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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.

Selection Method

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.
			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	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.
			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).

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		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

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.

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 ( 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%).