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Parameters of the investigated designs.

 Dimensions Positions in the field Analyzed models Design Plot Block Design m by m no. Four dummy treatments 1 R† 2 by 2 2 by 8 8 by 8 357 9 2 2 by 2 8 by 2 32 by 2 120 8 3 R 2 by 2 2 by 8 2 by 32 180 8 4 2 by 8 8 by 8 32 by 8 30 8 5 2 by 8 8 by 8 8 by 32 51 9 6 2 by 8 2 by 32 8 by 32 51 9 7 8 by 2 32 by 2 32 by 8 42 9 8 R 8 by 2 8 by 8 32 by 8 42 9 9 R 8 by 2 8 by 8 8 by 32 45 8 Ten dummy treatments 10 2 by 2 20 by 2 40 by 4 23 9 11 2 by 8 20 by 8 40 by 16 5 9 12 2 by 8 20 by 8 20 by 32 33 9 13 R 8 by 2 8 by 20 32 by 20 30 9
R = recommended.

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Analyzed covariance models.

 Model Covariance function† Variance–covariance parameters Analyzed correlation options and corresponding model abbreviation no. Basic σ2f(d) = σ2 if d= 0= 0 if d ≠ 0 1 no correlation B Correlation For the whole trial Only within blocks Spherical σ2f(d) = σ2[1 – 3/2(d/θ) + 1/2(d/θ)3] if d ≤ θ= 0 if d > θ 2 S s Power‡ σ2f(d) = σ2ρd 2 P p Gaussian σ2f(d) = σ2{exp[–(d/θ)2]} 2 G g First-order random walk Cov(i,j) = σd2min(i, j) 1 r Anisotropic power σ2f(dx,dy) = σ2ρxdx ρydy 3 A
d, distance between the midpoints of two plots; σ2, variance for d = 0; θ, range for the spherical model, the range parameter for the Gaussian model; ρ, correlation between plots if d = 1; i and j, plot numbers within a block, σd2, variance of the difference of neighboring plots; dx and dy, distances of two plots in the x and y directions, respectively; ρx and ρy, corresponding correlations if dx = 1 and dy = 1.
In the notation of SAS PROC MIXED.

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Basic characteristics of the crop yields for the investigated plot sizes.

 Parameter Plots Hemp 1954 Rye Beet Oat Hemp 1958 no. Mean, Mg ha−1 480 17.70 2.85 55.44 1.49 23.89 Variance of plots 2 by 2 m 480 22.00 0.12 121.08 0.39 13.70 2 by 8 m 120 16.84 0.11 75.62 0.33 9.08 8 by 2 m 120 15.97 0.11 68.31 0.31 8.27 Coefficients of variation, % 2 by 2 m 26.49 11.97 19.85 41.83 15.49 One-lag omnidirectional correlation 2 by 2 m 0.63 0.90 0.42 0.80 0.56

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Mean, total variance, and variance components of the ranked corrected Akaike Information Criterion values as a demonstration of the limited information value of the mean and of the consistent results for the variance components of plans using Design 12, rye and beet, as an example. The minimum value for each crop is in bold type.

 Crop Model† Mean Total variance Variance components Position Plan % Rye B 7.02 1.48 79.81 0.00 r 2.45 1.92 64.51 0.45 P 6.76 6.00 41.78 0.00 S 1.74 0.91 27.10 0.34 G 4.24 2.65 26.47 1.79 A 6.44 6.17 51.75 0.30 p 7.30 2.39 29.65 0.49 s 3.73 1.21 31.55 1.02 g 5.29 2.55 26.91 1.66 Beet B 7.89 5.18 58.38 0.50 r 3.10 5.74 61.49 0.04 P 3.18 3.41 45.92 0.25 S 4.70 5.88 54.28 0.00 G 5.98 3.21 30.24 0.81 A 5.70 5.24 44.23 0.46 p 3.86 2.95 36.88 0.31 s 4.12 4.06 40.72 0.00 g 6.47 3.74 38.05 0.87
B, basic; r, random walk; P and p, power; S and s, spherical; G and g; Gaussian; A, anisotropic power model.

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Best-fit models† with a frequency >10% sorted by descending frequency.

 Design Case Hemp 1954 Rye Beet Oat Hemp 1958 1 (A)‡ combinations B/r§ G/r/S = B B/r B = r/G/S B/r/G position B/r G/r/S B/r B/r/S = G B/r plan B G B r/B r/B 2 combinations B/r r/B B/r r/B B/r position B/r r/B B/r r/B B/r plan B/r r B r r/B 3 combinations B/r r/G/B B/r r/B B = r position B/r r/G/B B/r r/B r/B plan B r B r r/B 4 combinations r/B r/g r = B r/g r/B position r/B r B/r r/B r/B plan r r B = r r r 5 (A) combinations r/B r/G r = B r/B B/r position r/B r B/r r/B B/r plan r r B/r r r = B 6 (A) combinations S = B = r = G G/S/B = r B/S/G = r S/G B/G/r = S position S/B/r/G S/G/B B/S S/G B/S/G/r plan S S/G S/B S S 7 (A) combinations S/G S/G r/B/S/G S = G S/B/G/r position S/G S/G r/S/B/G S/G S/B/r = G plan S S S/B S/G S 8 (A) combinations r/B r B/r r/B/G r/B position r/B r B/r r/B/G r/B plan r r B r r 9 combinations r/B r/G B/r r/B/g r/B position r/B r/G B/r r/B r/B plan r r B/r r r 10 (A) combinations G/P/S S/A/G P/B/r S/P/G P/S/G/A position G/P/S S/G = A P/B = r S/P S/P/G plan P/G S P S P/S 11 (A) combinations A/G/r S/g/A r/P/S S/g S/A/r position A = r/G S/r P/r S/g S/r = p = g plan A/S S r/S/P S P/S = A 12 (A) combinations A/r S/r/G r/P/A S/r r/G = A/P/s position A/r/s S/r P/r/s r/S r/P/G = A = s plan A S r/P r P 13 (A) combinations r/A = s/S S/G/s = r/g r/P r/S/G S/r position r/s/S/A S/s/G = r/g r/P/p S/r/G/s S/r plan r r/S s/p S/r S
B, basic; r, random walk; P and p, power; S and s, spherical; G and g, Gaussian; A, anisotropic power model.
(A) indicates that the anisotropic model was analyzed.
§Bold type indicates 50% ≤ frequency ≤ 90%; bold type and underline indicates 90% < frequency; nearly the same frequency is indicated by =.

