ST 524NCSU - Fall 2007
Augmented Analysis
Augmented designs
Augmented designs also use grids or incomplete blocks to remove some field variation from the plot residuals. In an augmented design, a large set of experimental lines is divided into small incomplete blocks. In each incomplete block, a set of checks is included; every check occurs in each incomplete block, but the experimental lines are included in only one block. Because the design is unreplicated, the repeated checks are used to estimate the error mean square and the block effect. The block effect is estimated from the repeated check means and then removed from the means of the test varieties. This reduces error and increases precision somewhat. However, the repeated checks used to estimate block effects add a substantial number of plots to the trial. Block effects could also be estimated as effectively from the means of the test varieties in each block. This would save considerable space and labor. In general, augmented designs have few advantages over unreplicated nurseries in which block effects are estimated without repeated checks ().
Check: Stork (St), Cimmarron (Ci), Waha (Wa) - Durum Wheat species – replicated in each block
6 Blocks – 30 selection lines – 3 checks – 8 plots per block
Number of observations: 30 + 6*3 = 48
Check
Selection
Mean difference
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Block as random effect – PROC GLM
procglmdata=a;
class block check selection;
model yield =block selection ;
random block /test;
lsmeans selection/stderr;
run;
The GLM Procedure
Class Level Information
Class Levels Values
block 6 I II III IV V VI
check 2 check lines
selection 33 1 10 11 12 13 14 15 16 17 18 19 2 20 21 22 23 24 25 26 27 28 29 3 30 4 5 6 7
8 9 Ci St Wa
Number of Observations Read 48
Number of Observations Used 48
The GLM Procedure
Dependent Variable: yield
Sum of
Source DF Squares Mean Square F Value Pr > F
Model 37 19594659.36 529585.39 5.81 0.0027
Error 10 911026.56 91102.66
Corrected Total 47 20505685.92
R-Square Coeff Var Root MSE yield Mean
0.955572 11.71349 301.8322 2576.792
Source DF Type I SS Mean Square F Value Pr > F
block 5 15498754.42 3099750.88 34.02 <.0001
selection 32 4095904.94 127997.03 1.40 0.2930
Source DF Type III SS Mean Square F Value Pr > F
block 5 6968486.444 1393697.289 15.30 0.0002
selection 32 4095904.944 127997.030 1.40 0.2930
The SAS System 10:02 Thursday, November 15, 2007 83
The GLM Procedure
Source Type III Expected Mean Square
block Var(Error) + 3 Var(block)
selection Var(Error) + Q(selection)
The GLM Procedure
Tests of Hypotheses for Mixed Model Analysis of Variance
Dependent Variable: yield
Source DF Type III SS Mean Square F Value Pr > F
block 5 6968486 1393697 15.30 0.0002
selection 32 4095905 127997 1.40 0.2930
Error: MS(Error) 10 911027 91103
Least Squares Means
Standard
selection yield LSMEAN Error Pr > |t|
1 2260.22222 341.18756 <.0001
10 2567.88889 341.18756 <.0001
11 3054.88889 341.18756 <.0001
12 1632.22222 341.18756 0.0007
13 2387.88889 341.18756 <.0001
14 2401.88889 341.18756 <.0001
15 2323.88889 341.18756 <.0001
16 2769.88889 341.18756 <.0001
17 2568.88889 341.18756 <.0001
18 2562.22222 341.18756 <.0001
19 2890.22222 341.18756 <.0001
2 2329.88889 341.18756 <.0001
20 2344.88889 341.18756 <.0001
21 2962.88889 341.18756 <.0001
22 2701.88889 341.18756 <.0001
23 2444.88889 341.18756 <.0001
24 2629.88889 341.18756 <.0001
25 2784.22222 341.18756 <.0001
26 2851.88889 341.18756 <.0001
27 2816.22222 341.18756 <.0001
28 1862.22222 341.18756 0.0003
29 2162.22222 341.18756 <.0001
3 2901.88889 341.18756 <.0001
30 2801.88889 341.18756 <.0001
4 2864.88889 341.18756 <.0001
5 2024.22222 341.18756 0.0001
6 1822.88889 341.18756 0.0003
7 2512.22222 341.18756 <.0001
8 2527.88889 341.18756 <.0001
9 1942.88889 341.18756 0.0002
Ci 2725.66667 123.22247 <.0001
St 2759.16667 123.22247 <.0001
Wa 2677.83333 123.22247 <.0001
Proc Mixed – Block random effect KR correction degrees of freedom and standard errors.
procmixeddata=a method=reml; ;
class block check selection idchk;
model yield = selection /ddfm=kr;
random block;
lsmeans selection ;
estimate "1 lsmn n" intercept 1 selection 100 ;
estimate "1 lsmn n" intercept 1 selection 100|block 000001 ;
estimate "St lsmn" intercept 1
selection 0000000000
0000000000
0000000000
010/divisor=1 ;
estimate "St lsmn n" intercept 6
selection 0000000000
0000000000
0000000000
060 |block 111111 /divisor=6 ; ;
estimate "1 vs 12" selection 100 -1;
estimate "1 vs 10" selection 1 -1;
estimate "1 vs 10 n" selection 1 -1 | block 0000 -11 ;
estimate "1 vs St broad"
selection 6000000000
0000000000
0000000000 0 -60 /divisor=6 ;
estimate "1 vs St narrow"
selection 6000000000
0000000000
0000000000
0 -60 |block -1 -1 -1 -1 -15/divisor=6;
estimate "Wa vs St broad"
selection 0000000000
0000000000
0000000000 0 -66/divisor=6 ;
run;
OUTPUT
The Mixed Procedure
Model Information
Data Set WORK.A
Dependent Variable yield
Covariance Structure Variance Components
Estimation Method REML
Residual Variance Method Profile
Fixed Effects SE Method Prasad-Rao-Jeske-
Kackar-Harville
Degrees of Freedom Method Kenward-Roger
Class Level Information
Class Levels Values
block 6 I II III IV V VI
check 2 check lines
selection 33 1 10 11 12 13 14 15 16 17 18
19 2 20 21 22 23 24 25 26 27
28 29 3 30 4 5 6 7 8 9 Ci St
Wa
idchk 2 0 1
Dimensions
Covariance Parameters 2
Columns in X 34
Columns in Z 6
Subjects 1
Max Obs Per Subject 48
Number of Observations
Number of Observations Read 48
Number of Observations Used 48
Number of Observations Not Used 0
Iteration History
Iteration Evaluations -2 Res Log Like Criterion
0 1 245.51933126
1 1 232.87821020 0.00000000
The Mixed Procedure
Convergence criteria met.
