The Auk 110(4):752-758, 1993
OPTIMAL
ALLOCATION
SAMPLING
OF POINT-COUNT
EFFORT
RICHARD J. BARKER
TM,JOHN R. $AUER3, AND WILLIAM A. LINK3
•FloridaCooperative
Fishand WildlifeResearch
Unit,
Departmentof WildlifeandRangeScience,
Universityof Florida,Gainesville,
Florida32611,USA;
2Department
of Statistics,
MasseyUniversity,Palmerston
North,
New Zealand; and
3U.S.Fishand WildlifeService,PatuxentWildlife
ResearchCenter,Laurel,Maryland 20708, USA
Ans13•,Acr.--Bothunlimited and fixed-radius point counts only provide indices to popu-
lationsize.Because
longercountdurationsleadto countinga higherproportionof individuals
at the point, proper design of these surveysmust incorporateboth count duration and
samplingcharacteristics
of populationsize.Usinginformationaboutthe relationshipbetween
proportion of individuals detectedat a point and count duration, we presenta method of
optimizinga point-countsurveygivena fixedtotaltimefor surveyingandtravellingbetween
count points. The optimization can be based on several quantities that measure precision,
accuracy,or power of testsbasedon counts,including (1) mean-squareerror of estimated
populationchange;(2) mean-square
errorof averagecount;(3) maximumexpectedtotalcount;
or (4) powerof a testfor differencesin averagecounts.Optimalsolutionsdependon a function
that relatescount duration at a point to the proportion of animals detected.We model this
functionusingexponentialand Weibull distributions,and usenumericaltechniquesto conduct the optimization.We provide an exampleof the procedurein which the function is
estimated
from data of cumulative
number
of individual
birds seen for different
count du-
rationsfor three speciesof Hawaiian forestbirds. In the example,optimal count duration at
a point can differ greatly dependingon the quantitiesthat are optimized.Optimizationof
the mean-squareerror or of testsbasedon averagecountsgenerally requireslonger count
durationsthan doesestimationof populationchange.A clearformulationof the goalsof the
studyis a criticalstepin the optimizationprocess.Received
7 February1992,accepted
25November
1992.
POINT COUNTSare a popular method for surveying birds (Dawson 1981), and the method is
used for extensive monitoring programs such
as the North American Breeding Bird Survey
(Robbins et al. 1986). Point counts are con-
ducted by recording the number of individuals
of the target speciesthat are observedduring a
specifiedtime interval at a sampling point. Because not all individual
animals
associated
with
the sampling point are observed during the
count period, the data provide only an index of
animal abundance (Dawson 1981). The count of
individuals at point/, c,, is related to the total
number of animals associatedwith that point,
N, by an unknown detectionprobabilityp, such
that the expected count at point i is
environmental
factors have been demonstrated
to influence the p,'s (Dawson 1981). Numerous
papers in Ralph and Scott (1981) were devoted
to documenting seasonal,temporal, observer,
and species-specific
influenceson the pi's.The
duration of the point count is one of the most
obvious factors influencing the p,'s. Functions
relating p, to count duration under a variety of
conditionshave been considered(e.g. Scottand
Ramsey 1981). In a recent analysis, Gutzwiller
(1991) used an empirical approach to examine
how the mean number of speciesdetectedduring unlimited distance point counts in winter
was affectedby factorssuch as count duration,
time of day, and environmental covariates,and
offered suggestionson allocation of sampling
effort.
E(c,) = p,N,.
(1)
To use point-count data to compare populations over time or space, assumptionsmust be
made about the constancyof p,. Unfortunately,
752
Verner (1988) hassimulatedthe optimization
of point-count surveysbasedon duration of the
point count. He maximized expectedtotal count
of individuals
as a function
of various
combi-
October1993]
Optimizing
Point-count
Data
753
nations of counting and noncounting times,
radius (p) of the observer. Within p of the observer,
where
individuals have some probability of detection less
than or equal to one, and detectionprobabilitiesmay
differ among birds. At somedistancep from the observer, the birds are not perceptible to the observer
number
of birds counted
was a function
of counting time. He derived detection functions from all speciesand individuals detected
during point countsconductedover varying inand probabilityof detectionbecomeseffectivelyzero.
tervals of time at two study sites in the Sierra
In the secondcase,exemplifiedby limited-distance
National
Forest, California.
