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Using Known Populations of Pronghorn to Evaluate
Sampling Plans and Estimators

Discussion


Selection Methods

Although SRS and SYS were, on average, more precise than PPS, differences were not large. Confidence interval coverage and cost were consistently better for PPS sampling, especially with stratification. Caughley (1979:10) stated that selecting sampling units without replacement (e.g., SRS and SYS) gives more precise results than sampling with replacement (e.g., PPS) at the same intensity, but when intensity is < 10% there is little difference between the methods.

We found estimates to be precise under systematic sampling but confidence interval coverage was never nominal (Cochran 1977:205). Inadequate confidence interval coverage may have resulted from poor estimates of variance. Estimation of the variance of N-hat can be a problem in systematic sampling. There is no unbiased variance estimate unless additional assumptions are met (Zinger 1980, Wolter 1984). We encountered both over- and underestimated standard errors (Cochran 1977:213-226) with biases ranging from -45 to 87% of the true standard error. The limited number of samples obtained under systematic sampling (2-72 in our study) also may account for variability in coverage probabilities. For example, with 50% sampling intensity, we sampled either every even- or every odd-numbered sampling unit, resulting in only 2 distinct samples. Therefore, there are only 3 possible coverage percentages—0, 50, or 100—so the nominal confidence interval coverage of 95% cannot be attained.

Stratification

Stratification increased precision in the 2 populations in which the allocation we used was close to the theoretical optimal allocation that yields minimum variance of the estimate. Stratification in the Bowman area in 1987 did not, on average, improve precision. There may be 2 reasons for lack of improvement. First, the pronghorn population in the Bowman area increased from 1979 to 1987, which may have induced animals to spread out from preferred habitat (grassland) into less preferred habitat (cultivated areas and badlands). Second, the habitat changed between surveys (Samuelson, unpubl. data); therefore, the 1974 LANDSAT information on vegetation we used to stratify the area was no longer current in 1987. These 2 changes resulted in strata having approximately the same number (Table 1) and distribution of pronghorn; consequently, our assumptions about the proportion of pronghorn in each stratum, and concomitantly our allocations, were no longer optimal.

Stratification generally increases precision (Siniff and Skoog 1964, Steel and Torrie 1980: 560-563), but precision can decrease with stratification if sample size allocation is far from optimal (Cochran 1977:99). The sample sizes we determined were close to optimal except for the sampling plans and estimators we evaluated using the Bowman area in 1987. In particular, the sample size results for SRS with the simple estimator indicate that, for these populations, in each stratum the population total is proportional to the population variance of the count. This relationship depends on the degree of aggregation (Taylor 1961) and may not hold for pronghorn populations during other seasons or for other species. The allocation method we used is strictly appropriate if sampling is SRS with the simple estimator, but it gave good results for all combinations of sampling plans and estimators. This may not always be the case, however, and other allocation methods (e.g., Cochran 1977:172) may be needed depending on the selection method, estimator, and knowledge of the population.

At smaller sample sizes, there was little difference between confidence interval coverage with or without stratification (both had low coverages). With larger sample sizes, we expected better confidence interval coverage, and found the coverage was close to the nominal value of 95% without stratification but was much lower with stratification.

Benefits of stratification are known, but little is known about its pitfalls. Stratification reduces sample sizes within each stratum. If small sample sizes are taken from a skewed distribution, confidence intervals based on an assumption of normally distributed counts may not be appropriate (Cochran 1977:27). Small sample sizes in strata also may bias standard error estimates, so Jolly (1969) suggested replacing each stratum's standard deviation by a single standard deviation calculated from the entire sample. We did not follow Jolly's suggestion because the standard deviation estimate is poor if an optimal allocation, such as our allocation method, is used and allocation is not proportional (i.e., mj = m(Mj/Σ Mj) (Cochran 1977:136).

Stratification increased costs in Slope County due to a large difference in transect areas. For simulations for the Bowman (with and without stratification) and Slope areas (without stratification) the percentage of units selected and the percentage of the area sampled were approximately the same because most transects had similar length. In the stratified Slope area, however, transects in the grassland stratum were longer than those in the mixed stratum (Table 1). Because the grassland stratum was sampled more, a greater percentage of the area was sampled than sampling intensity indicated.

Estimators

The estimators we evaluated are widely used, require no assumptions about population distribution, and are easy to calculate, but their precision in simulations was not compelling except when sampling intensity was high. Caughley (1977) found the pps and ratio estimators to be more precise than the simple estimator when transects had unequal lengths, but all 3 performed equally well when transects had equal areas. In simulations, transect areas were not equal, but ratio and simple estimators had similar precision and confidence interval coverage with or without stratification. The simple estimator's variance was easier to calculate, and ratio estimators and their variances may be biased (Cochran 1977:160-161). We did not compare the pps estimator directly with simple and ratio estimators because the selection methods are different; therefore, effects of estimators and the selection method cannot be separated. Instead, we considered the pps estimator in association with PPS sampling. Jolly (1969) recommended the pps estimator for aerial surveys because he thought it was more precise when sampling units are unequal in area and because the formulas are simpler than those for the ratio estimator. Probability proportional to size sampling and ratio estimators may be more precise when the sampling unit area and the count are highly correlated (Cochran 1977:258), which we would expect for randomly or uniformly distributed animals. Our results suggest that when distributions of animals are clumped, perhaps due to habitat heterogeneity or the animals' behavior, the correlation between transect area and the count on that transect may be weak.

We observed that, on average and with only 2 strata, the simple, separate ratio, and combined ratio estimators had similar precision and confidence interval coverage except at 50% intensity, for which the coverage for the simple and separate ratio was much less. This disparity largely resulted from using simple and separate ratio estimators when sampling stratified transects under SRS at 50% intensity in the Slope area. The low percentages occurred because the grassland stratum was completely sampled and, therefore, contributed zero as the variance estimate from this stratum. The mixed stratum had few pronghorn, and many samples included zero values; thus, the variance and ratio estimates were zero, so the simple and separate ratio estimators gave a variance estimate equal to zero and a confidence interval that was a single point. The combined ratio estimator, however, combined the information from both strata to calculate the ratio estimate, and gave a positive standard error estimate.

In simulations, the ratio estimators had small bias, but the separate ratio estimator may have higher bias than the combined ratio estimator when the number of strata is large. The separate ratio has smaller variance if the population density differs markedly among strata (Cochran 1977:165-167).

Research has been conducted on estimators that take into account factors such as large numbers of zero counts in a population; these might be appropriate for the highly skewed populations typical of animals that aggregate (Aitchison 1955, Pennington 1983). These estimators are difficult to calculate, have not been widely used, and assume a specific population distribution. If assumptions are met for these estimators, then confidence interval coverage should improve, but it is not clear that they would be more precise than estimators that make no assumptions about population distribution. Thompson (1992) discussed adaptive cluster sampling, which may give more precise estimates. Little is known about the procedure's effect on confidence interval coverage. There is some indication that the simple variance estimator may have a large bias when used with systematic sampling (Kraft, unpubl. data).


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