A Hierarchical Model for Estimating Change in American Woodcock Populations

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Management and Conservation Article

A Hierarchical Model for Estimating Change in American
Woodcock Populations
JOHN R. SAUER,1 United States Geological Survey, Patuxent Wildlife Research Center, 12100 Beech Forest Road, Laurel, MD 20708, USA
WILLIAM A. LINK, United States Geological Survey, Patuxent Wildlife Research Center, 12100 Beech Forest Road, Laurel, MD 20708, USA
WILLIAM L. KENDALL, United States Geological Survey, Patuxent Wildlife Research Center, 12100 Beech Forest Road, Laurel, MD 20708, USA
JAMES R. KELLEY, United States Fish and Wildlife Service, Division of Migratory Bird Management, BH Whipple Federal Building, 1 Federal Drive,
Fort Snelling, MN 55111, USA
DANIEL K. NIVEN, National Audubon Society, Illinois Natural History Survey, 607 East Peabody Drive, Champaign, IL 61820, USA

ABSTRACT The Singing-Ground Survey (SGS) is a primary source of information on population change for American woodcock
(Scolopax minor). We analyzed the SGS using a hierarchical log-linear model and compared the estimates of change and annual indices of
abundance to a route regression analysis of SGS data. We also grouped SGS routes into Bird Conservation Regions (BCRs) and estimated
population change and annual indices using BCRs within states and provinces as strata. Based on the hierarchical model–based estimates, we
concluded that woodcock populations were declining in North America between 1968 and 2006 (trend ¼ 0.9%/yr, 95% credible interval:
1.2, 0.5). Singing-Ground Survey results are generally similar between analytical approaches, but the hierarchical model has several
important advantages over the route regression. Hierarchical models better accommodate changes in survey efficiency over time and space by
treating strata, years, and observers as random effects in the context of a log-linear model, providing trend estimates that are derived directly
from the annual indices. We also conducted a hierarchical model analysis of woodcock data from the Christmas Bird Count and the North
American Breeding Bird Survey. All surveys showed general consistency in patterns of population change, but the SGS had the shortest
credible intervals. We suggest that population management and conservation planning for woodcock involving interpretation of the SGS use
estimates provided by the hierarchical model. (JOURNAL OF WILDLIFE MANAGEMENT 72(1):204–214; 2008)

DOI: 10.2193/2006-534
KEY WORDS American woodcock, hierarchical model, route regression, Scolopax minor, Singing-ground Survey, trend analysis.

Population status of the American woodcock (Scolopax
minor) is monitored by the Singing-Ground Survey (SGS),
a roadside survey coordinated by the United States Fish and
Wildlife Service and the Canadian Wildlife Service. The
SGS is one of several major roadside, count-based surveys
that index change of bird populations in North America.
Other count-based surveys include the Call-Count Survey
(Sauer et al. 1994) for mourning doves (Zenaida macroura)
and the North American Breeding Bird Survey (BBS; Sauer
et al. 2003). These surveys share a common design: counts
of birds are collected along roadsides without any attempt to
estimate the proportion of animals missed during counts; a
variety of studies have indicated that it is likely that the
assumption of consistent proportion of birds counted is at
least sometimes invalid (e.g., Dwyer et al. 1988).
To minimize the consequences of design deficiencies,
analyses of SGS data have tended to be model based, using
covariates in analyses to model factors that influence
detectability of birds. Observers often differ in their ability
to count woodcock, hence covariates to accommodate
observer-associated differences in counts have been included
in most analyses of SGS data. Analysis methods have
evolved from base-year methods that estimate change from
ratios of counts from comparable routes (Tautin et al. 1983),
to route regression methods in which observers are treated as
covariates and change is estimated by averages of routespecific regressions (Sauer and Bortner 1991), to a modified
route regression in which Poisson regression with log links
1

E-mail: [email protected]

204

is used to estimate change on individual routes (Link and
Sauer 1994, Kelley and Rau 2006). The route regression
approach, although cumbersome, appears to provide reasonable estimates of population change (Thomas 1996, Link
and Sauer 1998) and has the great advantage that it can be
fit to almost any data set, no matter how unbalanced the
data in terms of changes in survey locations and missing
data. Criticisms of the route regression method generally
attack the ad hoc nature of the weighted average used to
estimate change (ter Braak et al. 1994), the lack of
goodness-of-fit methods to assess when it is inappropriate,
and the limited view of the population dynamics provided by
a trend estimate (James et al. 1990). Methods now exist that
provide a more coherent view of population change and
provide new opportunities for controlling for detection of
animals (e.g., Link and Sauer 2002).
Link and Sauer (2002) suggested the use of hierarchical
models to estimate regional population change from count
data. These models incorporate the complex structure of the
data provided by count surveys, allowing analysts to
explicitly incorporate model-based assumptions regarding
the distribution of observer effects, stratum effects, effort,
and other features over space and time. These methods have
been applied to both BBS (Link and Sauer 2002) and
Christmas Bird Count (CBC; Link et al. 2006) analyses and
provide new opportunities for estimation of population
change for species with limited data. The hierarchical model
also can include spatial associations and covariates for
abundance and change (Thogmartin et al. 2004) to better
meet regional conservation needs.
The Journal of Wildlife Management  72(1)

