ST 524
Homework 6
NCSU - Fall 2007
1. Variety Trial. Data from a number of yield trials on spring barley conducted at the principal research station of the
Oregon Agricultural Experiment Station during the 1983-1984 growing season. Each trial consisted of 12 entries
replicated two times in a randomized block design. The three trials have two checks in common: “Benton” and
“Steptoe”. The remaining 10 entries in each trial were new selections, and these differed from trial to trial.

Compute analysis of variance for the checks

Calculate check means

Calculate adjustment factor for each trial

Calculate adjusted mean yield for new selections in each trial.

Get the analysis of variance for the new selections for each trial,

Get a combined analysis, and pooled error means square.

Get the standard error for check least squares mean, and new selections least squares mean.

Get the standard error for the difference between a new selection and a check.
Linear Model Yijk     i  k i   j   ijk
where, i = 1, 2, 3 trials,
j = 1, 2, . . ., 30 selections + 2 checks, k = 1, 2 blocks within each trial
Results from example in book (Petersen, 1994)
Table 6.39
Variety
Trial
2
3317.0
3249.5
1.5
1
3395.5
3437.5
134.75
Benton
Steptoe
Adjustment
3
3324.5
2966.5
-136.75
Mean
3345.67
3217.83
3395.5  3317.0  3324.5
 3345.67
3
3437.5  3249.5  2966.5
y StepToe 
 3217.83
3
Table 6.40
y Benton 
Source
Total
Blocks
Check
Error
df
11
5
1
5
SS
2039994.2
883623.7
49024.0
1107436.5
MS
176724.7
49024.0
221469.3
1. Complete information left blank in table 6.41
Table 6.41
Trial
Selection
1
2
3
4
5
6
7
8
9
10
Adj. factor aj
Benton+LSI
Steptoe+LSI
1
2

y
y
y
3133.0
3200.0
3087.5
4025.0
3678.5
3474.5
3579.0
3599.5
2804.0
2737.0
134.75
Thursday November 1, 2007
2998.2
3065.25
2953.8
3890.2*
3543.8
3339.8
3444.2
3464.8
2669.2
2602.25
3

y
3350.0
3348.5
3516.0
3514.5
2845.5
2844.0
4087.0
4085.5++
3095.5
3094.0
2833.0
2831.5
2574.5
2673.0
2937.0
2935.5
2733.0
3731.5
3362.0
3360.5
1.50
663.72+ 3345.667 = 4009.387 (4009.867)
663.72+ 3217.833 = 3881.553 (3882.033)

y
y
3395.5
3178.0
2633.0
3208.0
3666.5
2562.0
2642.5
3241.5
3783.0
2691.5
-136.25
3531.8
3314.25
2769.2
3344.2
3802.8
2698.2
2598.8
3377.8
3919.2*
2827.75
1
ST 524
Homework 6
NCSU - Fall 2007
Table 6.42
Sums of Squares
Trial
Source
Total
Block
Selection
Error
df
19
1
9
9
1
709153.2
1472073.8
3005593.2
2613486.2
2
4326438.5
19406.4
3643529.0
708393.1
3
5256706.9
342434.4
3983850.4
930422.1
Pooled Variance
Ep 

SSEc  SSESel ,1  SSESel ,2  SSESel ,3
Error df c  Error df Sel ,1  Error df Sel ,2  Error df Sel ,3
SSEc  SSESel ,1  SSESel ,2  SSESel ,3
 p  r  1 c  1  p  s  1 r  1
p = 3 trials,
Ep 
s = 30 selections ,
c = 2 checks, r = 2 blocks within each trial
1107346.5  2613486.2  708393.05  930422.05
 3  2  1 2  1  3 10  1 2  1
5359648
 167489
32
Means and Standard Errors

