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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‡ 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.