We applied the Link and Sauer (2002) model to estimate
regional population change from SGS data, compared the
results to the estimating equation-based route regression
estimation methods used in earlier analyses, and extended
the analysis by conducting an analysis of woodcock
population change for Bird Conservation Regions (BCRs;
Sauer et al. 2003). Bird Conservation Regions have become
a primary geographic unit for regional bird conservation
planning in the North American Bird Conservation
Initiative. Summary of the survey results by BCRs is needed
for regional prioritization activities, including development
of the American Woodcock Conservation Plan, and for
development of models that associate woodcock populations
with environmental features that influence population
change. Sauer et al. (2003) implemented an analysis of the
BBS within BCRs by using the intersection of states or
provinces and BCRs as the strata for analysis. Results of
these stratum-level analyses can be aggregated for inference
about either BCRs or states and provinces. We implemented
a similar analysis for the SGS and documented the
consequences of implementing the hierarchical analysis
using state or provinces and strata relative to using BCRs
within states or provinces as strata.
The CBC (Link et al. 2006) and the BBS (Sauer et al.
2003) collect woodcock data, but analyses of woodcock data
from these surveys has been controversial (Straw et al. 1994)
due to limitations imposed by survey protocols and timing.
However, it is of interest to determine whether patterns of
population change are similar among surveys. Comparisons
of survey results do not provide insights into which (if any)
survey is providing unbiased estimates when results differ
among surveys (Sauer et al. 1994), but systematic differences
between roadside and nonroadside results would be reflected
in consistent differences between CBC and SGS or BBS
results. We compared the results of the SGS at the surveywide scale with continent-scale population changes estimated from the CBC (Link et al. 2006) and the BBS (Link
and Sauer 2002) using hierarchical models similar to those
applied to the SGS.

STUDY AREA
The SGS and BBS provided information from most of the
breeding range of the American woodcock. The SGS was
conducted in 25 states and provinces, grouped into Eastern
and Central Management Units that were coincident with
administrative Flyways. The BBS provided information for
a similar region, although we noted that neither the SGS
nor the BBS provided information from the northern
portions of Quebec and Ontario, Canada. The CBC survey
area included the wintering range of woodcock in the central
and southern United States.

METHODS
The Singing-Ground Survey
The SGS is based on roadside surveys conducted once each
year during early spring, timed to permit counting after
Sauer et al.  American Woodcock Population Change

woodcock migration has occurred but while courtship is still
occurring. Routes contain 10 counting stops along a 5.4-km
segment of secondary roads, at which an observer records all
woodcock heard during a 2-minute count period during
twilight. The SGS was begun in its present form in 1968,
although surveys have not been conducted in all years in
some of the states and provinces. To maximize efficiency,
not all routes are run each year, and some routes are only
occasionally surveyed (Sauer and Bortner 1991).
Analysis Methods
Estimating equation trend analysis and residual indices.—
In the analysis of SGS data presently used for setting harvest
regulations, Poisson regressions with log links are used to
estimate rates of change for individual survey routes (Kelley
and Rau 2006). Observer effects are included, allowing a
different intercept for each observer in the analysis (Link
and Sauer 1994). Change for a region is estimated as an
average of the route slopes, in which the route-specific
slopes are weighted by mean abundance of woodcock on the
route and by a route-specific measure of survey consistency
(a variance wt; Link and Sauer 1994). For management unit
estimates, an additional area weight is included to
accommodate regional differences in sampling. Variances
of these trend estimates are estimated by bootstrapping. See
Geissler and Sauer (1990) and Link and Sauer (1994) for
additional information about the route regression method
and the weights.
Residual indices are an approach for displaying year-toyear variation around an estimated regional trend. We
estimated observer effects on each route after subtracting the
effect of the regional trend from the yearly count. We then
estimated the residual distance from the yearly counts and
the trend and observer-adjusted predicted counts and
averaged these residuals by year for all routes in the region.
We then added these yearly average residuals to a regional
predicted trajectory for the year, which we estimated by
projecting the estimated regional yearly trend from the
regional mean count for the midyear of the survey. For a
complete method description, see Sauer and Geissler (1990).
To assess continuity with historical analysis methods, we
conducted an estimating equations analysis of trends and
annual indices by state or province and management units
for the period 1968–2006 and present these results for
comparison with hierarchical model results.
Hierarchical log-linear model.—Hierarchical models
provide a means for directly estimating regional population
change from the SGS. In hierarchical models, factors are not
governed by fixed parameters. Instead, distributions of
attributes such as year, stratum, and observer effects at the
level of states (or other strata) are conditional on parameters
that are also random variables. This hierarchical formulation
permits us to specify how attributes such as population
change are distributed over large regions that are divided
into strata and allows us to make statements about the
regional collections of population attributes such as trends
and abundances that are based on the underlying regional
parameters, not on the region-specific estimates. We
205

modeled the influence of regions, observers, and other
factors on the distributions of the parameters influencing
counts, rather than on the counts themselves. This
eliminates the need for ad hoc procedures for accommodating regional differences in precision of counts in summary
analyses. This approach to modeling, and its value for
analysis of bird populations, is discussed in Link and Sauer
(2002).
Hierarchical models are generally fit using Bayesian
methods, in which inference is based on the distributions
of parameters conditional on the data (the posterior
distributions). A Bayesian analysis requires that both the
prior distributions of parameters and the sampling distribution of the data conditional on the parameters be
specified. From these distributions, the posterior distribution can be found through integration (a difficult computation for most problems of interest), or through a
simulation approach known as the Markov chain Monte
Carlo method (MCMC; Gilks et al. 1996). The MCMC
methods use first-order Markov chains simulated based on
partially specified versions of the posterior distributions,
allowing approximation of the distribution; sample mean,
variance, and percentiles, when appropriately drawn from
the simulations, approximate the true mean, variance, and
percentiles.
The hierarchical model is an overdispersed Poisson
regression with fixed and random effects. Counts Yi,j,t
(i indexes stratum, j for unique combinations of route and
observer, and t for yr) are independent Poisson random
variables with means ki,j,t that are described by log-linear
functions of explanatory variables,
logðki; j;t Þ ¼ Si þ bi ðt  t  Þ þ xj þ ci;t þ gI ð j; tÞ þ ei; j;t
ð1Þ
stratum-specific intercepts (S), slopes (b), and effects for
observer–route combinations (x), year (c), start-up [g; I( j,t)
is an indicator for first year of survey for an observer], and
overdispersion (e). t * is a baseline year (set to 19) from
which change is measured. See Link and Sauer (2002) for
discussion of the role of parameters and hyperparameters in
Bayesian analyses. Hyperparameters Si and bi are given
diffuse (essentially flat) normal distributions. Other effects
had mean zero normal distributions, but observer-route
effects were identically distributed, all having the same
variance r2x; overdispersion effects (e) were identically
distributed with common variance r2e; and we allowed
variance of the year effects (c) to vary among strata (r2c,i).
We assumed all these variances to have flat inverse gamma
distributions.
Combining information among regions.—Stratum-specific annual indices of abundance are the year effects, scaled
by the stratum and trend effects:
ni;t ¼ exp½Si þ bi ðt  t  Þ þ ci;t þ 0:5r2 x þ 0:5r2 e  ð2Þ
ni,t is an index to the number of birds per route in stratum i
at year t (Link and Sauer 2002). Variance components are
added to accommodate asymmetries in the log-normal
206