Compare with MSError in combined ANOVA
Mean
Benton
StepToe
Difference two check means
Standard Error
3345.667
167489  3  2   167.08
3217.833
167489  3  2   167.08
127.834
2 167489  3  2   236.28
p = number of trials; r = number of blocks per trial; c = number of check varieties
s = number of new selections per trial.
2. Get the standard error for the difference between two new selections .
Adjusted Means and Standard Errors
Mean
Standard Error
Standard Error
Adjusted jth Selection in ith trial
E p  c  1  r  c 
167489  2  1  2  2   354.42
Difference two adjusted means
in same trial ith
Difference two adjusted means
in different trials
Difference jth adjusted mean in
trial ith and a check mean
2  Ep r
2 167489 2  409.25
2E p  c  1  r  c 
2 167489  2  1  2  2   501.23
E p  pc  p  c 
167489  3  2  3  2   3  2  2   391.83
Thursday November 1, 2007
 p  r  c
2
ST 524
Homework 6
NCSU - Fall 2007
Least Significant Increase: LSI  t1 , pr 1 c 1 p r 1 s 1  E p  pc  p  c   p  r  c  ,


LSI  t10.05,32 df  167489   3  2  3  2   3  2  2 
 1.6939  391.8311  663.72
Used to determine which, if any, of the new selections outyield a given check.
3. Rank the new selections.
Modify the previous report indicate analyses done with proc mixed and ranking of selections.
Report of the statistical analysis (Petersen, 1994)
A group of yield trials on new spring barley selections was conducted on the principal
research farm of the Oregon Agricultural Experiment Station near Corvallis. A
combined Analysis was conducted on a subset of data from three of these trials. There
were two checks in common in each trial. These were SteptToe, a long-term baseline
variety, and Benton, a recent release. Each trial contained 10 new selections, which
differed from trial to trial. The entries in each trial were replicated twice in a
randomized block design.
The combined analysis produced adjustment factors that could be used to adjust the
mean yields of the new selections for yield differences from one trial to another.
The mean yield of the baseline, long-term check, Steptoe, was 3217.833 , while that of
the recent release, Benton, was 3345.667, For these trials the 5% LSI (least significant
increase) for comparing an adjusted selection mean with a check mean was 663.72,
Hence any adjusted selection mean greater than3881.553 significantly outyields StepToe,
while any adjusted selection mean greater than 4009.387 significantly outyields Benton.
Selections that outyield Benton are line 4 in trial 2, while selections 4 in trial1, selection 4
in trial 2 and selection 9 in trial 3 significantly outyield StepToe.
4. Combined Anova: Checks and Selections all three trials
The GLM Procedure
Class Level Information
Class
Levels
trial
block
selection
3
2
32
entry
idcheck
12
2
Values
1 2 3
1 2
10_1 10_2 10_3 1_1 1_2 1_3 2_1 2_2 2_3 3_1 3_2 3_3 4_1 4_2 4_3 5_1 5_2 5_3
6_1 6_2 6_3 7_1 7_2 7_3 8_1 8_2 8_3 9_1 9_2 9_3 Benton Steptoe
1 10 2 3 4 5 6 7 8 9 Benton Steptoe
0 1
Number of Observations Read
Number of Observations Used
72
72
Dependent Variable: yield
Source
DF
Sum of
Squares
Mean Square
F Value
Pr > F
Model
39
14065384.16
360650.88
2.15
0.0141
Error
32
5359647.72
167488.99
Corrected Total
71
19425031.88
R-Square
Coeff Var
Root MSE
yield Mean
0.724086
12.67123
409.2542
3229.792
Thursday November 1, 2007
3
ST 524



