OPTIMAL ALLOCATION OF POINT-COUNT SAMPLING EFFORT

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 LI'TERA•RE

CITED

COCHRAN,W. G., 1977. Sampling techniques, 3rd ed. Wiley, New York. D^WSON,D. G. 1981. Counting birds for a relative measure(index)of density.Stud.Avian Biol.6:1216.

GUTZWlLLER, K. J. 1991. Estimating winter species richness with unlimited-distance point counts. Auk

108:853-862.

RALPH,C. J.,ANDJ. M. SCOTT.1981. Estimatingnumbers of terrestrial

birds. Stud. Avian

Biol. 6.

ROBBINS,C. S., D. BYSTRAK,AND P. H. GEISSLER. 1986.

The BreedingBird Survey:Its first fifteen years, 1965-1979.

U.S. Fish Wildl.

Serv., Resour. Publ.

157.

ScoTT,J. M., AND F. L. RAMSEY. 1981. Length of count period as a possiblesourceof bias in estimating bird densities. Stud. Arian Biol. 6:409413.

SE•ER,G. A.F.

1982. Estimation of animal abundance

and related parameters,2nd ed. Charles Griffin

and Co., London. sumptionsabout the population parameters,alVERNER, J. 1988. Optimizing the duration of point lowing for the examinationof the effectsof these countsfor monitoring trendsin bird populations. assumptions on optimal sampling schemes. U.S. Dep. Agric. ForestServ. Pacific.South. For. Consequently,the use of f(T•) in optimization Range Exp. Station, Res.Note PSW-395. of point countsmakes explicit the importance

of our assumptionsaboutf(T•) in the analysis. ACKNOWLEDGMENTS

We thank D. K. Dawson, S. Droege, M. L. Morrison, J.D. Nichols,and C. J.Ralph for reviewing the manuscript.