He concluded that
efficiencyof the surveyincreasedwith duration
of count, but recommendedan upper limit of
10 min to minimize double counting of individuals (Verner 1988).
Although total count may be a reasonableindicator of survey performance,it is important
when designinga survey to rememberthat the
goal is not to maximize total count, but rather
to provide datawith which relevant hypotheses
canbe addressed.
Usually,formaltestingof these
hypotheseswill be conductedwithin somestatisticalframework;thus, it may be usefulto allocate sampling effort on the basisof the potential performanceof the statisticalprocedures
that will be used to analyze the data. The allocation of sampling effort that leads to maximum total countmay not optimize performance
of the chosenstatisticalprocedure.
The allocation of sampling effort involves a
trade-off between time spent at each point and
the number of pointssampled.In this paper we
proposean objectivemeansof finding the "best"
trade-offpossiblegiven the total available time
for surveying, travel time between counts,certain population characteristics,and detection
functions. We illustrate the method using optimization criteriaderived from commonlyused
methods of statisticalanalysis, and apply the
method to published data on Hawaiian birds
(Scottand Ramsey1981).
METHODS
THE MODEL
We assumethat the animal population sizes(N,) at
distinctpointsareindependentwith mean• and variance a2. Conditional on N, the number counted at
the ith point c, is a binomial random variable with
parameterp, the detectionprobability. The detection
probability p is assumedto be a function of count
duration, Ts.For simplicity, p = f(Ts) is assumedto be
the same at all points. Some of the consequencesof
variation in p will be discussedlater in the paper.
There are two closely related ways to consider the
binomial counting process.In the first, exemplified
by unlimited-distance methods, counts can be con-
sidered as arising from a population of individual
birdsof a singlespeciesthat are locatedwithin a fixed
point-countmethods,the circleof radiusp* describes
the radius within which a bird must be present to be
counted.
All
birds
within
this distance
are counted
with probability 1, and the binomial detectionprobability p in the abovemodel correspondsto the probability that a bird located within p of the observer,
occurs within
p* of the observer at the time of the
count.
Despite the difference in the two sampling approaches,they can be modelled in the sameway. In
each case a bird that is in some sense "located"
at the
point being surveyedis either heard or not heard, at
random,during the survey.In the unlimited-distance
method they are not heard becausethey failed to call
or simply were missedby the observer.In the limiteddistancemethod they were missedbecausethey were
not locatedwithin the samplingradiusat the time of
the survey.
We assumethat survey cost can be expressedin
units of time, of which a fixed total T must be allocated
to travel time between points and sampling time at
points. Let T, denote the average traveling time between points. The number of points sampled,n, and
the duration of counts at individual points, Ts,are
constrainedby the relationship
T=(n-
1) T, + nT,.
(2)
Thus, either T, can be large and n small,or vice versa.
In practice,actualcostscanbe assignedfor personnel
time and travel for eachof these components.
OPTIMIZATION
CRITERIA
An important problem in sampling theory is the
optimizationof estimatorperformancegiven a fixed
sampling effort. The first step in the optimization
processis choosing a criterion for measuring estimatorperformance.Because
thischoicemayaffectthe
outcomeof the optimization, as we demonstratebelow, it is important that a reasonablechoicebe made.
This choice is itself determined by the goals of the
studyand the methodsthat will be usedto meet these
goals.We discussseveralalternative optimization criteria.
Mean-square
errorof counts.--Foroptimal allocation
of sampling effort, one traditional measureof estimator performanceis the samplevariance(Cochran
1977).Many of the estimatorsof populationsizebased
on point counts are biased; thus, the mean-square
error (MSE) is a more appropriate measureof estimator performancethan the samplevariance.For example, consider the mean count as an estimatorof
754
BARKER,
S^UER,
ANDLINK
[Auk,Vol.110
the mean number of animals presentat each possible
survey point. Under the model describedin the pre-
nativehypothesis
(H,:/•, ->/•j),and/• = (1 - k)/•, The
vious section, the mean and variance of the counts
the means. The standard
are given by
in mean countsunder the null and alternativehy-
E(c,) = E[E(c, I /,)] = E(/,p) = p•
(3)
and
Var(c,)
= E[Var(c
] N,,p)]+ Var[E(c
] N,,p)] (4)
= E[N,p(1 - p)] + Var(N,p).
factork representsa proportionaldifferencebetween
deviation
of the difference
pothesescan be computedusing expression(5).