Homework 6
NCSU - Fall 2007
Source
DF
Type I SS
Mean Square
F Value
Pr > F
idcheck(trial)
block*idcheck(trial)
selection(idcheck)
5
6
28
821744.43
2570642.95
10672996.78
164348.89
428440.49
381178.46
0.98
2.56
2.28
0.4443
0.0388
0.0130
Source
DF
Type III SS
Mean Square
F Value
Pr > F
idcheck(trial)
block*idcheck(trial)
selection(idcheck)
2
6
28
146895.50
2570642.95
10672996.78
73447.75
428440.49
381178.46
0.44
2.56
2.28
0.6488
0.0388
0.0130
Error SS = 1107346.417 + 2613486.200 + 708393.050+ 930422.050
> 5359648
> Error.ss/32
> 167489
Selection (idcheck) SS = 49024.0833 + 3005593.200 + 3634529.050 + 3983850.450
> 10672997
Alternative Analysis PROC MIXED – ADJUSTED Selection Means
Analysis:
3 trials * (2 Blocks each trial * 10 Selections each trial) +
3 trials * 2 Blocks each trial * 2 Checks
Total of 30 Selections + 2 Checks = 32 ENTRIES
Source
Trial
Block(trial)
Selection
Error
- Entry*block 31
- Check*Trial 2
- Check vs Selection(trial) =
3 - 1 = 2
Total
d.f.
2
3
31
35 =
(32-1)*(2-1) + 2 = 31
+ (2-1)*(3-1)
= 2
+ 1*3 – 1
= 2
72-1 = 71
PROC MIXED
Get Adjusted Means for Selections within each trial - LSMEANS
proc mixed data=barleychk method=reml;
class trial block selection idcheck entry;
model yield= trial
selection
/ddfm=kr;
random block (trial);
lsmeans trial selection ;
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.BARLEYCHK
yield
Variance Components
REML
Profile
Prasad-Rao-JeskeKackar-Harville
Kenward-Roger
Class Level Information
Class
trial
block
selection
Thursday November 1, 2007
Levels
3
2
32
Values
1 2 3
1 2
10_1 10_2 10_3 1_1 1_2 1_3 2_1
4
ST 524
Homework 6
idcheck
entry
NCSU - Fall 2007
2_2 2_3 3_1 3_2 3_3 4_1 4_2
4_3 5_1 5_2 5_3 6_1 6_2 6_3
7_1 7_2 7_3 8_1 8_2 8_3 9_1
9_2 9_3 Benton Steptoe
0 1
1 10 2 3 4 5 6 7 8 9 Benton
2
12
Steptoe
Dimensions
Covariance Parameters
Columns in X
Columns in Z
Subjects
Max Obs Per Subject
2
36
6
1
72
Number of Observations
Number of Observations Read
Number of Observations Used
Number of Observations Not Used
72
72
0
Iteration History
Iteration
Evaluations
0
1
-2 Res Log Like
Criterion
1
599.33857463
1
593.71674291
Convergence criteria met.
0.00000000
Covariance Parameter
Estimates
Cov Parm
Estimate
block(trial)
Residual
52997
158484
Fit Statistics
-2 Res Log Likelihood
AIC (smaller is better)
AICC (smaller is better)
BIC (smaller is better)
593.7
597.7
598.1
597.3
Type 3 Tests of Fixed Effects
Effect
trial
selection
Effect
trial
trial
trial
selection
selection
selection
selection
selection
selection
trial
1
2
3
10_1
10_2
10_3
1_1
1_2
Thursday November 1, 2007
Num
DF
Den
DF
F Value
Pr > F
2
31
11
35
0.28
2.18
0.7627
0.0134
Least Squares Means
Standard
Estimate
Error
3358.05
3224.80
3087.05
2602.25
3360.50
2827.75
2998.25
3348.50
235.13
235.13
235.13
338.36
338.36
338.36
338.36
338.36
DF
t Value
Pr > |t|
8.06
8.06
8.06
37.8
37.8
37.8
37.8
37.8
14.28
13.71
13.13
7.69
9.93
8.36
8.86
9.90
<.0001
<.0001
<.0001
<.0001
<.0001
<.0001
<.0001
<.0001
5
ST 524
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
Homework 6
1_3
2_1
2_2
2_3
3_1
3_2
3_3
4_1
4_2
4_3
5_1
5_2
5_3
6_1
6_2
6_3
7_1
7_2
7_3
8_1
8_2
8_3
9_1
9_2
9_3
Benton
Steptoe
3531.75
3065.25
3514.50
3323.25
2952.75
2844.00
2769.25
3890.25
4085.50
3344.25
3543.75
3094.00
3802.75
3339.75
2831.50
2698.25
3444.25
2673.00
2598.75
3464.75
2935.50
3377.75
2669.25
3731.50
3919.25
3345.67
3217.83
338.36
338.36
338.36
338.36
338.36
338.36
338.36
338.36
338.36
338.36
338.36
338.36
338.36
338.36
338.36
338.36
338.36
338.36
338.36
338.36
338.36
338.36
338.36
338.36
338.36
187.74
187.74
NCSU - Fall 2007
37.8
37.8
37.8
37.8
37.8
37.8
37.8
37.8
37.8
37.8
37.8
37.8
37.8
37.8
37.8
37.8
37.8
37.8
37.8
37.8
37.8
37.8
37.8
37.8
37.8
21.7
21.7
10.44
9.06
10.39
9.82
8.73
8.41
8.18
11.50
12.07
9.88
10.47
9.14
11.24
9.87
8.37
7.97
10.18
7.90
7.68
10.24
8.68
9.98
7.89
11.03
11.58
17.82
17.14
<.0001
<.0001
<.0001
<.0001
<.0001
<.0001
<.0001
<.0001
<.0001
<.0001
<.0001
<.0001
<.0001
<.0001
<.0001
<.0001
<.0001
<.0001
<.0001
<.0001
<.0001
<.0001
<.0001
<.0001
<.0001
<.0001
<.0001
Estimates of LSMEANS for L1T1 broad and narrow sense
Estimates
Label
Selection 1 trial 1
L1 T1 AdjMN
s.e.(LSMEAN L1 T1 narrow)
s.e.(LSMEAN L1 T1) =
=
Estimate
3133.00
281.50
2998.25
338.36
Standard
Error
35
37.8
DF
11.13
8.86
t Value
Pr > |t|
<.0001
<.0001
158484
 281.4996
2
52997  52997+158484 

