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Using Known Populations of Pronghorn to Evaluate
Sampling Plans and Estimators

Results


Repeatability of simulations was excellent (most CV <l%, with a few ≤3%). Pronghorn counts ranged from 0 to 28 ([mean of x] = 4.2, median = 0) for Bowman 1979, 0-72 ([mean of x] = 13.1, median = 3) for Bowman 1987, and 0-32 ([mean of x] = 5.6, median = 1) for Slope 1986. For grassland strata, pronghorn counts ranged from 0 to 28 ([mean of x] = 6.2, median = 1) for Bowman 1979, 0-40 ([mean of x] = 11.3, median = 0) for Bowman 1987, and 0-32 ([mean of x] = 7.2, median = 4) for Slope 1986. For mixed strata, pronghorn counts ranged from 0 to 7 ([mean of x] = 0.3, median = 0) for Bowman 1979, 0-60 ([mean of x] = 5.8, median = 2.5) for Bowman 1987, and 0-3 ([mean of x] 0.1, median = 0) for Slope 1986.

Selection Method

Without stratification, SRS was less precise (Table 2) than SYS sampling ([mean of x] CV = 33 and 31, respectively), but the results were opposite with stratification ([mean of x] CV = 23 and 26, respectively). Probability proportional to size sampling was the least precise ([mean of x] CV = 35 without stratification, 30 with stratification). Correlation coefficients between sampling unit area and pronghorn count on the unit were 0.003-0.46, explaining why PPS did not result in substantially more precise estimates.

Table 2.  Coefficient of variation (%) of estimators of abundance (N) determined from sampling plans at 3 intensities for known distributions of pronghorn in Bowman (1979 and 1987) and Slope counties (1986), North Dakota.
  Bowman 1979 Bowman 1987 Slope 1986 Average
  Sampling intensity (%) Sampling intensity (%) Sampling intensity (%) Sampling intensity (%)
Selection method a Estimator b Stratified 16 33 50 [mean of x] 16 33 50 [mean of x] 16 33 50 [mean of x] 16 33 50 [mean of x]
SRS c Simple no 56 35 25 39 48 30 21 33 41 26 18 28 48 30 21 33
    yes 38 20 11 23 48 29 20 32 22 10 2 11 36 20 11 22
  Ratio no 52 33 23 36 46 30 21 32 38 25 17 27 46 29 20 32
  Separate ratio yes 39 20 11 23 51 30 21 34 24 10 2 12 38 20 11 23
  Combined ratio yes 41 22 13 25 49 30 21 33 26 14 5 15 39 22 13 25
PPS d pps e no 52 37 30 40 45 32 26 34 39 28 22 30 45 32 26 35
    yes 41 28 23 31 50 33 26 36 29 20 17 22 40 27 22 30
SYS f Simple no 50 38 13 34 39 27 19 28 44 32 13 30 44 32 15 31
    yes 50 16 9 25 54 29 29 37 32 12 2 15 45 19 13 26
a The method used to select the sampling units.
b The estimator for the population count.
c Simple random sampling without replacement.
d Probability proportional to size with replacement sampling.
e Probability proportional to size estimator.
f Systematic sampling.

Confidence interval coverage (Table 3) of PPS without stratification was higher ([mean of x] = 92%) than SRS ([mean of x] = 91%) but lower than SYS ([mean of x] = 94%). With stratification, PPS sampling gave higher coverage ([mean of x] = 92%) than SRS and SYS ([mean of x] = 86 and 78%, respectively). Coverages under systematic sampling were erratic (Table 3); for example, in the Slope area, the confidence interval coverage ranged between 50 and 100%, depending on sampling intensity.

Table 3. Coverage of nominal 95% confidence intervals (%) of estimators of abundance (N-hat) at 3 intensities for known distributions of pronghorn in Bowman (1979 and 1987) and Slope counties (1986), North Dakota.
  Bowman 1979 Bowman 1987 Slope 1986 Average
  Sampling intensity (%) Sampling intensity (%) Sampling intensity (%) Sampling intensity (%)
Selection method a Estimator b Stratified 16 33 50 [mean of x] 16 33 50 [mean of x] 16 33 50 [mean of x] 16 33 50 [mean of x]
SRS c Simple no 87 91 94 91 87 92 93 91 90 93 94 92 88 92 94 91
    yes 89 93 88 90 85 87 87 86 94 94 46 78 89 91 74 85
  Ratio no 88 91 93 91 85 88 93 89 90 93 95 93 88 91 94 91
  Separate ratio yes 89 92 85 89 83 83 86 84 93 95 46 78 88 90 72 84
  Combined ratio yes 91 91 93 92 84 87 88 86 94 93 89 92 90 90 90 90
PPS d pps e no 88 92 94 91 88 94 93 92 90 92 93 92 89 93 93 92
    yes 91 93 93 92 84 90 91 88 94 93 96 94 90 92 93 92
SYS f Simple no 83 100 100 94 83 100 100 94 83 100 100 94 83 100 100 94
    yes 71 100 50 74 69 75 83 76 100 100 50 83 80 92 61 78
a The method used to select the sampling units.
b The estimator for the population count.
c Simple random sampling without replacement.
d Probability proportional to size with replacement sampling.
e Probability proportional to size estimator.
f Systematic sampling.