distribution. Stratum indexes are Ni,t ¼ Aini,t, where Ai is the
area of the stratum, and we defined composite indices for
collections of strata as sums of Ni,t divided by the total areas.
Because the n are not area-specific population estimates, we
did not present the Ni,t as population totals; they are simply
a weighted total of the route indexes. We defined trend as
an interval-specific geometric mean of proportional changes
in population size, expressed as a percentage (cf., Link and
Sauer 1998). Thus the trend from year ta to year tb for
stratum i is 100(Bi  1)%, where
 t t1
ni;tb b a
Bi ¼
ð3Þ
ni;ta
The composite trend B is calculated analogously
X as
Ni;t ,
100(B  1)%, using the composite indices Nt ¼
i
to calculate
 t t1
Nt b b a
B¼
ð4Þ
Nta
For presentation and comparison with residual index year
effects, we scaled the composite indices Nt by the total area,
obtaining a summary on the scale of birds per route.
Logistics of Analyses
Fitting the hierarchical model.—We used the program
WinBUGS (Spiegelhalter et al. 1995) to fit this model for
states and strata. WinBUGS is a user-friendly program for
analysis of hierarchical models and contains a variety of
model diagnostics to assess stability of the estimates (Link
and Sauer 2002). The program conducts the MCMC
analysis, presents summary statistics to allow users to
determine when the Markov chains become stationary,
and summarizes results based on the MCMC replicates. It
also allows users to define and estimate derived parameters
(such as the composite indices) and their variances and
output the MCMC replicates for additional summaries.
From the replicates, both estimates and credible intervals
can be calculated.
Scales of summary.—Historically, SGS trend and annual
indices were estimated at the scale of states or provinces,
flyway-based Eastern and Central regions, and survey-wide.
Consequently, we first conducted the hierarchical model
analysis using states and provinces as the fundamental strata
for comparison with historical results (the hierarchical state
or province analysis).
We also used SGS data to estimate woodcock population
change at the scale of BCRs. Because SGS routes have
historically been administered and stratified within states
and provinces, we retained the political units as a component
of the stratification but redefined ‘‘strata’’ in the second
analysis as the 55 regions formed by the intersection of
BCRs and states or provinces (i.e., the strata are the BCRs
within each state and province). Results from these strata
can be aggregated within states or provinces and also can be
aggregated within BCRs or larger regions. Hereafter, we call
this analysis the hierarchical-BCR analysis.
The Journal of Wildlife Management  72(1)

Summary data for the survey.—The SGS differs greatly
in consistency of information over its range, and these
differences affect the quality of the estimates of population
change. To summarize survey consistency for states,
provinces, and BCRs, we calculated total number of routes,
mean number of routes surveyed each year, the mean
number of years routes were surveyed, and the mean
duration (span of yr) of routes in each region. Duration of
routes indicates the portion of the interval covered by an
average route.
Christmas Bird Count Analysis
The CBC is coordinated by the National Audubon Society.
Bird observations are collected by variable numbers of
volunteer observers on a selected day in mid-late December
within predefined 24.13-km (15-mile) diameter circles. See
Link et al. (2006) for information regarding the design and
analysis of the CBC. To apply a hierarchical log-linear
model used for the SGS to the CBC, model components
must be added to accommodate variation in participation
(no. of counters) and the methods of data collection among
circles and through time. Consequently, the CBC model
includes an additional component, a stratum-specific effect
of effort Bi (ni,j,t pi  1)/pi (Link et al. 2006; Bi is the coeff.
relating transformed effort to counts, ni,j,t is the scaled effort
for circle j in stratum i at time t, and pi is the exponent that
defines the shape of the effort relationship). Effort values are
scaled to an overall mean. If the effort expended in
producing count Yi,j,t is equal to the overall mean, then
ni,j,t ¼ 1, and the effort effect is zero. Effort parameters pi
and Bi are normally distributed with means lp and lB and
variances sp and sB; the means and variances are fixed effects.
We used standard noninformative priors; for lp, we used a
uniform prior on the interval [4, 4] (Link et al. 2006). As
our goal in this analysis was to provide a comparative time
series from CBC data, we do not present regional
summaries or evaluation of effects of effort in this report.
We used all CBC data for woodcock collected between the
1965 CBC (the 66th yr of the count) and the 2003 count
(the 104th count) and conducted the analysis using BCRs
within states as strata, compositing results as described in
Link et al. (2006) to form survey-wide results.
North American Breeding Bird Survey Analysis
The BBS is an extensive roadside survey of breeding birds,
conducted in June along 4,000 roadside survey routes.
Unfortunately, the 50 3-minute counts are collected early in
the morning along the routes, and few woodcock are
encountered on the survey. Although observed on 572
routes over the survey interval, the mean abundance on
routes in the analysis is 0.03 woodcock per route within the
woodcock’s range, indicating that the species is only
observed occasionally. The formal description of the
hierarchical model for BBS data is given in Link and Sauer
(2002). We applied the hierarchical model to estimate
population change for the species, using BCRs within states
and provinces as strata. We only present regional estimates
for comparison with other survey results.
Sauer et al.  American Woodcock Population Change

