ST 524
NCSU - 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
y. j 
y.St
Y. j

 b1  b2  b3  b4  b5  b6  
j
6
6
b  b  b  b  b  b 
Y
 .St    1 2 3 4 5 6  St
6
6
 
Var Y .st 
 b2
6

 e2
6
Selection
Y. j '

b j '
 j '
1
1
Y.1    b6  1 Var Y.1    b2   e2
y. j ' 
Mean difference
Y.1  y.St  Y.1 

 b  b
1
2


b  b  b  b  b  b 
Y.St
    b6  1      1 2 3 4 5 6  St 
6
6


 b3  b4  b5  b6  6  1   6  1  St 
6
 
Var y. j . 
?  b2( L )
6
12 e2

?
6
Block as random effect – PROC GLM
proc glm data=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
Thursday November 15, 2007 Augmented Analysis
1
ST 524
NCSU - Fall 2007
Augmented Analysis
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
Number of Observations Used
48
48
The GLM Procedure
Dependent Variable: yield
Source
DF
Sum of
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
selection
5
32
15498754.42
4095904.94
3099750.88
127997.03
34.02
1.40
<.0001
0.2930
Source
DF
Type III SS
Mean Square
F Value
Pr > F
block
selection
5
32
6968486.444
4095904.944
1393697.289
127997.030
15.30
1.40
0.0002
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
selection
5
32
6968486
4095905
1393697
127997
15.30
1.40
0.0002
0.2930
Error: MS(Error)
10
911027
91103
Thursday November 15, 2007 Augmented Analysis
2
ST 524
NCSU - Fall 2007
Augmented Analysis
Least Squares Means
selection
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
yield LSMEAN
Standard
Error
Pr > |t|
2260.22222
2567.88889
3054.88889
1632.22222
2387.88889
2401.88889
2323.88889
2769.88889
2568.88889
2562.22222
2890.22222
2329.88889
2344.88889
2962.88889
2701.88889
2444.88889
2629.88889
2784.22222
2851.88889
2816.22222
1862.22222
2162.22222
2901.88889
2801.88889
2864.88889
2024.22222
1822.88889
2512.22222
2527.88889
1942.88889
2725.66667
2759.16667
2677.83333
341.18756
341.18756
341.18756
341.18756
341.18756
341.18756
341.18756
341.18756
341.18756
341.18756
341.18756
341.18756
341.18756
341.18756
341.18756
341.18756
341.18756
341.18756
341.18756
341.18756
341.18756
341.18756
341.18756
341.18756
341.18756
341.18756
341.18756
341.18756
341.18756
341.18756
123.22247
123.22247
123.22247
<.0001
<.0001
<.0001
0.0007
<.0001
<.0001
<.0001
<.0001
<.0001
<.0001
<.0001
<.0001
<.0001
<.0001
<.0001
<.0001
<.0001
<.0001
<.0001
<.0001
0.0003
<.0001
<.0001
<.0001
<.0001
0.0001
0.0003
<.0001
<.0001
0.0002
<.0001
<.0001
<.0001
Thursday November 15, 2007 Augmented Analysis
3
ST 524
NCSU - Fall 2007
Augmented Analysis
Proc Mixed – Block random effect KR correction degrees of freedom and standard errors.
proc mixed data=a method=reml; ;
class block check selection idchk;
model yield = selection /ddfm=kr;
random block;
lsmeans selection
;
estimate "1 lsmn n" intercept 1 selection 1 0 0 ;
estimate "1 lsmn n" intercept 1 selection 1 0 0|block 0 0 0 0 0 1 ;
estimate "St lsmn" intercept 1
selection 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 1 0/divisor=1 ;
estimate "St lsmn n" intercept 6
selection 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 6 0 |block 1 1 1 1 1 1 /divisor=6 ; ;
estimate "1 vs 12" selection
1 0 0 -1;
estimate "1 vs 10" selection
1 -1;
estimate "1 vs 10 n" selection
1 -1 | block 0 0 0 0 -1 1 ;
estimate "1 vs St broad"
selection 6 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 -6 0 /divisor=6 ;
estimate "1 vs St narrow"
selection 6 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 -6 0 |block -1 -1 -1 -1 -1 5/divisor=6;
estimate "Wa vs St broad"
selection 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 -6 6/divisor=6 ;
run;
OUTPUT
The Mixed Procedure
Model Information
Data Set