 338.49
6
2
Estimates of LSMEANS for L1T1, L2T1, L2T3 Benton and pairwise differences.
Estimates
Label
L2 T1 AdjMN
L2 T3 AdjMN
L1-L2 , T1 AdjMNDiff
L1_T1-L2_T3 AdjMNDIFF
Benton LSMEAN
AdjMN L1 T1 - Benton
Estimate
3065.25
3323.25
-67.00
-325.00
3345.67
-347.42
Standard
Error
338.36
338.36
398.10
487.57
187.74
363.41
DF
37.8
37.8
35
35
21.7
35
t Value
9.06
9.82
-0.17
-0.67
17.82
-0.96
Pr > |t|
<.0001
<.0001
0.8673
0.5094
<.0001
0.3456
Standard Error for Selection means and pairwise mean differences
s.e.(Difference between two adjusted means within the same trial) =
2
158484
 398.10
2
s.e.(Difference between two adjusted means in different trials) =
Thursday November 1, 2007
6
ST 524
2
Homework 6
NCSU - Fall 2007
158484   2+1
 487.57
2 2
s.e.(Benton Check mean) =
52997 158484

 187.74
6
6
s.e.(Difference between an adjusted means and a check mean) =
3  52997 4 158484

 363.53
6
6
Considering PROC MIXED results
Standard Errors
Mean
Standard Error (CLASSICAL)
Difference two
adjusted means in
same trial ith
Difference two
adjusted means in
different trials
Difference jth
adjusted mean in
trial ith and a check
mean
2 167489 2  409.25
2 167489  2  1  2  2   501.23
167489  3  2  3  2   3  2  2   391.83
Standard Error (PROC MIXED)
2
158484
 398.10
2
2
158484   2+1
 487.57
2 2
363.41
(from Estimate or LSMEAN
Differences)
Least Significant Increase: LSI  t1 , pr 1 c 1 p r 1 s 1  E p  pc  p  c   p  r  c  ,


LSI  t10.05,35 df  363.41
 1.6896  363.41  614.02
Used to determine which, if any, of the new selections outyield a given check
Report of the statistical analysis (Petersen, 1994)
A group of yield trials on new spring barley selections was conducted on the principal
research farm of the Oregon Agricultural Experiment Station near Corvallis. A
combined Analysis was conducted on a subset of data from three of these trials. There
were two checks in common in each trial. These were SteptToe, a long-term baseline
variety, and Benton, a recent release. Each trial contained 10 new selections, which
differed from trial to trial. The entries in each trial were replicated twice in a
randomized block design.
The combined analysis produced adjustment factors that could be used to adjust the
mean yields of the new selections for yield differences from one trial to another. The
variation among blocks within reps is 52997, while the residual variance is 52997.
The mean yield of the baseline, long-term check, Steptoe, was 3217.833 , while that of
the recent release, Benton, was 3345.667, For these trials the 5% LSI (least significant
increase) for comparing an adjusted selection mean with a check mean was 614.02,
Hence any adjusted selection mean greater than3831.85 significantly outyields StepToe,
while any adjusted selection mean greater than 3959.69 significantly outyields Benton.
Thursday November 1, 2007
7
ST 524
Homework 6
NCSU - Fall 2007
Selections that outyield Benton are line 4 in trial 2, while selections 4 in trial1, selection 4
in trial 2 and selection 9 in trial 3 significantly outyield StepToe.
Trial
Selection
1
2
3
4
5
6
7
8
9
10
Adj. factor aj
Benton+LSI
Steptoe+LSI
Thursday November 1, 2007
1
2
2
3



