36 The Auk 119(1):36���45, 2002 DOUBLE SAMPLING TO ESTIMATE DENSITY AND POPULATION TRENDS IN BIRDS JONATHAN BART1 AND SUSAN EARNST U.S. Geological Survey Forest and Rangeland Ecosystem Science Center, Snake River Field Station, 970 Lusk Street, Boise, Idaho 83706, USA ABSTRACT.���We present a method for estimating density of nesting birds based on double sampling. The approach involves surveying a large sample of plots using a rapid method such as uncorrected point counts, variable circular plot counts, or the recently suggested double-observer method. A subsample of those plots is also surveyed using intensive meth- ods to determine actual density. The ratio of the mean count on those plots (using the rapid method) to the mean actual density (as determined by the intensive searches) is used to adjust results from the rapid method. The approach works well when results from the rapid method are highly correlated with actual density. We illustrate the method with three years of shorebird surveys from the tundra in northern Alaska. In the rapid method, surveyors covered 10 ha h21 and surveyed each plot a single time. The intensive surveys involved three thorough searches, required 3 h ha21, and took 20% of the study effort. Surveyors using the rapid method detected an average of 79% of birds present. That detection ratio was used to convert the index obtained in the rapid method into an essentially unbiased estimate of density. Trends estimated from several years of data would also be essentially unbiased. Other advantages of double sampling are that (1) the rapid method can be changed as new methods become available, (2) domains can be compared even if detection rates differ, (3) total population size can be estimated, and (4) valuable ancillary information (e.g. nest success) can be obtained on intensive plots with little additional effort. We suggest that dou- ble sampling be used to test the assumption that rapid methods, such as variable circular plot and double-observer methods, yield density estimates that are essentially unbiased. The feasibility of implementing double sampling in a range of habitats needs to be evaluated. Received 31 January 2001, accepted 21 September 2001. RESUMEN.���Presentamos un me ��todo para estimar la densidad de aves nidificantes con base en un muestreo doble. La metodolog�� ��a incluye censos de una muestra grande de par- celas usando un me ��todo ra ��pido como conteos de punto no corregidos, conteos en parcelas circulares variables (������variable circular plot counts,������ VCP) o el me ��todo de doble observador sugerido recientemente. Para determinar las densidades reales, una submuestra de estas par- celas tambie ��n es censada usando me ��todos intensivos. Para ajustar los resultados de los cen- sos ra ��pidos se utiliza la razo ��n entre el conteo promedio obtenido con el me ��todo ra ��pido y la densidad media real (determinada a trave ��s de bu ��squedas intensivas). Esta metodolog�� ��a fun- ciona bien cuando los resultados del me ��todo ra ��pido esta ��n altamente correlacionados con la densidad real. Aqu�� �� ilustramos el uso del me ��todo basa ��ndonos en datos de tres an ��os de censos de aves playeras en la tundra del Norte de Alaska. Utilizando el me ��todo ra ��pido, censamos cada parcela so ��lo una vez, cubriendo 10 ha h21. Los censos intensivos incluyeron tres bus-�� quedas exhaustivas que llevaron 3 h ha21 y comprendieron el 20% del esfuerzo del estudio. Los censos realizados con el me ��todo ra ��pido detectaron en promedio el 79% de las aves pre- sentes. Esta tasa de deteccio ��n fue utilizada para convertir el �� ��ndice obtenido con el me��todo ra ��pido en un estimador no sesgado de la densidad. Del mismo modo, las tendencias esti- madas con base en varios an ��os de datos tambie ��n estar�� ��an esencialmente libres de sesgos. Otras ventajas del muestreo doble son: (1) el me ��todo ra ��pido puede modificarse conforme otros me ��todos se hagan disponibles, (2) las a ��reas de muestreo pueden ser comparadas au��n si las tasas de deteccio ��n difieren, (3) permite estimar el taman ��o poblacional total y (4) se puede obtener informacio ��n adicional de intere ��s (e.g. e ��xito de anidacio ��n) en las parcelas in- tensivas con poco esfuerzo adicional. Sugerimos que el muestreo doble se utilice para poner a prueba el supuesto de que los me ��todos ra ��pidos como el de VCP y el de doble observador 1 E-mail: jbart@eagle.boisestate.edu

