Northern Prairie Wildlife Research Center

Sampling Plans and Estimators

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

Without stratification, SRS was less precise (Table 2) than SYS sampling ( CV = 33 and 31, respectively), but the results were opposite with stratification ( CV = 23 and 26, respectively). Probability proportional to size sampling was the least precise ( 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. |
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Bowman 1979 |
Bowman 1987 |
Slope 1986 |
Average |
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Sampling intensity (%) |
Sampling intensity (%) |
Sampling intensity (%) |
Sampling intensity (%) |
|||||||||||||||

Selection method ^{a} |
Estimator ^{b} |
Stratified |
16 |
33 |
50 |
16 |
33 |
50 |
16 |
33 |
50 |
16 |
33 |
50 |
||||

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 ( = 92%) than SRS ( = 91%) but lower than SYS ( = 94%). With stratification, PPS sampling gave higher coverage ( = 92%) than SRS and SYS ( = 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 ()
at 3 intensities for known distributions of pronghorn in Bowman (1979
and 1987) and Slope counties (1986), North Dakota. |
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Bowman 1979 |
Bowman 1987 |
Slope 1986 |
Average |
|||||||||||||||

Sampling intensity (%) |
Sampling intensity (%) |
Sampling intensity (%) |
Sampling intensity (%) |
|||||||||||||||

Selection method ^{a} |
Estimator ^{b} |
Stratified |
16 |
33 |
50 |
16 |
33 |
50 |
16 |
33 |
50 |
16 |
33 |
50 |
||||

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).

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. |
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Bowman |
Slope |
Average |
|||||||||||

Sampling intensity (%) |
Sampling intensity (%) |
Sampling intensity (%) |
|||||||||||

Selection method ^{a} |
Stratified |
16 |
33 |
50 |
16 |
33 |
50 |
16 |
33 |
50 |
|||

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 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 ( = 3%). In the Slope area, there was an increase in cost ( = 17%) due to stratification.

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 ( CV = 33 and 32, respectively). With stratification, the simple estimator was slightly more precise ( CV = 22) than the separate ratio ( CV = 23) or combined ratio estimators ( CV = 25).

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

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