RESULTS
Singing-Ground Survey Analysis
We estimated population trajectories for 25 states and
provinces, for 12 BCRs, for Eastern and Central Management Units, and for the entire surveyed area. The number of
survey routes varied greatly among states (Table 1), ranging
from 3 (DE and RI) to 153 (MI). Overall, we used data
from 1,232 routes in the analysis. For the hierarchical model
analyses, results were based on 10,000 simulations after a
burn-in (initial iterations to allow simulations to stabilize) of
40,000–90,000 iterations. We assessed stability of results
from observation of graphs of results and autocorrelations.
Comparison of Markov chain error with standard deviations
suggested that convergence had occurred (i.e., the Markov
chain error estimated from binned estimates of the replicates
is generally ,5% of the SDs [Spiegelhalter et al. 1995]; for
the trend estimates of states and provinces the Markov chain
errors were 2.8% of the SDs). The hierarchical analysis was
based on 28,754 counts conducted by 7,290 observers in 55
BCR-state or BCR-province units within the 25 states and
provinces.
Regions vary greatly in the consistency of information
(Table 1). In Manitoba, Canada, the survey was initiated in
1992, hence the number of years of survey and duration of
survey are much lower than other regions; thus estimates of
long-term change from Manitoba have low credibility. Of
the states and provinces providing data from the entire
survey interval (1968–2006), Illinois, USA, and Quebec,
Canada, both had mean number of years ,10 surveyed per
route, and Quebec also had mean range of ,20 years
covered for the survey interval (Table 1). Many regions also
experienced a large amount of turnover of observers of
survey routes, with Delaware having 8 observers surveying
an average route over the interval. We summarize attributes
of some early surveys in North Carolina, USA (N ¼ 16),
which we did not include in the analysis.
In conducting summary analyses for BCRs, it was evident
that few data exist at the northern edge of the survey region.
In particular, only the southern edge of the Boreal Softwood
Shield was sampled in either province, and only 21 routes
were surveyed in this BCR (Table 1). Because the historical
analysis only covered the areas actually surveyed in the
provincial-level summary of the SGS, this Boreal Softwood
Shield cannot be considered a part of the area sampled by
the SGS. Although we present estimates for the region, we
follow the historic precedent and do not include it from our
regional summaries in the hierarchical-BCR analyses. We
also note that 5 routes in the Boreal Taiga Plains in
Manitoba were also surveyed.
Trend estimates.—Trends (yearly % changes) for states
and the Eastern, Central, and survey-wide summary regions
estimated over 1968–2006 (Table 2) were very consistent
between the 2 hierarchical modeling approaches. Estimating
equation trends tended to be larger in magnitude (21/25
states and provinces, based on state or province strata) and
more variable (CI width larger in 20/25 states and
provinces) than trends based on hierarchical models.
207

Table 1. Summary by Bird Conservation Region (BCR), state, and provinces of Singing-Ground Survey data used in the hierarchical model analysis, 1968–
2006. Sample sizes, mean number of years of survey on each route, mean number of observers that surveyed each route, and mean range of years of coverage
for routes are presented for each region.
BCR, state, or province

No. routes

No. yr

SE

No. observers

SE

Mean of range

SE

Boreal Taiga Plains
Boreal Softwood Shield
Prairie Potholes
Boreal Hardwood Transition
Lower Great Lakes–St. Lawrence Plain
Atlantic Northern Forest
Eastern Tallgrass Prairie
Prairie Hardwood Transition
Central Hardwoods
Southeastern Coastal Plain
Appalachian Mountains
Piedmont
New England–Mid-Atlantic Coast
CT
DE
IL
IN
ME
MB
MD
MA
MI
MN
NB
NH
NJ
NY
NC
NS
OH
ON
PA
PE
PQ
RI
VT
VA
WV
WI

5
21
16
292
161
290
95
165
35
13
200
41
72
10
3
42
56
67
25
25
24
153
119
66
18
18
117
16
68
72
146
75
13
60
3
22
72
54
116

7.2
10.6
10.8
24.9
20.6
24.2
11.7
21.9
12.7
11.8
17.2
13.3
19.8
20.6
19.3
9.3
14.2
28.6
7.8
16.6
21.8
27.4
20.6
26.8
29.1
18.4
21.8
4.2
19.0
17.0
19.3
15.3
25.5
8.6
14.3
29.5
13.2
17.4
23.9

0.92
1.23
1.59
0.61
0.78
0.60
0.62
0.76
0.76
1.77
0.54
1.01
0.96
3.34
6.36
0.49
0.80
1.02
0.62
1.45
1.73
0.78
0.91
1.22
1.66
1.88
0.86
0.23
1.26
0.98
0.82
0.86
2.50
0.69
3.48
1.57
0.70
1.12
0.93

2.4
4.3
4.3
5.5
5.4
5.3
4.5
6.3
5.5
3.9
4.3
4.2
5.4
6.1
8.0
4.9
5.3
5.4
2.1
6.2
4.7
6.2
5.3
6.0
6.3
3.9
4.9
2.4
4.4
4.5
5.7
4.0
5.5
2.9
3.0
6.3
4.1
4.1
6.0

0.24
0.54
0.69
0.16
0.22
0.14
0.18
0.19
0.31
0.42
0.12
0.35
0.29
0.71
2.08
0.26
0.23
0.25
0.15
0.41
0.50
0.21
0.26
0.33
0.50
0.37
0.19
0.26
0.28
0.20
0.27
0.21
0.51
0.24
1.00
0.46
0.19
0.23
0.20

13.6
17.1
22.4
31.1
30.7
32.4
28.6
34.5
32.1
29.8
32.9
30.4
32.6
35.0
32.3
25.3
32.9
36.0
11.4
32.2
30.2
33.5
31.3
32.0
37.3
28.6
32.6
4.3
28.7
33.1
26.6
30.8
30.5
17.3
37.0
36.8
32.2
32.4
35.5