The average distance flown with PPS Sampling without stratification was 461 km-9% lower than the average for SRS and SYS (509 and 507 km, respectively). With stratification, the average distance flown with PPS sampling (574 km) was 21% lower than with SRS (728 km) and 20% lower than with SYS (715 km).

Sampling Intensities

Precision and confidence interval coverage generally increased with increasing intensities and costs (Tables 2, 3, and 4) with some exceptions. The average confidence interval coverages at the 3 intensities (16, 33, and 50%) without stratification were 87, 94, and 95%, respectively, and with stratification were 87, 91, and 78%, respectively.

Table 4. Cost (distance in km) of sampling plans at 3 intensities for known distributions of pronghorn in Bowman (1979 and 1987) and Slope counties (1986), North Dakota. 
  Bowman Slope Average
  Sampling intensity (%) Sampling intensity (%) Sampling intensity (%)
Selection method a Stratified 16 33 50 [mean of x] 16 33 50 [mean of x] 16 33 50 [mean of x]
SRS b no 185 352 514 350 350 657 994 667 268 505 754 509
  yes 195 403 587 395 560 1,115 1,505 1,060 378 759 1,046 728
PPS c no 198 342 454 331 337 603 832 591 268 473 643 461
  yes 189 348 464 334 511 837 1,093 814 350 593 779 574
SYS d no 190 353 514 352 351 661 972 661 271 507 743 507
  yes 192 453 568 404 539 966 1,572 1,026 366 710 1,070 715
a The method used to select the sampling units.
b Simple random sampling without replacement.
c Probability proportional to size sampling.
d Systematic sampling.

Standard errors were generally underestimated at all intensities with the percent bias of the underestimated standard errors ranging from −45 to −1%. Only 6 standard error estimates had zero bias and a few under systematic sampling had a large positive bias. Without stratification and excluding systematic sampling, the percent bias of the standard errors consistently decreased as sample size increased with −6% bias at 16% sampling intensity to −0.9% bias at 50% sampling intensity. With stratification and excluding systematic sampling, percent bias increased from −9% at 16% sampling intensity to −11% at 50% sampling intensity.

Stratification

Stratification generally increased precision (Table 2) but reduced average confidence interval coverage (Table 3) and usually increased costs (Table 4). Except in the Bowman area in 1987, the method we used to allocate sample sizes yielded results close to actual optimal sample sizes; therefore, for a given combination of sampling plan and estimator, the greatest possible precision was nearly achieved.

The overall average confidence interval coverage was 86% with stratification and 92% without stratification (Table 3), but this difference was not consistent at all intensity levels. At 16 and 33% intensity, average coverages were similar (87 and 91% with stratification and 87 and 94% without stratification). At 50% intensity without stratification, estimates were normally distributed and the average coverage was the nominal 95%, but with stratification the coverage was only 78%.

The gain in precision due to stratification for the Bowman area in 1979 came without a substantial increase in cost ([mean of x] = 3%). In the Slope area, there was an increase in cost ([mean of x] = 17%) due to stratification.

Estimators

We compared simple and ratio estimators for sampling plans in which SRS was used to sample transects, both stratified and not stratified. When the study area was not stratified, the simple estimator (Table 2) and the ratio estimator were similarly precise ([mean of x] CV = 33 and 32, respectively). With stratification, the simple estimator was slightly more precise ([mean of x] CV = 22) than the separate ratio ([mean of x] CV = 23) or combined ratio estimators ([mean of x] CV = 25).

The percentage of confidence intervals containing the actual pronghorn count (Table 3) was the same ([mean of x] = 91%) for the ratio and simple estimators without stratification. The combined ratio estimator gave better coverage ([mean of x] = 90%) than either the separate ratio ([mean of x] = 84%) or simple estimator ([mean of x] = 85%).


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