An Introduction to Determining Optimum Quadrat Size and Shape

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Table 1. Effect of quadrat size on SD for measurements of basal area of trees in an oak-hickory forest in North Carolina. The data are from Bormann (1953). Quadrat size Observed SD (m) per 4 m 2 Sample size a SE of the mean for sample size 4 x 4 50.7 70 6.06 4 x 10 47.3 28 8.94 4 x 20 44.6 14 11.92 4 x 70 41.3 4 20.65 4 x 140 34.8 2 24.61 a 2 The number of quadrats of a given size needed to sample 1,120 m . Two methods are available for choosing the best quadrat size statistically. Wiegert (1962) proposed a general method that can be used to determine optimal size or shape. Hendricks (1956) proposed a more restrictive method for estimating optimal size of quadrats. In both methods it is essential that data from all quadrats be standardized to a single unit area—for example, per m 2 . This conversion is simple for means, SDs, and SEs: divide by the relative area. For example, Mean number per m 2 2 Mean number per 0.25m = 0.25 SD per m 2 2 SD per 4m = 4 For variances, the square of the conversion factor is used: Variance per m 2 = 2 Varianceper 9 m 2 9 For both Wiegert’s and Hendrick’s methods, you should standardize all data to a common base area before testing for optimal size or shape of quadrat. They both assume further that you have tested for and eliminated quadrat sizes that give an edge effect bias. We will only deal with Wiegert’s method here. Wiegert’s method Wiegert (1962) proposed that 2 factors were of primary importance in deciding on optimal quadrat size or shape: relative variability and relative cost. In any field study, time or money would seem to be the limiting resource, and we must consider how to optimize with respect to sampling time. We will assume that time = money, and in the formulas that follow, either unit may be used. Costs of sampling have 2 components (in a simple world anyway): C = C0 + Cx, 4
where C = Total cost for 1 sample C0 = Fixed costs or overhead Cx = Cost for taking 1 sample quadrat of size x. Fixed costs involve the time spent walking or flying between sampling points and the time spent locating a random point for the quadrat; these costs may be trivial in an open grassland; they can be enormous when sampling the ocean in a large ship. The cost for taking a single quadrat may or may not vary with the size of the quadrat. Consider a simple example from Wiegert (1962) of grass biomass in quadrats of different sizes: Quadrat size (area) 1 3 4 12 16 Fixed cost ($) 10 10 10 10 10 Cost per sample ($) 2 6 8 24 32 Total cost for 1 quadrat ($) 12 16 18 34 42 Relative cost for 1 quadrat 1 1.33 1.50 2.83 3.50 We need to balance these costs against the relative variability of samples taken with quadrats of different sizes: Quadrat size (area) 1 3 4 12 16 Observed variance per 0.25 m 2 0.97 0.24 0.32 0.14 0.15 The operational rule is, Pick the quadrat size that minimizes the product of (Relative cost) x (Relative variability). In this example, we begin by disqualifying quadrat size 1 because it showed a strong edge effect bias (Figure 1). Figure 2 shows that the optimal quadrat size for grass in this particular study is 3, although there is relatively little difference between the products for quadrats of size 3, 4, 12, and 16. In this case, the size 3 quadrat gives the maximum precision for the least cost. 5
Page 1 and 2: Problem Set #2: An Introduction to
Page 3: Figure 1. Edge effect bias in small
Page 7 and 8: Relative variance = (SD) 2 (Minimum
Page 9 and 10: Table 3. Optimal quadrat size for p

An Introduction to Determining Optimum Quadrat Size and Shape

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