0.51
1.49
3.20
0.59
0.79
0.49
0.93
0.50
1.12
3.20
0.51
1.54
0.87
1.06
4.41
1.31
0.92
0.76
0.59
2.01
2.02
0.73
0.77
1.06
0.32
2.31
0.88
0.24
1.29
0.77
0.86
1.25
2.56
1.41
0.58
0.40
0.90
0.94
0.58

Estimates from states or provinces with small sample sizes
(CT, DE, RI) tended to be quite different between
hierarchical and route regression methods, but with large
confidence and credible intervals, reflecting the low quality
of the estimates. However, several other states were also
poorly estimated (NJ, MA, IL, OH) in the estimating
equation results. Note that hierarchical structure, by
assuming that trend parameters are random effects varying
among strata, results in fewer extreme estimates, a clear
benefit of a hierarchical analysis (Link and Sauer 2002). At
the regional scale, precision does not consistently vary
among analyses, although the hierarchical model estimates
from the state–province analysis are always most precise.
Aside from differences in magnitude of trend and size of
confidence (or credible) intervals among analyses, patterns of
population change from the hierarchical models were similar
to historical analyses. Woodcock populations are declining
range-wide; a primary distinction between analyses is the
consequence of partitioning Quebec and Ontario into BCR
units, leading to changes in size of credible intervals in both
208

the provinces and the regional results. Bird Conservation
Regions results (Table 3) suggested declining populations in
the eastern portions of the range and imprecisely estimated
trends in the northern and southwestern parts of the
woodcock range. We present results for the Boreal Softwood Shield, although we caution (as noted above) that
coverage only extends to the southern portion of the BCR.
Population annual indices.—We present population
annual indices for selected states and provinces in the
Eastern (Fig. 1) and Central (Fig. 2) Regions, as well as in
the larger regions (Fig. 3). Precision of indices can only be
estimated for the hierarchical model–based estimates and are
reflected by the credible intervals around the hierarchical
analyses. Because trends were defined as the ratios of the
population indices in the hierarchical models, description of
change and its precision was simple to calculate for any
interval.
Because the relative abundance scaling (i.e., the level of the
time series) is based on slightly different values (the residual
index value is scaled to the relative abundance in the midyear
The Journal of Wildlife Management  72(1)

Table 2. Estimated population trends (% change/yr) for 1968–2006 for 3 analyses of American woodcock population data from the Singing-Ground Survey.
A hierarchical Bird Conservation Region (BCR) analysis, a hierarchical BCR state or province analysis, and an Estimating Equations analysis are presented.
We present estimated trends, credible intervals (2.5 and 97.5 percentile) for the hierarchical model analyses, and confidence intervals for the estimating
equations analysis.
Hierarchical model
State or provincial scale

BCR within state or province scale

Credible interval

Estimating equations

Credible interval

CI

Region

Trend

2.5%

97.5%

Trend

2.5%

97.5%

Trend

2.5%

97.5%

CT
DE
ME
MD
MA
NB
NH
NJ
NY
NS
PA
PE
PQ
RI
VT
VA
WV
Eastern
IL
IN
MB
MI
MN
OH
ON
WI
Central
Survey

4.4
1.0
1.3
4.0
2.4
0.9
0.0
6.2
1.4
1.1
1.7
1.4
0.3
11.6
0.3
5.3
2.8
0.9
3.1
4.2
3.5
1.1
0.0
2.0
0.8
0.8
0.9
0.9

6.7
7.4
2.0
5.7
3.5
1.9
1.2
7.9
2.0
2.1
2.6
3.1
1.2
17.8
1.6
6.5
3.8
1.4
0.1
5.9
6.6
1.5
0.7
2.9
1.4
1.4
1.3
1.2

2.1
5.2
0.6
2.3
1.1
0.2
1.3
4.3
0.9
0.2
0.8
0.3
1.7
6.1
1.0
4.1
1.9
0.4
6.4
2.7
0.4
0.6
0.7
1.1
0.0
0.2
0.6
0.5

5.0
0.5
1.2
4.3
2.1
0.9
0.2
5.8
1.5
1.1
1.2
1.5
0.3
11.6
1.0
4.4
2.9
0.8
2.6
4.0
6.1
0.9
0.1
2.0
1.0
0.9
0.7
0.7

7.1
6.8
1.9
6.1
3.6
2.0
1.5
7.5
2.0
2.1
2.2
3.1
2.6
17.7
2.2
5.8
3.9
1.5
0.4
5.6
12.7
1.7
0.7
2.9
0.6
1.5
1.1
1.1

2.6
5.6
0.6
2.4
0.4
0.2
1.1
3.5
0.9
0.2
0.1
0.3
3.7
5.8
0.3
2.9
1.9
0.3
6.1
2.4
5.6
0.0
1.0
1.1
2.8
0.3
0.3
0.2

10.4
2.9
1.9
9.7
4.6
0.5
1.2
8.9
2.5
0.2
3.4
1.6
1.3
16.3
0.7
11.1
2.7
1.9
24.5
7.1
2.4
1.7
1.0
6.2
1.9
1.9
1.8
1.9

18.4
10.7
2.9
18.2
9.5
1.8
1.3
11.2
3.7
2.0
5.7
3.7
5.0
25.4
2.7
15.6
4.4
2.5
13.2
13.1
6.1
2.6
2.0
9.6
2.8
2.7
2.3
2.3

2.5
16.4
1.0
1.2
0.2
0.8
3.7
6.6
1.2
1.6
1.1
0.4
2.3
7.2
1.2
6.6
0.9
1.3
62.3
1.0
1.4
0.7
0.0
2.8
1.0
1.1
1.3
1.5

of the survey; the hierarchical model yr effects are scaled by
estimated stratum effects in the context of the model), time
series tended to be scaled to different levels. We note that,
generally, the pattern of population change as estimated by
Table 3. Estimated population trends (yearly % change) of American
woodcock for 1968–2006, based on the hierarchical Bird Conservation
Region state or province analysis. We present trend and credible intervals.
Credible interval
BCR

No.

