Elongated sample plot shapes produce higher measurement precision in clustering populations

Monitoring the density of natural populations is crucial for ecosystem management decision making and natural resource management. The most widely used method to measure the population density of animal and plant species in natural habitats is to count organisms in sample plots. Yet evaluation of survey performance by different sample plot shapes, e.g. quadrats compared with transects, has been largely neglected since the 1990s and has not been undertaken using simulation. Simulating populations and surveys, we evaluated population density measurement precision for 900 cases, testing 30 sample plot survey designs in each of 30 spatially clustered populations. We varied three design options: elongation of plot shape while keeping sample area constant, systematic or random plot allocation, and sample size. Survey design performance varied markedly: elongating the plot shape always improved survey precision; allocating plots systematically sometimes did. (i) Averaged across all tested populations, elongated (1:100) transect plot shapes were 2-to-3 times more precise than square (10:10) quadrats. (ii) The precision of systematic surveys accelerated with sample plot number, increasing faster than the (known) linear increase under simple random sampling. This non-linear, concave upward, dependence of systematic precision on sample size has not previously been reported. (iii) The most precise design we evaluated used long narrow transects allocated systematically. Averaging among all 30 tested populations, a researcher would need 600 random square (10:10) quadrats to equal the precision achieved by 100 systematic (1:100) transects. Finding this average efficiency difference of 600% for a survey sample size of 100 plots, these simulation results imply that field trips requiring five sampling days using random quadrats could achieve equal precision in one or two days using systematic elongated transects. For all clustered populations we tested, long narrow transects resulted in a more efficient design for sample plot survey.

240 In the first subsection to follow, we evaluate the precision effect of elongating sample 241 plot shape and specifically compare long narrow 1:100 transects with square 10:10 quadrats.
242 Second, we compare systematic with simple random plot allocation, focusing on the effects 243 of increasing sample plot number and varying the autocorrelation scale (and so also spatial 244 extent) of population clustering. Third, we compare the best and worst performing designs.
245 Summary statistics of these design performance comparisons are given in Table 1.

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(summarizing variance ratios shown in Fig 3); Section 2: random over systematic 289 sample allocation (variance ratios of Fig 4); Section 3: least precise design (random  298 and (one-start aligned) systematic, we graphed the ratios of their sampling error variances as 299 (var(random)/var(systematic)), one variance ratio for each combination of population 300 parameters psill and range, and of survey design sample size and plot shape (Fig 4). Most 301 random-over-systematic variance ratios being either equal to or exceeding 1 (bars near to or 302 greater than 1 in Fig 4) imply equal or higher precision by the systematic design. Where 303 systematic precision was higher, that was generally evident for all tested plot shapes (all bar 304 colors in Fig 4), but the precision gain under a systematic allocation was highest for 305 elongated transects (L100 and L50).

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Previous studies have identified systematic designs as being more precise. However, we 307 observed two previously unreported results that refine this general trend: (i) higher 308 systematic precision was much more strongly evident for the larger sample sizes among those 309 evaluated here, and (ii) systematic allocation yielded better precision than random sampling 15 310 only in populations distributed with longer-range spatial autocorrelation (broader clusters). 311 We examine these two outcomes in the next two subsections.  (Fig 4D-F) and again to n = 100 (Fig 4A-C). This was primarily 325 evident for elongated plot shapes (L100 and L50) and longer autocorrelation range ≥ 20-50.
326 Means, medians and quartiles of the random-over-systematic variance ratios (Table 1, 327 Section 2) all increased with sample size, for all range values and more strongly for range ≥ 328 50. Thus, systematic precision increased more rapidly with n than random precision.
329 Knowing that random precision increases linearly with n as , this result indicates that 330 systematic precision increases faster than linearly with sample size.

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To examine this result directly, we plotted survey precision against sample size as 332 number of sample plots (n). We separately consider both random and systematic allocations 333 (Fig 5). 18 test populations were examined, six levels of autocorrelation range (0.001, 5, 20,   The trend for systematic precision dependence on sample size (blue lines in Fig 5) 347 confirms non-linear, accelerating dependence, here notably for n ≥ 25. The rate of this 348 acceleration increased strongly with longer autocorrelation range (greater divergence of blue 349 plots from red plots in descending rows of graphs in Fig 5). As basic statistics predicts, the 350 precision of random sampling increased strictly linearly with sample size (red lines in all 351 graphs of Fig 5). Overall, a more rapid increase of systematic than random precision, 352 systematic precision accelerating with increasing number of sample plots, was evident for 353 both 2:50 transects ( Fig 5A) and square quadrats (Fig 5B). We could find no previous report 354 of systematic precision increasing faster than linearly with sample size.  1; finer detailed mapping of spatial clusters are shown in Figs S1.1-S1.6, S1.7-S1.12, 360 S1.13-S1.18

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For range ≤ 10, random-over-systematic variance ratios are all near 1 (bar heights ~ 1 364 in Fig 4), implying similar precision by systematic and random sampling. Higher precision by 365 the systematic design (Fig 4 bar Fig 4), random-over-systematic variance ratios did not consistently rise or fall.