasi-Poisson regression is conducted.
The above analysis makes no allowance for the estimation of the offset, so that
both the analytic standard errors and AIC may be unreliable guides to model selection.
We therefore calculated bootstrap standard errors and 95% and 99% percentile
confidence intervals for parameters based on 999 bootstrap resamples. To generate
nonparametric bootstrap resamples, points were resampled within plots, and each
resample analysed using the same methods as for the real data. The bootstrap analyses
allow for any dependence between counts at the same point across years, and for
estimation of the offset, and we see that the standard errors are larger on average than the
analytic ones (Table 3).
If prescribed burning affected bird density, we would hope to detect a treatment
by year interaction, because the treatment covariate distinguished between treatment and
control plots, but did not reflect whether the treatment had yet been applied. Just two of
the bootstrap standard errors of Table 3 are (slightly) smaller than the analytic ones, but
these both correspond to the treatment by year interaction.
Inference is therefore
unaffected in this case; we have strong evidence of reduced bird densities on burned
plots in the year following burning (the year2×treatment2 term of Table 3), and
moderate evidence that the effect continues into a second year (the year3×treatment2
term of Table 3). Our results provide evidence that the among-year differences were not
consistent across sites, and that an interaction between type of plot (treatment or control)
and site existed. Although the analytic results gave strong evidence of a difference
between treatment and control plots (see the treatment2 term in Table 3), when
uncertainty arising from estimation of the offset and possible dependence in repeat counts
at a site across years were taken into account, the 95% bootstrap confidence interval for
the difference between treatment and control plots (just) included zero.
4. DISCUSSION
Ecological experiments can be difficult and costly to implement, but inference
from these studies is stronger than from correlative or observational designs. Adequate
replication is often difficult due to the inherent spatial variability in ecosystems, and the
costs and logistics that limit numbers of replicate plots. Attempts should be made to
maximize the number of replicates given available resources, and to minimize ecological
differences in study locations.
Developing methods to analyse data resulting from BACI experiments is vital for
providing land management agencies with guidelines regarding the impacts of their
management decisions (Block et al. 2001). In this context, and including the data we
analysed, Dickson et al. (in press) demonstrated the utility of a BACI design to quantify
the multi-scale (i.e. plot and point level) response of bird density to prescribed fire using
spatial mixed effect models. Similarly, the methods presented in this paper extend
readily to generalized additive models, and random effects are easily accommodated
through the use of generalized linear mixed models (an extension of the count models
advocated by McDonald et al., 2000, to allow for uncertain detectability) or generalized
additive mixed models. In our example, we might prefer to make site a random effect,
which can be achieved by loading the glmm library in R. To gain full benefit from such
an approach, it might be preferable to have more pairs of plots in the design, with fewer
points sampled per plot. However, the extra costs involved in accessing and burning
more sites can be substantial.
In implementing the first stage of the bootstrap resampling to quantify precision
of H k in the prescribed fire study, we conditioned on the hazard-rate detection function
model with variables tree and snag as continuous covariates and year as a factor, because
the other models considered all had a ΔAIC value exceeding 4. Where the choice of
detection function model is less clear-cut, model uncertainty is readily accommodated in
the bootstrap by selecting the detection function model with smallest AIC from each
resample analysis, instead of conditioning on the model selected for the original data.
Similarly, at stage two of the bootstrap, we could reselect from the possible count models
using say AIC, although this complicates inference as most parameters would be included
in the model used to analyse some resamples, but absent from the model used for others.
It is simpler to use the model selected by AIC based on the real data, then to use the
bootstrap confidence intervals to assess whether any of the parameters of that model
could be omitted.
We have demonstrated a flexible, efficient, and quantitatively robust method of
analysing ecological experiments to evaluate changes in wildlife abundance in response
to habitat alterations or other interventions.
ACKNOWLEDGMENTS
We thank Stephanie Jentsch for key contributions to the design and supervision of
the data collection effort. Lisa Bate summarized the vegetation data. Personnel with the
Gila, Apache-Sitgreaves, Coconino, and Kaibab National Forests provided logistical
support for the project and implemented the treatments. Funding was provided by the
Joint Fire Science Program and U.S. Forest Service, Rocky Mountain Research Station,
research work unit numbers 4156, 4852, and 4251. We thank Rudy King, Barry Noon,
Eric Rexstad and Len Thomas, who provided internal reviews of an earlier draft, and the
referee and associate editor. Their comments led to a much improved paper.