MSE of ratio estimatorof populationchange.--Estimation of populationchangeat n pointscountedon
two occasions(time periods t and t + 1) can be done
usingthe ratio of the total countfor all pointsat time
t + 1 and the total count for all points at time t or
Thus,
Var(g) = (t•p + t•p2d)/n,
(5)
where d is (•r2/•) - 1.
The theoretical MSE for this estimator under our mod-
ChoosingT, to minimize the varianceof the counts
leads to choosing Tsso that p is zero, in which case
variance
el can be obtainedusing the expressionsfor biasand
all counts are zero, and the variance of the counts is
zero. Obviously,minimizing the varianceof the counts
is an inappropriate criterion becausethis maximizes
the bias. Note that the bias (i.e. E[c,] - •) increasesas
p decreases.
An alternative approachis to minimize the meansquare error (MSE, where MSE = bias: + variance).
Bias(fi)
= (fi/(nt•)){[1
- f(T•)]/f(T•)•
Var(fi)
=(11n)([fi(1
+fi)/M{[1
- f(T,))lf(Ts)]
Given that p = f(Ts)and the resultsabove,MSEof the
mean
count
+ •rf1+
is
MSE(Q = •211 - f(T•)] 2 + [•f(T•) + i•f(Ts)2d]/n. (6)
By incorporatingbiasand error into the optimization,
we simultaneouslyaddressthe issuesof precisionand
accuracyof the counts.
Maximum expectedtotal count.--It has been suggested that total count is a valid criterion for optimization; studiesbasedon small sample sizesof individuals are difficult to analyze and lead to statistical
testsof low power (Verner 1988). The expectedvalue
of the total number of animals counted is given by
the product of the total number of individuals at a
point, the probabilityof detectingeachindividual for
the count duration, and the number of points, or
(10)
and
,
(11)
where fi and %• denote the conditional mean and
varianceof N,+• IN, amongpoints,and •r2the variance
of the number of individual animalsamongpoints
in the firsttimeperiod.Here,fi denotes
an estimate
of the true populationchangefl. Equations(10) and
(11) were derived from (9) using the method of statistical differentials (Seber 1982).
OPTIMIZATION
PROCEDURE
From calculustheory, the maximum and minimum
valuesof a function with respectto one variableoccur
at the point where the slope of the function is zero.
To find out if this is a maximum
or minimum
value
we look at the rate of changeof the slope(the second
derivative) at this point. If this is negative (i.e. slope
is going from being positive to negative) then we
E(,•
c,)=nt•f(Ts).
(7)
Powerof testfor difference
in means.--Ifpoint counts
are usedfor comparisonof countsbetween studysites
or within study sitesover time, the strongassumption
of no differencesin meandetectionprobabilitiesmust
be made. Under our model, power of a one-sided
z-test for a difference
in means between
two sets of
counts(denoted by subscriptsi and j) with identical
detection probabilities and equal sampling effort is
given by
Power = •[z•r0 - f(Ts)k•,l•h],
(8)
where ae(z)denotesthe upper-tail probability of the
standard normal distribution, a0denotes the standard
deviation
of the difference
in means under
the null
know we have found a maximum.
Where our function depends on more than one
variable, the gradient vector correspondsto the slope,
and the matrix of second partial derivatives corresponds to the second derivative. Thus, the values of
n and T, that correspondto either maximum or minimum values of the function describingefficiencyof
our estimation or testing procedures, occur at the
points where the gradient of that function with respect to n and T• equals zero. If the matrix of second
partial derivatives(Hessian)is positive-definiteat these
points then they correspondto local minima. If the
Hessian is negative-definite,then the points correspond to local maxima. For simple functions, closed-
hypothesis(H0: •, = •j), a• denotesthe standardde-
form solutions can often be obtained, but in more
viation
complex cases(such as those described above) nu-
of the difference
in means
under
the alter-
October1993]
Optimizing
Point-count
Data
755
TABLE1. Maximum-likelihood estimatesof exponential (r) and Weibull parameters(a and b), and resultsof
generalizedlikelihood-ratio testbetween modelsfor three speciesof Hawaiian forestbirds (data from Scott
and Ramsey 1981).