y
y
y
2998.2
3065.25
2953.8
3890.2*
3543.8
3339.8
3444.2
3464.8
2669.2
2602.25
3348.5
3514.5
2844.0
4085.5++
3094.0
2831.5
2673.0
2935.5
3731.5
3360.5
3531.8
3314.25
2769.2
3344.2
3802.8
2698.2
2598.8
3377.8
3919.2*
2827.75
614.02 + 3345.67 =3959.69
614.02 + 3217.83 = 3831.85
8
ST 524
Homework 6
NCSU - Fall 2007
Question 2. Alpha Design
Design Layout
Thursday November 1, 2007
„-----------------------…-----------------------†
|
|
block
|
|
‡---…---…---…---…---…---‰
|
| 1 | 2 | 3 | 4 | 5 | 6 |
‡-----------…-----------ˆ---ˆ---ˆ---ˆ---ˆ---ˆ---‰
|replication|plot
|
|
|
|
|
|
|
‡-----------ˆ-----------‰
|
|
|
|
|
|
|1
|1
| 11| 21| 23| 13| 17| 6|
|
‡-----------ˆ---ˆ---ˆ---ˆ---ˆ---ˆ---‰
|
|2
| 4| 10| 14| 3| 15| 12|
|
‡-----------ˆ---ˆ---ˆ---ˆ---ˆ---ˆ---‰
|
|3
| 5| 20| 16| 19| 7| 24|
|
‡-----------ˆ---ˆ---ˆ---ˆ---ˆ---ˆ---‰
|
|4
| 22| 2| 18| 8| 1| 9|
‡-----------ˆ-----------ˆ---ˆ---ˆ---ˆ---ˆ---ˆ---‰
|2
|1
| 8| 24| 12| 5| 2| 19|
|
‡-----------ˆ---ˆ---ˆ---ˆ---ˆ---ˆ---‰
|
|2
| 20| 15| 11| 9| 18| 7|
|
‡-----------ˆ---ˆ---ˆ---ˆ---ˆ---ˆ---‰
|
|3
| 14| 3| 21| 10| 13| 6|
|
‡-----------ˆ---ˆ---ˆ---ˆ---ˆ---ˆ---‰
|
|4
| 4| 23| 17| 1| 22| 16|
‡-----------ˆ-----------ˆ---ˆ---ˆ---ˆ---ˆ---ˆ---‰
|3
|1
| 11| 2| 17| 12| 21| 3|
|
‡-----------ˆ---ˆ---ˆ---ˆ---ˆ---ˆ---‰
|
|2
| 1| 15| 18| 13| 22| 5|
|
‡-----------ˆ---ˆ---ˆ---ˆ---ˆ---ˆ---‰
|
|3
| 14| 9| 4| 10| 16| 20|
|
‡-----------ˆ---ˆ---ˆ---ˆ---ˆ---ˆ---‰
|
|4
| 19| 8| 6| 23| 24| 7|
Š-----------‹-----------‹---‹---‹---‹---‹---‹---Œ
9
ST 524
Homework 6
NCSU - Fall 2007
Data
„-----------------------…-----------------------------------------------†
block
‡-------…-------…-------…-------…-------…-------‰
1
2
3
4
5
6
‡-----------…-----------ˆ-------ˆ-------ˆ-------ˆ-------ˆ-------ˆ-------‰
-replication-plot
‡-----------ˆ-----------‰
-1
-1
- 4.1172- 4.6540- 4.2323- 4.2530- 4.7876- 4.7085‡-----------ˆ-------ˆ-------ˆ-------ˆ-------ˆ-------ˆ-------‰
-2
- 4.4461- 4.1736- 4.7572- 3.3420- 5.0902- 5.2560‡-----------ˆ-------ˆ-------ˆ-------ˆ-------ˆ-------ˆ-------‰
-3