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Table 1. AIC scores for selected models for the detection function. Potential continuous
covariates were shrub stem density, large tree density, snag density, and a burn severity
index, while factor covariates considered for inclusion were year at three levels (2003,
2004, 2005) and species at two levels (Grace’s warbler, yellow-rumped warbler). A
stepwise procedure was adopted for inclusion of covariates; the covariate giving rise to
the greatest improvement in AIC was selected at each step, until no additional covariate
reduced the AIC score. The fitted detection function corresponding to the model selected
by AIC was gˆ ( y ) = 1 − exp[−( y / σˆ ) −1.63 ] where
σˆ = 56.2 exp[−0.0024t i − 0.0100u i + 0.37ci + 0.22d i ] , and ti is tree density, ui is snag
density, ci is 1 if observation i was for year 2003 and 0 otherwise, and di is 1 if
observation i was for year 2004 and 0 otherwise. Median tree density was 140 km-2, and
median snag density 2.5 km-2.
Key function
Continuous covariates
Factor covariates
ΔAIC
Half-normal
Half-normal
Half-normal
Half-normal
Hazard-rate
Hazard-rate
Hazard-rate
Hazard-rate
None
None
Tree
Tree, Snag
None
Tree
Tree, Snag
Tree, Snag
None
Year
Year
Year
None
None
None
Year
25.26
20.63
16.94
16.89
16.57
10.14
4.69
0.00
Table 2. Analysis of deviance of warbler analyses, treating the values of the offset as
known constants, Poisson error model. Note that the factor ‘treatment’ refers to
whether a plot is a prescribed fire treatment or a control plot.
df deviance resid. df resid. dev.
Null model
+ site
+ year
+ treatment
+ site×year
+ site×treatment
+ year×treatment
+ site×year×treatment
3
2
1
6
3
2
6
923.34
69.13
11.33
81.64
109.51
10.70
6.04
23
20
18
17
11
8
6
0
1211.69
288.35
219.23
207.89
126.25
16.74
6.04
0.00
AIC
191.1
184.4
190.4
Table 3. Poisson model: estimated coefficients and standard errors. Linear predictor:
intercept + site + year + treatment + site×year + site×treatment +
year×treatment. Levels of site are Apache-Sitgreaves (site1), Coconino (site2), Gila
(site3) and Kaibab (site4); levels of year are 2003 (year1), 2004 (year2) and 2005
(year3); levels of treatment are control (treatment1) and burn (treatment2).
Term
Estimate
Intercept
site2
site3
site4
year2
year3
treatment2
site2×year2
site3×year2
site4×year2
site2×year3
site3×year3
site4×year3
site2×treatment2
site3×treatment2
site4×treatment2
year2×treatment2
year3×treatment2
-11.999
1.485
0.607
1.176
0.729
0.947
-0.523
0.004
-0.591
-0.041
-0.209
-1.680
-0.637
1.105
1.618
0.387
-0.378
-0.242
Analytic
std. error
0.212**
0.223**
0.257*
0.230**
0.248**
0.236**
0.199**
0.259
0.290*
0.268
0.246
0.297**
0.258*
0.188**
0.232**
0.199
0.116**
0.116*
Bootstrap
std. error
0.317**
0.310**
0.345*
0.336**
0.311*
0.273**
0.264
0.305
0.329
0.323
0.264
0.340**
0.292*
0.268**
0.314**
0.289
0.111**
0.111*
*Estimate differs significantly from zero (5% level) using z-test (analytic standard error)
or percentile c.i. (bootstrap)
**Estimate differs significantly from zero (1% level) using z-test (analytic standard error)
or percentile c.i. (bootstrap)
0.8
0.4
0.2
0.0
0.0
0
20
40
60
80
100
Distance from point (m)
1.0
0.8
0.6
0.4
0.2
0
20
40
60
Distance from point (m)
0
20
40
60
Distance from point (m)
0.0
Detection probability
0.6
Detection probability
0.6
0.4
0.2
Detection probability
0.8
1.0
1.0
1.2
Figure 1. Estimated detection function by year for Grace’s and yellow-rumped warblers,
pooling over values of continuous covariates tree and snag density. Years 2003 (top left),
2004 (top right) and 2005 (bottom left). Note that the histogram bars represent number of
detections divided by distance from the point; this transformation allows the fit of the
detection function to the data to be assessed visually (Buckland et al., 2001:147-150).
80
100
80
100