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Mean standard errors of differences (SED) (est = estimated SED and obs = observed SED) for Designs 4, 9, 12, and 13 and all analysis models, as well as for the best model for each combination of position and plan. The smallest SED in each row is indicated in bold type; the largest SED in each row is indicated by bold italic type.

 Mean SED Crop Design SED type Best Basic r† P‡ S§ G¶ p‡ s§ g¶ A# Hemp 1954 4 est 1.467 1.769 1.497 1.531 1.504 1.626 1.527 1.591 1.624 obs 1.642 1.779 1.538 1.599 1.593 1.627 1.567 1.578 1.631 9 est 1.068 1.330 1.106 1.122 1.096 1.208 1.138 1.162 1.211 obs 1.178 1.330 1.121 1.146 1.136 1.250 1.210 1.145 1.244 12 est 1.236 2.297 1.521 1.561 1.490 1.552 1.604 1.542 1.555 1.227 obs 1.398 2.290 1.562 1.550 1.531 1.646 1.583 1.579 1.645 1.285 13 est 1.109 1.860 1.162 1.205 1.173 1.275 1.204 1.183 1.275 1.123 obs 1.226 1.859 1.188 1.185 1.188 1.402 1.162 1.192 1.197 1.401 Rye 4 est 0.056 0.082 0.061 0.059 0.060 0.070 0.062 0.061 0.060 obs 0.059 0.081 0.055 0.058 0.056 0.070 0.063 0.057 0.062 9 est 0.051 0.082 0.062 0.063 0.060 0.056 0.061 0.062 0.058 obs 0.056 0.082 0.056 0.058 0.057 0.059 0.065 0.058 0.060 12 est 0.060 0.146 0.067 0.066 0.064 0.052 0.067 0.067 0.052 0.060 obs 0.058 0.145 0.057 0.055 0.055 0.068 0.056 0.056 0.068 0.054 13 est. 0.056 0.109 0.065 0.068 0.065 0.055 0.067 0.064 0.055 0.065 obs 0.062 0.109 0.057 0.058 0.056 0.068 0.059 0.058 0.056 0.068 Beet 4 est. 3.549 4.035 3.609 3.624 3.592 4.015 3.727 3.798 4.015 obs 4.025 4.017 3.891 3.890 3.938 4.172 3.917 3.939 4.154 9 est 3.060 3.422 3.123 3.110 3.089 3.420 3.154 3.263 3.479 obs 3.428 3.422 3.392 3.350 3.416 3.548 3.343 3.390 3.583 12 est 3.772 5.482 3.768 3.964 3.808 4.433 3.975 3.894 4.434 3.806 obs 4.298 5.481 4.142 4.139 4.140 4.693 4.152 4.239 4.693 4.126 13 est 3.277 4.464 3.202 3.461 3.305 3.775 3.395 3.347 3.775 3.321 obs 3.714 4.461 3.553 3.563 3.588 3.901 3.613 3.563 3.687 3.901 Oat 4 est 0.139 0.195 0.149 0.152 0.145 0.159 0.166 0.149 0.147 obs 0.149 0.197 0.140 0.144 0.144 0.165 0.167 0.144 0.153 9 est 0.127 0.179 0.139 0.141 0.134 0.141 0.127 0.138 0.137 obs 0.138 0.180 0.131 0.135 0.134 0.144 0.130 0.133 0.141 12 est 0.144 0.315 0.154 0.146 0.148 0.145 0.146 0.154 0.146 0.140 obs 0.141 0.313 0.139 0.133 0.136 0.173 0.132 0.139 0.172 0.134 13 est 0.132 0.281 0.149 0.155 0.146 0.125 0.155 0.148 0.126 0.143 obs 0.134 0.281 0.133 0.136 0.132 0.143 0.133 0.136 0.131 0.143 Hemp 1958 4 est 1.203 1.414 1.225 1.232 1.216 1.348 1.286 1.311 1.366 obs 1.349 1.412 1.277 1.280 1.294 1.376 1.313 1.301 1.385 9 est 0.936 1.172 0.965 0.983 0.967 1.055 1.007 1.031 1.054 obs 1.061 1.179 0.977 1.010 1.009 1.102 1.057 1.020 1.089 12 est 1.300 1.903 1.337 1.402 1.336 1.440 1.418 1.396 1.440 1.318 obs 1.475 1.903 1.398 1.400 1.382 1.510 1.417 1.431 1.510 1.388 13 est 0.994 1.657 1.024 1.041 1.007 1.104 1.067 1.040 1.105 1.028 obs 1.070 1.647 1.044 1.035 1.036 1.172 1.082 1.055 1.056 1.171
r, random walk model.
P and p, power model; results based on the Satterthwaite method.
§S and s, spherical model; results for the non-cereals based on the Kenward–Roger method, for the cereals on the Satterthwaite method.
G and g, Gaussian model; results based on the Kenward–Roger method.
#A, anisotropic power model; results based on the Satterthwaite method.