- 5.8757- 4.0141- 4.4906- 4.7269- 4.1505- 4.9577‡-----------ˆ-------ˆ-------ˆ-------ˆ-------ˆ-------ˆ-------‰
-4
- 4.5784- 4.3350- 3.9737- 4.9989- 5.1202- 3.3986‡-----------ˆ-----------ˆ-------ˆ-------ˆ-------ˆ-------ˆ-------ˆ-------‰
-2
-1
- 3.9926- 3.9039- 5.3127- 5.1202- 5.1566- 5.3148‡-----------ˆ-------ˆ-------ˆ-------ˆ-------ˆ-------ˆ-------‰
-2
- 3.6056- 4.9114- 5.1163- 4.2955- 5.0988- 4.6297‡-----------ˆ-------ˆ-------ˆ-------ˆ-------ˆ-------ˆ-------‰
-3
- 4.5294- 3.7999- 5.3802- 4.9057- 5.4840- 5.1751‡-----------ˆ-------ˆ-------ˆ-------ˆ-------ˆ-------ˆ-------‰
-4
- 4.3599- 4.3042- 5.0744- 5.7161- 5.0969- 5.3024‡-----------ˆ-----------ˆ-------ˆ-------ˆ-------ˆ-------ˆ-------ˆ-------‰
-3
-1
- 3.9205- 4.0510- 4.3234- 4.1746- 4.4130- 2.8873‡-----------ˆ-------ˆ-------ˆ-------ˆ-------ˆ-------ˆ-------‰
-2
- 4.6512- 4.6783- 4.2486- 4.7512- 4.2397- 4.1972‡-----------ˆ-------ˆ-------ˆ-------ˆ-------ˆ-------ˆ-------‰
-3
- 4.3887- 3.1407- 4.3960- 4.0875- 4.3852- 3.7349‡-----------ˆ-------ˆ-------ˆ-------ˆ-------ˆ-------ˆ-------‰
-4
- 4.5552- 3.9821- 4.2474- 3.8721- 3.5655- 3.6096Š-----------‹-----------‹-------‹-------‹-------‹-------‹-------‹-------Œ
Model
3 Replicates , 6 blocks of size 4 each, 24 entries, 3 repetitions per entry
Linear Model Yijh     j  bh j    i  eijh
k=4 plots per block, r=3 replications, t=24 entries, blocks=6 per replication
entries by block table: trial 1 | trial 2 | trial 3
1
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
2
3
4
5
*
6
1
2
3
4
*
5
6
1
*
*
*
*
2
3
4
5
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
6
*
*
*
*
Entry 1 in blocks 1_5, 2_4 and 3_1
Block 1 in trial 1 : Entries 1 , 7, 15, 17
Some entries are in the same block once,   1
Thursday November 1, 2007
10
ST 524
Homework 6
NCSU - Fall 2007
Some entries do not occur together in any of the blocks,   0
  1 for Entry 1 vs entries : 7, 15, 17 , 5, 9, 10 , 12, 14, 19
Q2.1 Calculate the following
PROC GLM
 Eb  Ee 
k  r  1 Eb
 0.240240  0.083463  0.081573
A
4  3  1 0.240240
A
Adjustment factor
Adjustment factor
Effective error mean square