Dependent Variable
Covariance Structure
Estimation Method
Residual Variance Method
Fixed Effects SE Method
Degrees of Freedom Method
WORK.A
yield
Variance Components
REML
Profile
Prasad-Rao-JeskeKackar-Harville
Kenward-Roger
Class Level Information
Class
Levels
block
check
selection
idchk
6
2
33
2
Values
I II III IV V VI
check lines
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
0 1
Dimensions
Covariance Parameters
Columns in X
Columns in Z
Thursday November 15, 2007 Augmented Analysis
2
34
6
4
ST 524
NCSU - Fall 2007
Augmented Analysis
Subjects
Max Obs Per Subject
1
48
Number of Observations
Number of Observations Read
Number of Observations Used
Number of Observations Not Used
48
48
0
Iteration History
Iteration
Evaluations
-2 Res Log Like
Criterion
0
1
1
1
245.51933126
232.87821020
0.00000000
The Mixed Procedure
Convergence criteria met.
Covariance Parameter
Estimates
Cov Parm
Estimate
block
Residual
434198
91103
Fit Statistics
-2 Res Log Likelihood
AIC (smaller is better)
AICC (smaller is better)
BIC (smaller is better)
Effect
232.9
236.9
237.9
236.5
Type 3 Tests of Fixed Effects
Num
Den
DF
DF
F Value
selection
32
10.1
Pr > F
1.38
0.3022
DF
t Value
Estimates
Label
Estimate
1 lsmn b
1 lsmn n
St lsmn
St lsmn n
1 vs 12
1 vs 10
1 vs 10 n
1 vs St broad
1 vs St narrow
Wa vs St broad
2309.43
3013.00
2759.17
2759.17
628.00
-175.12
1720.00
-449.74
253.83
-81.3333
Standard
Error
434.86
301.83
295.89
123.22
426.86
493.69*
426.86
363.21
326.02
174.26
14.5
10
6.34
10
10
10.3
10
10.3
10
10
5.31
9.98
9.33
22.39
1.47
-0.35
4.03
-1.24
0.78
-0.47
Pr > |t|
<.0001
<.0001
<.0001
<.0001
0.1720
0.7299
0.0024
0.2432
0.4542
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
Thursday November 15, 2007 Augmented Analysis
5
ST 524
NCSU - Fall 2007
Augmented Analysis
Least Squares Means
The Mixed Procedure
Least Squares Means
Effect
selection
selection
selection
selection
selection
selection
selection
selection
selection
selection
selection
selection
selection
selection
selection
selection
selection
selection
selection
selection
selection
selection
selection
selection
selection
selection
selection
selection
selection
selection
selection
selection
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
Estimate
Standard
Error
DF
t Value
Pr > |t|
3076.14
1681.43
2388.09
2402.09
2333.90
2686.55
2569.09
2564.89
2939.43
2246.55
2366.14
2879.55
2702.09
2466.14
2639.90
2786.89
2852.09
2818.89
1864.89
2211.43
2911.90
2811.90
2874.90
2026.89
1844.14
2561.43
2444.55
1964.14
2725.67
2759.17
2677.83
434.86
434.86
434.86
434.86
434.86
434.86
434.86
434.86
434.86
434.86
434.86
434.86
434.86
434.86
434.86
434.86
434.86
434.86
434.86
434.86
434.86
434.86
434.86
434.86
434.86
434.86
434.86
434.86
295.89
295.89
295.89
14.5
14.5
14.5
14.5
14.5
14.5
14.5
14.5
14.5
14.5
14.5
14.5
14.5
14.5
14.5
14.5
14.5
14.5
14.5
14.5
14.5
14.5
14.5
14.5
14.5
14.5
14.5
14.5
6.34
6.34
6.34
7.07
3.87
5.49
5.52
5.37
6.18
5.91
5.90
6.76
5.17
5.44
6.62
6.21
5.67
6.07
6.41
6.56
6.48
4.29
5.09
6.70
6.47
6.61
4.66
4.24
5.89
5.62
4.52
9.21
9.33
9.05
<.0001
0.0016
<.0001
<.0001
<.0001
<.0001
<.0001
<.0001
<.0001
0.0001
<.0001
<.0001
<.0001
<.0001
<.0001
<.0001
<.0001
<.0001
0.0007
0.0001
<.0001
<.0001
<.0001
0.0003
0.0008
<.0001
<.0001
0.0004
<.0001
<.0001
<.0001
> sqrt((1*434198+91103)/6)
[1] 295.8888
Thursday November 15, 2007 Augmented Analysis
6
ST 524
NCSU - Fall 2007
Augmented Analysis
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.
Thursday November 15, 2007 Augmented Analysis
7
ST 524
NCSU - Fall 2007
Augmented Analysis
(
).
Thursday November 15, 2007 Augmented Analysis
8