ST 524
RCBD and Missing observations
NCSU - Fall 2008
LSMEAN Calculation
Oil Content of Redwing Flaxseed inoculated at different stages of growth with S. linicola, Winnipeg, 1947 (in percentage)
Treat
block_1
block_2
block_3
block_4
Mean
Early_Bloom
33.3
31.9
34.9
37.1
34.300
Full_Bloom
34.4
34.0
34.5
33.1
34.000
Full_Bloom_P
36.8
36.6
37.0
36.4
36.700
Ripening
36.3
34.9
35.9
37.1
36.050
Seedling
34.4
35.9
36.0
34.1
35.100
Uninoculated
36.4
37.3
37.7
36.7
37.025
Mean
35.2667
35.1000
36.0000
35.7500
35.52917
Linear Model
Y  Xβ  e
F

, eij ~ iidN 0,  e2

Yij     i   j  eij ,
t
Treatment effect and Block effect. Under the assumption that

i 1
 i  i.  ..
 j  . j  ..
ˆi  y i.  y..
ˆ j  y. j  y..
i
 0,
r
and

j 1
j
0,
Analysis of variance table (Decomposition of Total Sum of Squares for Y)
The GLM Procedure
Class Level Information
Class
block
treat
Levels
4
6
Values
1 2 3 4
Early_Bloom Full_Bloom Full_Bloom_P Ripening Seedling uninoculated
Number of Observations Read
Number of Observations Used
24
24
The GLM Procedure
Dependent Variable: y
DF
Sum of
Squares
Mean Square
F Value
Pr > F
Model
8
34.79333333
4.34916667
3.31
0.0219
Error
15
19.71625000
1.31441667
Corrected Total
23
54.50958333
Source
SS  Block |  
Source
SS Treat | Block ,  
SS  Block | Treat 
SS Treat | Block 
block
treat
Source
block
treat
Tuesday September 16, 2008
R-Square
Coeff Var
Root MSE
y Mean
0.638298
3.226870
1.146480
35.52917
DF
Type I SS
Mean Square
F Value
Pr > F
3
5
3.14125000
31.65208333
1.04708333
6.33041667
0.80
4.82
0.5147
0.0080
DF
Type III SS
Mean Square
F Value
Pr > F
3
5
3.14125000
31.65208333
1.04708333
6.33041667
0.80
4.82
0.5147
0.0080
1
ST 524
RCBD and Missing observations
NCSU - Fall 2008
Balanced Design:
Type I and Type III SS are the same: Block and Treatments are uncorrelated.
Solution to linear model
Parameter
Intercept
block
block
block
block
treat
treat
treat
treat
treat
treat
Predicted value for observations
Standard
Error
t Value
Pr > |t|
0.70207282
0.66192061
0.66192061
0.66192061
.
0.81068387
0.81068387
0.81068387
0.81068387
0.81068387
.
53.05
-0.73
-0.98
0.38
.
-3.36
-3.73
-0.40
-1.20
-2.37
.
<.0001
0.4765
0.3417
0.7110
.
0.0043
0.0020
0.6941
0.2477
0.0313
.
Estimate
1
2
3
4
Early_Bloom
Full_Bloom
Full_Bloom_P
Ripening
Seedling
uninoculated
37.24583333
-0.48333333
-0.65000000
0.25000000
0.00000000
-2.72500000
-3.02500000
-0.32500000
-0.97500000
-1.92500000
0.00000000
B
B
B
B
B
B
B
B
B
B
B
y11 , y23 , y64
 37.2458 
 0.4833


 0.6500 


 0.2500 
 yˆ11  1 1 0 0 0 1 0 0 0 0 0   0  37.2458  0.4833  2.725 34.0375
 yˆ   1 0 0 1 0 0 1 0 0 0 0   2.725    37.2458  0.25  3.025   34.4708
 
 23  

 

 yˆ 64  1 0 0 0 1 0 0 0 0 0 1   3.025  
 37.2458
37.2458


 0.3250 
 0.9750 


 1.9250 


 0 
Note that, predicted value is a function of the effects
ˆi and ˆ j
34.0375 = 35.52917 + (34.3000 - 35.52917) + (35.2667 - 35.52917)
34.4708 = 35.52917 + (34.0000 - 35.52917) + (36.0000 - 35.52917)
37.2458 = 35.52917 + (
- 35.52917) + (
- 35.52917)
Least Squares Mean for Treatment - Treatment LSMEAN
Average for each treatment level over the block effects
 37.2458 
 0.4833


 0.6500 


 0.2500 

 37.2458  1 4   0.4833  0.6500  0.2500  0    2.725    34.3 
ˆ

1
1
4
1
4
1
4
1
4
1
0
0
0
0
0
0
 1.  

 ˆ   1 1 4 1 4 1 4 1 4 0 1 0 0 0 0   2.725   37.2458  1 4 0.4833  0.6500  0.2500  0  3.025    34.0 
 
 
 
2.