Exponential r
Species
Red-billed
Omao
• + SE
Leiothrix
Apapane
Weibull b
d + SE
• + SE
X2
P
0.149 + 0.024
0.234 + 0.031
0.143 + 0.018
0.236 + 0.033
1.255 + 0.144
0.984 + 0.095
3.324
0.027
0.068
0.869
0.061 + 0.008
0.058 + 0.011
0.962 + 0.080
0.233
0.629
likelihood
function
because MLEs
or other
efficient
merical techniques such as the Newton-Raphson
method (Seber 1982:17-18) must be used.A computer
program that will optimize the allocation of effort
using the above estimatorsand measuresof survey
detectedin the observationperiod of duration r, then
performance is available in executable code from the
the likelihood
estimators are not available in closed form. If tl, t2,
..., tkare the distinct times to detection for all animals
function
to be maximized
is
authors.
L=•I [ab(t,a)
b•e",•)•]/[1
- e('"•]. (16)
ESTIMATION
01•
Maximum
The functionf(•E,)describesthe way is which the
proportion of birds counted increaseswith count duration. Ideally, f(T•) could be estimatedby explicitly
estimating p using a known population of animals.
Unfortunately, this is rarely possible,so alternative
estimation proceduresmust be sought.
with
mean
r. In this case
f(t) = 1 - e ".
(12)
In practice,the simple curve provided by the exponential model may be too restrictiveto mimic the
shapeof f(T•). A more flexible model is the Weibull,
in which the probability that the time to first detection is lessthan t is given by
f(t) = 1 - e ,a)•.
(13)
This model includesthe exponentialmodelas a special casecorrespondingto a b of one.
Given estimatesof the parameters,the percentage
of animalssightedin the samplingperiod T• can be
estimatedby
f(L) = I - e •,',
(14)
under the exponential model, or by
/(L) = 1 - e(•-)",
under
the Weibull
(15)
model.
Under the Weibull model, parameterestimatesmust
be obtained using numerical maximization of the
can also be used
in the
ex-
for theexponential
parameter
canbeobtained.
Let[
denote the average length of time to detection, and
S2 the variance of these times, then
Associated with each bird is a random variable x,
denoting the time until first sighting of the bird. The
function f(Ts) is the cumulativedistribution function
of the randomvariablex;f(T•) shouldincreaserapidly
at first and then slowly approachone.
We considertwo models for f(Ts). In the first, times
until first detection are modeled as independent and
identically distributedexponentialrandom variables
likelihood
ponential model using the likelihood function above
by constrainingthe parameterb to equal one. Alternatively, a closed-formmethod of momentsestimator
? = ([/]• + S2
rF) / (2[- r)
(17)
is a consistent estimator of r, and is almost as efficient
as the maximum-likelihood estimator. Consistency
means
that
the
error
of estimation
tends
to zero
as
the sample size becomeslarge. If the maximum-likelihood estimator is used, then a likelihood-ratio
test
can be used to test between the Weibull and exponential models. Executablecomputer code for fitting
the models
is available
from
the authors.
EXAMPLE
Scottand Ramsey(1981), in a study of Hawaiian
forest birds presented data on the cumulative proportion of Red-billed Leiothrix (Leiothrixlutea),Omao
(Myadestesobscurus),
and Apapane (Himationesanguinea) counted as a function of time spent counting
within a 32-rain interval. Using their data we fitted
the exponential and Weibull models. For the Redbilled
Leiothrix
the
likelihood-ratio
test indicated
marginal evidenceof the need for the Weibull model
(P = 0.068; Table 1). For the Omao and Apapane,
however, the likelihood-ratio testswere not significant (P = 0.869 and 0.629, respectively),indicating
that the Weibull model did not lead to an improved
fit over the exponential model.
Supposethat the Omao studywas to be usedas the
basisfor planning a future study. We illustrate the
optimization procedure, using the exponential-pa-
rameterestimatefrom the Omao data,for a studyarea
756
BARKER,
SAtmR,
ANDLINK
TABLE
2. Optimalsamplingallocationand associated
expectedvaluesof f(T•) for hypotheticalstudywith
meannumberof animalspresentat eachpoint of
20, varianceof numberof animalsat eachpoint of
20, total samplingtime availableof 180min, 10 min
required for travel betweenpoints,and function
relating detectionprobabilityat eachpoint (p) to
time spent sampling at each point (Ts)given by
p = 1 - errs, where r is 0.23.