Trend

2.50%

97.50%

Boreal Softwood Shield
Prairie Potholes
Boreal Hardwood Transition
Lower Great Lakes–
St. Lawrence Plain
Atlantic Northern Forest
Eastern Tallgrass Prairie
Prairie Hardwood Transition
Central Hardwoods
Southeastern Coastal Plain
Appalachian Mountains
Piedmont
New England–MidAtlantic Coast

8
11
12

5.0
0.5
1.2

7.1
6.8
1.9

2.6
5.6
0.6

13
14
22
23
24
27
28
29

4.3
2.1
0.9
0.2
5.8
1.5
1.1
1.2

6.1
3.6
2.0
1.5
7.5
2.0
2.1
2.2

2.4
0.4
0.2
1.1
3.5
0.9
0.1
0.1

30

1.5

3.1

0.34

Sauer et al.  American Woodcock Population Change

hierarchical models and estimating equations were very
similar (e.g., NB, Fig. 1; MI, Fig. 2). Regions with
inconsistent coverage or small samples (e.g., CT, PQ, Fig.
1; IL, Fig. 2) tended to show larger differences in magnitude
of year-to-year changes between the residual indices and the
hierarchical model–based indices. Hierarchical model indices were similar except in Ontario and Quebec, where the
large areas associated with each BCR influenced the overall
level of the composite index. State and regional indices
reflected the slightly higher magnitudes of declines from the
route regression trend indices; extreme yet imprecise trend
estimates in the route regression method tended to have
residual indices that suggested large population changes. In
contrast, hierarchical model indices suggested more moderate population change.
Population annual indices and credible intervals from each
BCR documented regional variation in precision of indices
(as shown by width of band formed by the CIs) and also
showed the regional patterns in population change (Fig. 4).
Boreal Softwood Shield results were based on one route (in
ON) prior to 1973, and small numbers of routes were added
incrementally until 1995 when the maximum number (21)
was attained. Consequently, credible intervals are very large
209

Figure 2. Annual indices of American woodcock populations in Michigan,
USA, and Ontario, Canada, from a hierarchical analysis of Singing-Ground
Survey data, 1968–2006, using state-provincial strata (¤), their 95%
credible intervals (m), from a hierarchical analysis using Bird Conservation
Regions as strata (*), and for a residual index analysis using state-provincial
strata (x).

that were initiated between 1969 and 1980. Other BCR
estimates portray more precise views of population change,
consistent with the increased precision of the estimated
trends (Table 3).

Figure 1. Annual indices of American woodcock populations in New
Brunswick, Canada; Connecticut, USA; and Quebec, Canada, based on
data from the Singing-Ground Survey, 1968–2006. Results are presented
for a hierarchical model analyses, conducted using state-provincial strata
(¤), (with 95% credible intervals m); a hierarchical model analysis using
Bird Conservation Regions as strata (*, Quebec only), and for a residual
index analyses using state-provincial strata (x).

in the early years and are not displayed in Figure 4. The
Prairie Pothole BCR also has imprecise estimates in early
years. It is largely peripheral to the survey, and is only
represented prior to 1992 by 10 routes in Minnesota, USA,
210

Comparative Survey Analysis
Although year-to-year changes were often inconsistent
between SGS (state or province), CBC, and BBS
continental-scale analyses of woodcock population changes,
overall patterns of population change were quite consistent
among surveys (Fig. 5). Estimated population change over
the intervals for the CBC (interval: 1965–2003; trend:
1.8%/yr; 95% CI: 2.8, 0.9) and the BBS (interval:
1966–2005; trend: 1.5%/yr, 95% CI: 2.8, 0.2) show
declining populations but have larger credible intervals than
the SGS estimates (interval: 1968–2006; trend: 0.9%/yr,
95% CI: 1.2, 0.5). Note that the scaled indices were
large for both the CBC and the BBS in 1965–1966 (Fig.
5).
The Journal of Wildlife Management  72(1)

results in documenting this decline. Our analysis does not
justify speculation about causes of regional declines; we refer
readers to the American Woodcock Conservation Plan (J.
R. Kelley, United States Fish and Wildlife Service,
unpublished document) for information regarding possible
causal factors influencing the declines, particularly changes
in amounts of successional habitats.

Figure 3. Composite annual indices of American woodcock for the Eastern
and Central Management Units and the entire survey area based on
Singing-Ground Survey data, 1968–2006, from a hierarchical stateprovincial analysis (¤) with 95% credible intervals (m), from a
hierarchical-BCR analysis (*), and from residual indices (x).

DISCUSSION
American woodcock populations are declining in North
America, and SGS results are consistent with other survey
Sauer et al.  American Woodcock Population Change

Using Hierarchical Log-Linear Models for SGS Analyses
Our analysis of SGS data using historical methods and the
hierarchical log-linear model indicates several benefits
associated with the hierarchical models. First, estimation is
generally more efficient, providing smaller credible intervals
for estimates of trend in most states and provinces. The
quality of information varies greatly among strata, and the
hierarchical model permits information from the collection
of regions to inform the estimation both for each region and
for the composite estimates. This use of information from
the ensemble to improve individual estimates is a wellknown phenomenon (Efron and Morris 1977). Second, the
model allows direct estimation of population trend (intervalspecific change) from composite annual indices, unlike the
2-step process used in earlier analyses of SGS data. Third,
credible intervals can be estimated directly for the annual
indices as part of the analysis. Although extremely
computer-intensive, computer programs are now readily
available for estimation of hierarchical models using
MCMC methods (e.g. Speigelhalter et al. 1995 and other
programs cited therein).
A referee suggested that a negative binomial distribution
would be an alternative to the Poisson log-normal mixture
distribution to model counts in the analysis (e.g., White and
Bennetts 1996). In our view, there is no compelling
statistical reason for preferring one over the other. The
marginal distribution of the Poisson-gamma model (i.e.,
negative binomial) can be written down in closed form,
which is useful if one chooses to use maximum likelihood as
a means for model fitting. However, when using Bayesian
methods for model fitting, the Poisson log-normal model is
easier to use, as it has both a natural hierarchical structure
and natural hyperprior distributions. Aside from these
operational considerations, we also note that the models
are extremely similar. Both are overdispersed relative to the
Poisson, with the overdispersion induced by a random mean
governed by a 2-parameter distribution. In many cases, the
log-normal and gamma distributions are virtually indistinguishable, hence the Poisson mixed versions will be even less
distinguishable. It is quite unlikely that inference will be
affected by the choice. To illustrate this, we fit an
overdispersed Poisson log-normal distribution with mean
¼ 5 and variance ¼ 25, along with the closest approximating
negative binomial. The Kullback–Leibler distance (Burnham and Anderson 1998) is very small (approx. 0.01).
Consequently, it would be difficult to distinguish the 2
distributions without a very large sample size.
Use of hierarchical models to estimate population change
from count surveys avoids many of the ad hoc aspects of the
route regression analysis. The route regression method has
211