January 2002] 37 Double Sampling to Estimate Density estiman la densidad esencialmente sin sesgos. La factibilidad de implementar el muestreo doble en una variedad de ha ��bitats necesita ser evaluada. THE NEED FOR accurate estimates of trends in avian abundance, and in some cases for esti- mates of absolute population size, is well ac- knowledged (Ralph et al. 1995, Carter et al. 2000, Beisinger et al. 2000, O���Connor et al. 2000). In statistical literature (e.g. Cochran 1977), ac- curacy is usually measured using the mean square error, defined as variance plus bias squared. ������Variance������ is a measure of precision, the degree to which estimates, drawn in the same manner from the same population, vary from sample to sample. ������Bias������ is the difference between the ������expected������ value of the estimate, its mean value based on a large number of samples, and the quantity being estimated. Precision is estimated by standard statistical methods whereas bias is not. It is thus imperative that methods be used that are unbiased, or in which the bias is small relative to precision. Most avian survey methods are indices���sur- veys in which ratio of the count to actual pop- ulation size is unknown. Indices cannot be used to estimate population size. In using them to es- timate trend, we must assume that there is no substantial long-term trend in the ������index ratio������ (Bart et al. 1998), defined as index result divided by parameter, actual population size in this study. In recent years, there has been increasing concern over assuming that no temporal trend exists in the index ratio (e.g. Nichols et al. 2000). As a result, more emphasis is being placed on estimating index ratios so that density estima- tors may be used rather than indices. We describe a method that yields essentially unbiased estimates of population size and thus of trend in population size. We use the qualifier ������es- sentially������ because few field methods, if any, are completely unbiased, but we believe that any bias in the method we describe is small enough to be ignored. The method is based on double sam- pling, a standard statistical method from the sur- vey sampling literature (Cochran 1977, Thomp- son 1992). Double sampling has been widely used to survey waterfowl and has been used in at least two other avian studies (Handel and Gill 1992, Anthony et al. 1999), but it has not been widely used to study other avian taxa. The meth- od involves an initial survey using a rapid meth- od such as area searches, point counts, or variable circular plot counts, and a subsample of those plots on which actual density is determined through intensive methods. The ratio of the rap- id-method result to actual density is then used to adjust results from the initial large sample of plots. The method yields unbiased estimates of density���and thus of trend in density���if the in- tensive methods provide accurate counts. No as- sumptions are required about how the index ratio in the initial surveys varies with observer, time of day, habitat, or other factors. We illustrate the method with several years of data from a study of shorebirds on the North Slope of Alaska. METHODS Estimating means and standard errors. The ap- proach below is from Thompson (1992) except that our r is his 1/r. Let n9 5 number of plots in the large sample surveyed with the rapid method n 5 number of plots in the subsample on which intensive methods are used x9 �� 5 xi9/n9 O 5 mean number of birds recorded per plot in the large sample using the rapid method x �� 5 xi/n O 5 mean number of birds recorded per plot in the subsample using the rapid method y �� 5 yi/n O 5 mean number of birds actually present per plot in the subsample The estimate of actual density, d, is obtained by ad- justing results from the rapid method using results from the subsample: x9 �� y �� d 5 5 x9 �� (1) (x/y) �� �� x �� The standard error of d may be estimated as s2 (yi ) 1 1 c SE( d) 5 1 2 (gi ) (2) 1 2s2 [ ]0.5 n9 n n9 where gi is calculated with the results from the sub- sample, y �� gi 5 yi 2 xi , x �� and n n 2 (yi 2 y) �� (gi 2 g)2 �� O O i i s2 (yi ) 5 , s2 (gi ) 5 n 2 1 n 2 1 are sample variances of yi and gi. The first term on