Statistic
Count MSE
Total count
Power
Ratio
Locations
4.863
11.307
11.390
12.958
[Auk,Vol.110
we change tr2 so that tr2/• = 0.5, but keep all
other parameters the same, optimal allocation
of sampling effort occursfor 11.3 points when
expected total count is maximized, but when
power is maximized the optimal number of
sampling points is 9.8. If tr2/• = 2, the optimal
number of points sampled when expectedtotal
count is maximized
T•
f(Ts)
29.073
6.803
6.681
4.663
0.999
0.798
0.792
0.666
remains at 11.3, but for
power the optimal number of sampling points
is 13.0.
In contrast
to the result
for the MSE
of the
average count, the minimum MSE of the population change estimator occurred for a combination of a relatively large number of points,
and much lesstime sampling at eachpoint, with
with meannumberof birdspresent(#) of 20, variance a predicted detection probability of 0.67.
of 20, total surveytime (T) of 180 (min), travel time
between stations(T,) of 10 (min), and exponential-
DISCUSSION
parameter r of 0.23 (as estimatedfrom the data). Note
that the exponentialparameteris the only value estimated from the pilot data,and the other valuescan
be varied for any particular experimentalsituation.
To compareoptimal sampling effort under the constraint (equation 2), we: (1) minimized MSE of the
averagecount;(2) maximizedpower of a one-sided
z-test for a difference
in means between
two sets of
countswith identicalf(Ts);(3) minimized MSEof the
ratio estimatorfor population change between two
time periodswith identicalf(Ts);and (4) maximized
expectedtotal count. We set k at 0.1 (proportional
differencein population means= 10%),and a at 0.05.
For the two-yearpopulation-change
analysis,we considered a casewhere •[•1+1
= •[•t'and where the coeffi-
cientof variationamongpointsof the ratioof animals
present during the two years (%/•/) was 0.2.
RESULTS
For the Hawaiian forest birds discussedby
Scottand Ramsey(1981),the MSE of the average
count
was
minimized
with
a combination
of
few points and a long sampling time at each
point (Table 2). A detectionprobabilityof 0.99
would occur under this combination. Thus, with
this particular combination of parametersand
optimality criterion, avoidanceof bias plays a
dominant role in determining the allocation of
sampling effort.
Optimal allocation of the sampling effort to
obtain maximum power of the one-sided z-test
and maximum total count were nearly coincident in this case.However, unlike the expected
value of total counts, the power of the z-test is
sensitive
to the variance
of the distribution
of
animals through space(i.e. tr2).For example if
Becauseestimator performance is samplingschemedependent, it is necessaryto have an
objectivemeansof bestallocatingsampling effort that explicitly incorporatesthe estimatorof
interest.
We
have
demonstrated
a method
of
allocating point-count sampling effort, basedon
a framework that relatesunderlying population
characteristicsand detection probabilitiesto the
numbers
of individual
animals
counted.
Our
results emphasize the importance of defining
clear objectives,becausethe results of the optimization can differ greatly depending on the
goals of the study and the methods used to
achieve these goals. If goals that require estimates of population size tend to require longduration countsat eachpoint asis suggestedby
our example, actual implementation of point
counts as surrogates of population size estimates is unlikely to be feasible.However, optimal point-counting periods for other objectives, such as the estimation of population
trends, seem reasonable. Of course, while some
generalizationscan be made from our example,
we recommend that investigators collect pilot
data and actually assessthe feasibility of their
study goalsusingthesemethodsbeforeactually
implementing a study.
Verner (1988) used total counts as an optimization criterion with the justification that
maximizing counts maximized statisticalpower. However,
a statistical test must be defined
beforepower can be assessed.
Also, many of the
estimation methods that use point-count data
are biased (e.g. ratio estimator for trend). Considering power alone and ignoring bias over-
October1993]
Optimizing
Point-count
Data
757
looks the fact that power is a function of type
I error rates. The type I error rate (a level) cor-
probability for a fixed-durationcount,and then
to use this estimated detection probability to
respondsto the power for the smallestpossible obtain an estimateof exactpopulation size. We
departure from the null hypothesis(i.e. H0 is hesitate to recommend this approach because
true). Thus,increasingthe a-level automatically of the difficulty in testing the necessaryasincreasestest power. Usually, the a-level is con-
sumptions. Incorrect model specification is li-
sideredto be pre-setby the experimenter;how- able to have far more seriousconsequencesfor
ever, biasedtestingproceduresusually lead to estimation than for planning of studies. The
a-levels higher than the nominal values.In this consequenceof error in the first case may be
way bias can lead to increasedpower simply biased inference, but in the second the same
error may only lead to weaker inference.
through increasingthe type I error rate.