212

The Journal of Wildlife Management  72(1)

Figure 5. Comparative analysis of population change in continental
American woodcock populations as surveyed by the Singing-Ground
Survey (¤), Christmas Bird Count (*), and North American Breeding Bird
Survey (&). Indices have been standardized by subtracting the mean of the
time series from each observation and dividing the difference by the
standard deviation of the time series.

been widely used because it provides a means for estimating
trend for complicated data sets that are often missing years
and require covariates. The route estimates can be easily
calculated then combined using abundance, relative precision, and area weights to form aggregate estimates at any
geographic scale. Unfortunately, this flexibility has a cost, in
that it is often unclear how data within the interval of
interest is used in the aggregation. Abundance and precision
quantities used in the procedure are also controversial, as
they are only proportional to the actual quantities of interest,
and estimation of variances require use of bootstrapping
procedures (ter Braak et al. 1994, see Sauer et al. 2003 for a
discussion of route-weighting). In the hierarchical model
analysis, the state or provincial year effects are estimated at
the stratum level, and trend and regional estimates are
summarized from these year effect models. This approach
simplifies the overall analysis, eliminates possible inconsistencies associated with distinct trend and annual index
estimation procedures, and provides a better theoretical basis
for estimation of regional population change.
For the SGS analysis, hierarchical model–based year
effects are similar in pattern to the residual method
estimates. Consequently, the results provided by earlier
management summaries of SGS data (e.g., Kelley and Rau
2006) are consistent with results from new methods.
However, hierarchical model approaches have several
important advantages relative to the residual index method.
Specification of a model that incorporates overdispersion

and other random effects allows estimation of variances and
covariances for the year effects. These estimates can be
displayed around the time series, allowing users to ascertain
the significance of population fluctuations. From these
estimates, reduced models such as trend can be directly
estimated from the year effects for any subinterval,
eliminating the possibility of inconsistencies between trend
estimates and patterns in the annual indices. The models
also can be extended to incorporate covariates at the scale of
survey routes, allowing for modeling of effects of habitats
and other environmental features on SGS counts, and these
more complex models can be compared to the basic model
presented here. Due to limited availability of covariate
information across the range of the SGS, we did not
construct more complicated models to compare with the
basic model.
Unfortunately, uncertainties remain with regard to the
value of information collected in the SGS. We cannot
directly address the question of biased estimation associated
with roadside sampling, and the possibility exists that other
factors (such as traffic noise) have had differential effects on
observers’ ability to hear birds over time. The hierarchical
models described here permit several opportunities for
modeling and controlling for these factors. Covariate
analyses, although model-based, can increase our understanding of roadside habitat changes, if habitat information
are associated with routes and introduced as explanatory
variables in the analyses. Thogmartin et al. (2007) have used
SGS data in spatial hierarchical models, and these
approaches allow us to partially address some of the
concerns about roadside surveys. However, collection of
direct data on factors influencing counting (e.g., Dwyer et al.
1988) in combination with modeling exercises is likely the
most effective way of increasing our understanding of how
best to sample and model bird populations (Sauer et al.
2003).
Analysis for Bird Conservation Regions
Analysis of the SGS data for BCRs highlights the limited
information from the northern edge of the survey range. In
earlier analyses, estimates have been provided for Ontario
and Quebec, and integrated into overall survey results
without a clear documentation of the coverage of the survey.
However, the area weights used in the analysis clearly
indicate that the biologists who designed the survey did not
include the Boreal Softwood Shield. Our analysis of BCRs
has clarified the northern edge of the survey, and
documented the limited information associated with the
Boreal Softwood Shield and other peripheral BCRs such as
the Prairie Potholes. Within the survey area, however,
woodcock appear to be well-surveyed in several BCRs.

‹
Figure 4. Composite annual indices 1968–2006 of American woodcock populations from a hierarchical analysis of Singing-Ground Survey data (¤) and their
95% credible intervals (m) in Boreal Softwood Shield, Prairie Potholes, Boreal Hardwood Transition, Lower Great Lakes–St. Lawrence Plain, Atlantic
Northern Forest, Prairie Hardwood Transition, Appalachian Mountains, and New England–Mid-Atlantic Coast Bird Conservation Regions. Upper credible
intervals were omitted for some regions and years.
Sauer et al.  American Woodcock Population Change

213

Comparisons with Other Surveys
Although anecdotal, it is useful to document general
consistency in results of continent-scale surveys. All of the
surveys have deficiencies, and a primary concern among
biologists is that estimates based on the surveys contain
directional biases that lead to flawed views of population
change. For example, roadside sampling in the SGS and
BBS may lead to declines in populations if roadsides are
developed faster than surrounding countryside or vehicle
traffic increases over time; CBC counts might indicate
unwarranted population increases if observer effort is not
appropriately accommodated in the analysis. Consistency
among estimates provides weak evidence that these possible
sources of bias in estimation are not influencing long-term
change estimates.