Covariance Parameter
Estimates
Cov Parm Estimate
block 434198
Residual 91103
Fit Statistics
-2 Res Log Likelihood 232.9
AIC (smaller is better) 236.9
AICC (smaller is better) 237.9
BIC (smaller is better) 236.5
Type 3 Tests of Fixed Effects
Num Den
Effect DF DF F Value Pr > F
selection 32 10.1 1.38 0.3022
Estimates
Standard
Label Estimate Error DF t Value Pr > |t|
1 lsmn b 2309.43 434.86 14.5 5.31 <.0001
1 lsmn n 3013.00 301.83 10 9.98 <.0001
St lsmn 2759.17 295.89 6.34 9.33 <.0001
St lsmn n 2759.17 123.22 10 22.39 <.0001
1 vs 12 628.00 426.86 10 1.47 0.1720
1 vs 10 -175.12 493.69* 10.3 -0.35 0.7299
1 vs 10 n 1720.00 426.86 10 4.03 0.0024
1 vs St broad -449.74 363.21 10.3 -1.24 0.2432
1 vs St narrow 253.83 326.02 10 0.78 0.4542
Wa vs St broad -81.3333 174.26 10 -0.47 0.6507
> sqrt((0*434198+91103)/6)
[1] 123.2227
> sqrt(2*91103/1)
[1] 426.8559
> sqrt(8*91103/3) * See formula in handout
[1] 492.8908
> sqrt(7*91103/6)
[1] 326.0166
> sqrt(19*91103/18)
[1] 310.1037
> sqrt(28*91103/18)
[1] 376.4516 * difference between an adjusted selection mean and a check mean. See formula in handout
Least Squares Means
The Mixed Procedure
Least Squares Means
Standard
Effect selection Estimate Error DF t Value Pr > |t|
selection 11 3076.14 434.86 14.5 7.07 <.0001
selection 12 1681.43 434.86 14.5 3.87 0.0016
selection 13 2388.09 434.86 14.5 5.49 <.0001
selection 14 2402.09 434.86 14.5 5.52 <.0001
selection 15 2333.90 434.86 14.5 5.37 <.0001
selection 16 2686.55 434.86 14.5 6.18 <.0001
selection 17 2569.09 434.86 14.5 5.91 <.0001
selection 18 2564.89 434.86 14.5 5.90 <.0001
selection 19 2939.43 434.86 14.5 6.76 <.0001
selection 2 2246.55 434.86 14.5 5.17 0.0001
selection 20 2366.14 434.86 14.5 5.44 <.0001
selection 21 2879.55 434.86 14.5 6.62 <.0001
selection 22 2702.09 434.86 14.5 6.21 <.0001
selection 23 2466.14 434.86 14.5 5.67 <.0001
selection 24 2639.90 434.86 14.5 6.07 <.0001
selection 25 2786.89 434.86 14.5 6.41 <.0001
selection 26 2852.09 434.86 14.5 6.56 <.0001
selection 27 2818.89 434.86 14.5 6.48 <.0001
selection 28 1864.89 434.86 14.5 4.29 0.0007
selection 29 2211.43 434.86 14.5 5.09 0.0001
selection 3 2911.90 434.86 14.5 6.70 <.0001
selection 30 2811.90 434.86 14.5 6.47 <.0001
selection 4 2874.90 434.86 14.5 6.61 <.0001
selection 5 2026.89 434.86 14.5 4.66 0.0003
selection 6 1844.14 434.86 14.5 4.24 0.0008
selection 7 2561.43 434.86 14.5 5.89 <.0001
selection 8 2444.55 434.86 14.5 5.62 <.0001
selection 9 1964.14 434.86 14.5 4.52 0.0004
selection Ci 2725.67 295.89 6.34 9.21 <.0001
selection St 2759.17 295.89 6.34 9.33 <.0001
selection Wa 2677.83 295.89 6.34 9.05 <.0001
> sqrt((1*434198+91103)/6)
[1] 295.8888
Alpha-lattice designs
Alpha-lattice designs are replicated designs that divide the replicate into incomplete blocks that contain a fraction of the total number of entries. Genotypes are distributed among the blocks so that all pairs occur in the same incomplete-block in nearly equal frequency. The design permits removal of incomplete-block effects from the plot residuals and maximizes the use of comparisons between genotypes in the same incomplete-block.
How effective are alpha-lattice designs in increasing the precision of genotype means estimated from rainfed rice variety trials? There are several ways to address this question. One way is to compare the SEM or a related statistic like the LSD for trials laid out as alpha-lattices, and analyzed both as alpha-lattices and RCBDs.
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Thursday November 15, 2007 Augmented Analysis1