If one can assume that bias is constant beThe feasibility of the methodswe suggestdetween comparisons,testing proceduressuchas pendson the actualfunctionf(Ts),which relates
a comparison of means can be used. Because the probability of detection to sampling effort
expectedmaximumtotal countis unaffectedby (time spent counting) at a sampling point. The
the variance of the distribution
of animals, it
shape of this function ultimately determines
doesnot alwaysactasa goodsurrogatefor max- the efficacyof any point-countstudy through
imum power, and can lead to a quite different the effects of detection probabilities on estiallocation of sampling effort than that which mators. Becausepoint counts rarely sample all
maximizes power.
the animals present,bias is an important comThe method
we have described above is most
easily applied where a single speciesis of interest, or where there is some simple way of
characterizing the collection of species,such as
speciesrichness (number of speciesassociated
with a point). In many studies,however, pointcount-samplingschemesprovide dataon a multitude of speciesand, in monitoring programs,
all speciesmay be of interest. In this latter case,
optimalallocationof samplingeffortmay be an
ill-defined concept because the allocation of
sampling effort that leads to optimal performance of estimatorsor testing proceduresis
speciesspecific.The processof allocatingsampling effort in a multispeciesprogrammustinvolve reconciling the differing sampling requirementsof the species.Many approachescan
be used to develop a compositeoptimization.
For example, it is conceivablethat a measureof
survey performance could be computed for the
entire assemblageof species(e.g. total MSE),
but in practice this will be extremely difficult
for anything other than a few species.Alternatively, key speciescould be picked from the
assemblage
associated
with the studyarea,and
effort optimized with respectto the hardestspeciesto sample.This will lead to a tendency to
spend more time samplingat eachpoint. If too
little time is spentat eachpoint, biasmay dominate estimator performance.
It is tempting to use the method we have
describedabove for modeling the manner in
which detectionprobabilitychangeswith count
duration as a method of estimatingdetection
ponent of estimation procedures.Consequent-
ly, the optimal allocationof samplingeffort involvesa trade-offbetweenbiasand the precision
of the estimate.This trade-offdependsin large
measureon f(T•). Therefore,optimizationbased
on f(T•) should be the basisof all point-count
studies and studies
based on similar
methods.
Given expressions
for bias,variance,f(T•), and
a function describing sampling constraints,it
is possibleto optimize samplingeffort usingthe
methods
we have outlined.
The solutions
will
depend on the model for f(T•), the estimator
used,and the optimalitycriterion.Consequently, it is difficultto makegeneralstatementsabout
how sampling effort should be allocated.
All modeling of animal populationinvolves
assumptionsthat mustbe carefully considered
in any application. For our model, we first assume
that the detection
function
is a realistic
reflection of the mechanicsof counting. This
assumesthat double counting of individuals
doesnot occur.Verner (1988)suggestedthat the
maximumtime at a point is limited by increased
probability of double counting after 10 min.
Disturbance
associated with
the observer
also
may attract or repel birds (Scott and Ramsey
1981). Finally, sexesof most songbirdspecies
have greatly different detection functions, as
the females are only occasionallyobserved.A
detection function basedon singing males can
only accountfor part of the population,and the
optimization depends on the validity of the
model.
It must be recognized that the assumption
758
BARKER,
SAUER,
ANDLINK
that f(Ts) is constantbetween points and over
time is unrealistic. However, two points must
be made. First, the assumption that some consistentf(Ts) existsis implicit to most existing
analyses of point-count data, becauseall hypothesistestswill be biased if basedon pointcount data in which f(Ts) is changing among
the items to be compared. Second, even if our
model for f(T•) is imperfectbecauseof violation
of this assumption,the consequence
is a slightly
less-than-optimalsamplingscheme,which will
still be more efficient than sampling basedentirely on a subjectively chosen regimen. The
procedureoutlined here is objective,and it places appropriate emphasis on the critical underlying assumptionof all point-count studies:the
probability of detecting a bird at a point increaseswith time spent counting at the point.
Furthermore, the procedure is flexible; optimization
can be carried
out under
various
as-
[Auk,Vol. 110
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ACKNOWLEDGMENTS
We thank D. K. Dawson, S. Droege, M. L. Morrison,
J.D. Nichols,and C. J.Ralph for reviewing the manuscript.