MANAGEMENT IMPLICATIONS
We recommend that the hierarchical model be used to
analyze Woodcock SGS data. Although estimation at the
scale of BCRs within states and provinces is possible, we
suggest that, unless BCR information is of explicit interest,
the present state and province strata be retained for yearly
analyses. Bird Conservation Regions are not part of the SGS
design, and little benefit accrues from subdividing the strata
unless future studies require the BCR-level results. However, we recommend that survey organizers formally define
the northern edge of the survey range, which we have
operationally defined as the northern boundary of the Boreal
Hardwood Transition BCR. Establishing new survey routes
and ensuring consistent coverage of existing routes should
be a priority of survey coordinators, with particular emphasis
on the northern edge of the survey region. As with any
survey, evaluation and improvement of the SGS should be
viewed as an ongoing process, requiring periodic review of
both analysis and survey methods.

LITERATURE CITED
Burnham, K. P., and D. R. Anderson. 1998. Model selection and inference:
a practical information-theoretic approach. Springer, New York, New
York, USA.
Dwyer, T. J., G. F. Sepik, E. L. Derlith, and D. G. McAuley. 1988.
Demographic characteristics of a Maine woodcock population and effects
of habitat management. U.S. Fish and Wildlife Service, Fish and Wildlife
Research 4, Washington, D.C., USA.
Efron, B., and C. Morris. 1977. Stein’s paradox in statistics. Scientific
American 238:119–127.
Geissler, P. H., and J. R. Sauer. 1990. Topics in route regression analysis.
Pages 54–57 in J. R. Sauer and S. Droege, editors. Survey designs and
statistical methods for the estimation of avian population trends. U.S.
Fish and Wildlife Service, Biological Report 90(1), Washington, D.C.,
USA.
Gilks, W. R., S. Richardson, and D. J. Spiegelhalter, editors. 1996. Markov
chain Monte Carlo in practice. CRC Press, New York, New York, USA.

214

James, F. C., C. E. McCulloch, and L. E. Wolfe. 1990. Methodological
issues in the estimation of trends in bird populations with an example: the
pine warbler. Pages 84–97 in J. R. Sauer and S. Droege, editors. Survey
designs and statistical methods for the estimation of avian population
trends. U.S. Fish and Wildlife Service, Biological Report 90(1),
Washington, D.C., USA.
Kelley, J. R., Jr., and R. D. Rau. 2006. American woodcock population
status, 2006. U.S. Fish and Wildlife Service, Laurel, Maryland, USA.
Link, W. A., and J. R. Sauer. 1994. Estimating equations estimates of
trends. Bird Populations 2:23–32.
Link, W. A., and J. R. Sauer. 1998. Estimating population change from
count data: application to the North American Breeding Bird Survey.
Ecological Applications 8:258–268.
Link, W. A., and J. R. Sauer. 2002. A hierarchical model of population
change with application to cerulean warblers. Ecology 83:2832–2840.
Link, W. A., J. R. Sauer, and D. K. Niven. 2006. A hierarchical model for
regional analysis of population change using Christmas Bird Count data,
with application to the American Black Duck. The Condor 108:13–24.
Sauer, J. R., and J. B. Bortner. 1991. Population trends from the American
woodcock singing- ground survey, 1970–88. Journal of Wildlife Management 55:300–312.
Sauer, J. R., D. D. Dolton, and S. Droege. 1994. Mourning dove
population trend estimates from Call-count and North American
Breeding Bird Surveys. Journal of Wildlife Management 58:506–515.
Sauer, J. R., J. E. Fallon, and R. Johnson. 2003. Use of North American
Breeding Bird Survey data to estimate population change for bird
conservation regions. Journal of Wildlife Management 67:372–389.
Sauer, J. R., and P. H. Geissler. 1990. Annual indices from route regression
analyses. Pages 58–62 in J. R. Sauer and S. Droege, editors. Survey
designs and statistical methods for the estimation of avian population
trends. U.S. Fish and Wildlife Service, Biological Report 90(1),
Washington, D.C., USA.
Spiegelhalter, D. J., A. Thomas, N. G. Best, and W. R. Gilks. 1995.
BUGS: Bayesian inference using Gibbs sampling, Version 0.50. MRC
Biostatistics Unit, Cambridge, United Kingdom.
Straw, J. A., Jr., D. G. Krementz, M. W. Olinde, and G. F. Sepik. 1994.
American woodcock. Pages 97–114 in T. C. Tacha and C. E. Braun,
editors. Migratory shore and upland game bird management in North
America. Revised and updated edition. International Association of Fish
and Wildlife Agencies, Washington, D.C., USA.
Tautin, J., P. H. Geissler, R. E. Munro, and R. S. Pospahala. 1983.
Monitoring the population status of American woodcock. Transactions of
the North American Wildlife and Natural Resources Conference 48:376–
388.
ter Braak, J. F., A. J. van Strien, R. Meijer, and T. J. Verstrael. 1994.
Analysis of monitoring data with many missing variables: which method?
Pages 663–673 in E. J. Hagemeijer and T. J. Verstrael, editors. Bird
numbers 1992. Distribution, monitoring and ecological aspects. Statistics
Netherlands, Voorburg–Heerlen and Samenwerkende Organisaties
Vogelonderzoek Nederland, Beek-Ubbergen, Netherlands.
Thogmartin, W. E., J. R. Sauer, and M. G. Knutson. 2004. A hierarchical
spatial model of avian abundance with application to cerulean warblers.
Ecological Applications 14:1766–1779.
Thogmartin, W., J. R. Sauer, and M. G. Knutson. 2007. Modeling and
mapping abundance of American woodcock across the Midwestern and
Northeastern United States. Journal of Wildlife Management. 71:376–
382.
Thomas, L. 1996. Monitoring long-term population change: why are there
so many analysis methods? Ecology 77:49–58.
White, G. C., and R. E. Bennetts. 1996. Analysis of frequency count data
using the negative binomial distribution. Ecology 77:2549–2557.
Associate Editor: White.

The Journal of Wildlife Management  72(1)