Ee,lattice   e2 1   rkA  k  1 



Ee,lattice  0.083463 1   3  4  0.081573  4  1   0.099803
pooled error Erb 
 SSB  SSE  
 t  1 r  1
pooled error Erb 
 3.603599  2.587355  0.134586
 24  1 3  1
Same as Error MS when ignoring blocks within reps , Taken as if design is RCB with 3
blocks.
Sometimes recommended to use RCBD as an alternative for analysis when Eb  Ee , and the
efficiency is not that high.
% relative precision 
RP 
Erb
100
Ee,lattice
0.134586
 134.85%
0.099803
*********************************************************;
PROC MIXED
Q2.2 Questions
a)
Find the standard error of the differences of two lines.
Standard
Label
Estimate
Error
DF
t Value
entry 1 vs 2
entry 1 vs 2
b)
0.3146
0.6292
0.1381
0.2763
38.2
38.2
2.28
2.28
Pr > |t|
0.0285
0.0285
**divisor=2
**divisor=1
Calculate the standard error of the LSMEAN for Line (Entry) 1, narrow and broad
sense. Explain the differences.
Standard
Label
Estimate
Error
DF
t Value
Pr > |t|
entry 1 LSMEAN
5.1077
0.1995
44.1
25.60
<.0001
entry 1 LSMEAN BLUP
5.1625
0.1685
29.7
30.63
<.0001
Narrow space
LSMEAN is calculated
directly included in
Broad Space
LSMEAN is calculated
in the estimation of
conditional in the observed effects of blocks, which are not
the estimation of the standard error of the entry mean
unconditional in the observed random effects, which are included
the standard error of the entry mean
Thursday November 1, 2007
11
ST 524
Homework 6
NCSU - Fall 2007
Other calculations
Variance for the difference of two treatment means in class   1 , they can be found in the same
block.

  2  4  0.08523 
2k e2
var y i..  y i '..  
  
  0.068184
 r  k  1  1   3   4  1  1 






s.e. y i..  y i '..  av. var y i..  y i '..  0.068184  0.2611
Variance for the difference of two treatment means in class
block.
  0,
they do not share a common

  2  4  0.08523 
2k e2
var y i..  y i '..  
  
  0.07576
 r  k  1  1   3   4  1  0 






s.e. y i..  y i '..  av. var y i..  y i '..  0.07576  0.2752
Compare with table below,
  1 for Entry 1 vs entries : 7, 15, 17 , 5, 9, 10 , 12, 14, 19
Differences of Least Squares Means
Effect
entry
_entry
Estimate
Standard
Error
DF
t Value
Pr > |t|
entry
1
2
0.6292
0.2763
38.2
2.28
0.0285
entry
1
3
1.6085
0.2749
37.7
5.85
<.0001
entry
1
4
0.6176
0.2749
37.7
2.25
0.0306
entry
1
5
0.07049
0.2619
34.8
0.27
0.7894
entry
1
6
0.5710
0.2749
37.7
2.08
0.0447
entry
1
7
0.9966
0.2619
34.8
3.81
0.0005
entry
1
8
0.5801
0.2750
37.7
2.11
0.0416
entry
1
9
1.6055
0.2631
35.3
6.10
<.0001
entry
1
10
0.7345
0.2643
35.9
2.78
0.0086
entry
1
11
0.8244
0.2621
35
3.15
0.0034
. . .
c) Run a pairwise comparison test for the lsmeans and use the macro pdmix80 to
represent the differences among the entries.
--------------------------- Effect=entry
Obs
entry
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
1
5
15
19
21
14
13
12
16
17
6
8
22
4
2
10
18
11
23
24
Thursday November 1, 2007
Method=LSD(P<.01)
no block in common
In block 1 Trial 1
Set=1 ----------------------------
Estimate
Standard
Error
Letter
Group
5.1077
5.0372
4.9691
4.8403
4.7950
4.7757
4.7579
4.7553
4.7301
4.6026
4.5367
4.5276
4.5275
4.4901
4.4785
4.3732
4.3617
4.2833
4.2524
4.1539
0.1995
0.1994
0.1994
0.1994
0.1994
0.1994
0.1994
0.1995
0.1995
0.1994
0.1994
0.1995
0.1994
0.1995
0.1995
0.1995
0.1995
0.1995
0.1994
0.1995
A
AB
ABC
ABCD
ABCDE
ABCDE
ABCDEF
ABCDEF
ABCDEF
ABCDEF
ABCDEF
ABCDEF
ABCDEF
ABCDEF
ABCDEF
BCDEF
ABCDEF
CDEF
CDEF
DEFG
12
ST 524
Homework 6
21
22
23
24
Entries 1 and 5
9 3
7
20
9
3
4.1111
4.0400
3.5022
3.4992
are significantly different
0.1995
0.1994
0.1994
0.1995
NCSU - Fall 2007
EFG
FG
G
G
(higher mean) from entries 11 23 24 7 20
One reason for alpha design approximated values when
using above formulas, although there might be better
approximations.
Look at the standard errors of mean differences.
Thursday November 1, 2007
13