APPENDIX 1. Avian Mean and Variance Parameters



Mean and variance parameter estimation methods

To estimate θ0and τ12, the mean response and coefficient of variation at each Calling Lake (Alberta, Canada) site across the 6-year period was calculated and averaged across the three sites. The response at a site in a given year was total detections across all point-count stations averaged across surveys. Sites for which no individuals of target species were detected over the 6-year period were excluded to avoid undefined coefficients of variation. To evaluate the effect of within-site sample effort on τ12 and θ0, subsamples of the data set were taken and the parameters were estimated using the reduced data set. Within-site effort levels of 4, 9, and 16 stations and 1, 2, 3, and 4 surveys were evaluated, resulting in 12 estimates of θ0 and τ12 for each monitoring target. Two strategies were used when subsampling surveys. To simulate a monitoring program in which the timing of site visits can be kept constant from year to year, survey 2 was always sampled, and surveys 3, 4, and 1 were added successively as the number of surveys increased from two to four. To simulate a monitoring program in which timing of site visits cannot be kept constant from year to year, the surveys were randomly selected each year, without replacement. This process was repeated 100 times to achieve 100 subsamples of randomly selected surveys across the six years. When subsampling to reduce the number of stations, multiple estimates of τ12 and θ0 were generated for each site to represent the various contiguous arrangements of the reduced number of stations. The mean of these estimates was taken as an overall estimate for each site.

We estimated τ22 as the response variance between BBS routes in the closed boreal forest region of Alberta within a year. This estimate reflected not only variability between routes, but also variability due to sample error at each route (Link et al. 1994). Sample error, already accounted for in τ12, was extracted to avoid double counting. To isolate spatial variation in abundance, we multiplied variance estimates by (1 - the proportion of between-route variance due to sampling error). Estimates of the proportion of BBS between-route variance due to sampling error (α) were taken from Link et al. (1994), who estimated this quantity using repeated counts at BBS routes within seasons. For species not evaluated by Link et al. (1994), α was estimated based on their finding that α is negatively related to abundance according to a regression of logit(α) on the natural logarithm of abundance. Sampling error could not be extracted for the community metrics that we examined because Link et al. (1994) did not evaluate sampling error at the community level. After extracting sampling error, we calculated the coefficient of variation in response across routes. Estimates were made using data collected from 1992 to 1998 (11–16 routes per year), and the average of these coefficients of variation was used to estimate τ22. To avoid undefined coefficients of variation, we excluded years with zero mean response across routes from the analysis.

We estimated τ32 as the variation in trend across BBS routes in the closed boreal forest region of Alberta between 1989 and 1998. Although counts from 1989 to 1991 were not used to estimate τ22 due to reduced numbers of routes sampled in this time frame, they were included here to achieve longer trajectories and to enhance τ32 estimates. No counts prior to 1989 were used, because of the infrequency of route visits in the region. For species, only routes satisfying the U.S. National Biological Service data selection criteria (Thomas 1997) were used in the analysis. After applying these criteria, data from four to 15 routes were available, depending on the species and community metric. The trend at each route was estimated by fitting the data to an exponential trend model, using software written by Thomas (1997). After estimating the variance of route trend, a coefficient of variation was calculated by dividing the square root of the variance estimate by the mean route trend.

When calculating species richness and Shannon-Weiner community metrics from the Calling Lake and BBS data sets, we used all species in the data sets. When calculating species richness of the ground-nesting guild, we included the following species: Black and White Warbler,Mniotilta varia; Blackpoll Warbler, Dendroica striata; Canada Warbler, Wilsonia canadensis; Clay-colored Sparrow, Spizella pallida; Connecticut Warbler, Oporornis agilis; Dark-eyed Junco, Junca hyemalis; Hermit Thrush, Catharus guttatus; LeConte’s Sparrow, Ammodramus leconteii; Lincoln’s Sparrow, Melospiza lincolnii; Mourning Warbler, Oporornis philadelphia; Northern Waterthrush, Seiurus noveboracensis; Orange-crowned Warbler, Vermivora celata; Ovenbird, Seirus aurocapillus; Palm Warbler, Dendroica palmarum; Ruffed Grouse, Bonasa umbellus; Song Sparrow,Melospiza melodia; Swamp Sparrow, Melospiza georgiana; Tennessee Warbler, Vermivora peregrina; Wilson’s Warbler, Wilsonia pusilla; White-throated Sparrow, Zonotrichia albicollis; and Yellow-bellied Flycatcher, Empidonax flaviventris.

Mean and variance parameter estimates

Table 2, Table 3, and Table 4 present variance parameter and mean abundance estimates for selected species and community metrics. To summarize the effect of within-site sample effort on τ12, the response of τ12 to number of surveys, number of stations, and timing of surveys was calculated for each species and community metric (Table 5).

Assumptions and data limitations

Our results are contingent on assumptions made during variance parameter estimation. Within-site temporal variance was estimated by calculating variance in abundance across years at three control sites. Control sites were assumed devoid of population trends, which would confound variance. This assumption appears to be valid, because a high level of sample effort failed to detect significant trends over the 6-year period for the eight species selected. As an alternative, Gibbs et al. (1998) estimated temporal variance from the residuals of a linear regression of counts against time, in order to remove trend from the counts and minimize the effect of non-natural variation. This method is only valid if a significant trend exists. Otherwise, it can underestimate temporal variance by allocating natural variation to a nonexistent trend. This may explain why the estimate of Gibbs et al. (1998) of temporal variation for small birds (coefficient of variation = 0.57) was lower than ours (mean coefficient of variation across species = 1.083). Underestimating temporal variance results in overly optimistic predictions of the effectiveness of proposed monitoring strategies. We evaluated how τ12 was affected by within-site sample effort. Two other factors that will also influence τ12 are consistency in sampling methodology and the person collecting data at the site. Both the Calling Lake study (Schmiegelow et al. 1997) and the BBS (Droege 1990) use consistent sampling methodology that includes specification of weather conditions and the time of day and year during which data can be collected. Similar strict requirements should be a component of all monitoring programs because they reduce sample error by reducing variation in the proportion of present birds that are singing. The ability to detect birds that are singing varies across people. Although some of this variation can be reduced through training, consistency in data collectors should be maintained as much as possible. Such consistency will be difficult to maintain in long-term monitoring programs, perhaps causing an increase in τ12.

Because sample error was accounted for in τ12, it was extracted from τ22 to avoid double counting, and therefore exaggeration, of sample error. Sample error was not extracted from τ32, however, because no estimates were available to do so. Although this may have caused exaggeration of τ32, the influence on power estimates is thought to be minimal due to its insensitivity to τ32 (see Table 6).

Data limitations required us to use BBS data to estimate between-site variance parameters. The BBS design differs from the sampling design assumed here, which may reduce the accuracy of related estimates due to differences in sample error associated with the different methodologies. However, results should be insensitive to this difference because, of the two between-site variance parameters, one had little influence on power (τ32) and sample error was extracted from the other (τ22). Nevertheless, the most effective way to improve the data would be to implement a long-term monitoring program with a design based on estimates presented here. Over time, data from the monitoring program could be used to re-estimate variance parameters and reassess program design.

One final caveat is that our power simulations assumed that the coefficient of variation remained stable across abundance levels within a species. Power will be exaggerated if coefficients of variation increase, or will be underestimated if they decrease, as abundance declines. However, sampling efficiencies will remain valid, unless spatial and temporal coefficients of variation are differentially affected.