 
  


 37.025 
 ˆ 6.  1 1 4 1 4 1 4 1 4 0 0 0 0 0 1   3.025  
37.2458  1 4  0.4833  0.6500  0.2500  0   0


 0.3250 
 0.9750 


 1.9250 


 0 
Balanced Design with no missing observations:
Simple Arithmetic Mean = Least Squares Mean
ErrorMS
1.3144
LSMEAN Standard Error =

 0.5732401
r
4
treat
Standard
Early_Bloom
Full_Bloom
Full_Bloom_P
Ripening
Seedling
uninoculated
Tuesday September 16, 2008
y LSMEAN
Error
Pr > |t|
34.3000000
34.0000000
36.7000000
36.0500000
35.1000000
37.0250000
0.5732401
0.5732401
0.5732401
0.5732401
0.5732401
0.5732401
<.0001
<.0001
<.0001
<.0001
<.0001
<.0001
2
ST 524
RCBD and Missing observations
NCSU - Fall 2008
Missing observations
treat
block_1
block_2
block_3
block_4
Mean
Early_Bloom
33.3
31.9
34.9
37.1
34.300
Full_Bloom
34.4
.
34.5
33.1
34.000
Full_Bloom_P
36.8
36.6
37.0
36.4
36.700
Ripening
36.3
34.9
35.9
37.1
36.050
Seedling
34.4
35.9
36.0
34.1
35.100
Uninoculated
36.4
37.3
.
36.7
36.80
Mean
35.2667
35.3200
35.6600
35.7500
35.5000
The GLM Procedure
Level of
treat
N
Early_Bloom
Full_Bloom
Full_Bloom_P
Ripening
Seedling
uninoculated
4
3
4
4
4
3
Level of
block
N
1
2
3
4
6
5
5
6
--------------y-------------Mean
Std Dev
34.3000000
34.0000000
36.7000000
36.0500000
35.1000000
36.8000000
2.23308456
0.78102497
0.25819889
0.91469485
0.98994949
0.45825757
--------------y-------------Mean
Std Dev
35.2666667
35.3200000
35.6600000
35.7500000
1.41938954
2.10760528
0.98640762
1.71551741
Analysis of Variance Table
The GLM Procedure
Dependent Variable: y
Sum of
DF
Source
SS Treat | Block ,  
SS  Block | Treat ,  
SS Treat | Block ,  
Mean Square
F Value
Pr > F
2.35
0.0820
Model
8
28.06797619
3.50849702
Error
13
19.37202381
1.49015568
Corrected Total
21
47.44000000
R-Square
Coeff Var
Root MSE
y Mean
0.591652
3.438646
1.220719
35.50000
Source
SS  Block |  
Squares
block
treat
Source
block
treat
Missing observations:
Tuesday September 16, 2008
Type I SS
DF
Type I SS
Mean Square
F Value
Pr > F
3
5
0.99166667
27.07630952
0.33055556
5.41526190
0.22
3.63
0.8795
0.0282
DF
Type III SS
Mean Square
F Value
Pr > F
3
5
2.87797619
27.07630952
0.95932540
5.41526190
0.64
3.63
0.6005
0.0282
 Type III SS
3
ST 524
RCBD and Missing observations
NCSU - Fall 2008
Solution
Parameter
Intercept
block
block
block
block
treat
treat
treat
treat
treat
treat
Estimate
37.21488095
-0.48333333
-0.76130952
0.20297619
0.00000000
-2.65446429
-3.12142857
-0.25446429
-0.90446429
-1.85446429
0.00000000
1
2
3
4
Early_Bloom
Full_Bloom
Full_Bloom_P
Ripening
Seedling
uninoculated
B
B
B
B
B
B
B
B
B
B
B
Standard
Error
t Value
Pr > |t|
0.81834165
0.70478263
0.75049541
0.75049541
.
0.94591683
1.03169613
0.94591683
0.94591683
0.94591683
.
45.48
-0.69
-1.01
0.27
.
-2.81
-3.03
-0.27
-0.96
-1.96
.
<.0001
0.5049
0.3289
0.7911
.
0.0149
0.0097
0.7921
0.3564
0.0717
.
Least Squares Mean for Treatment - Treatment LSMEAN
Average for each treatment level over the block effects
 37.2149 
 2.6545


 3.1214 


 0.2545
 ˆ1.  1 1 0 0 0 0 0 1 4 1 4 1 4 1 4   0.9045 37.2149   2.6545   1 4   0.4833  0.7613  0.2030  0    34.3 
 ˆ   1 0 1 0 0 0 0 1 4 1 4 1 4 1 4   1.8545  37.2149  3.1214  1 4 0.4833  0.7613  0.2030  0   33.8331

  
 
 
 2.  


 36.9545 
 ˆ 6.  1 0 0 0 0 0 1 1 4 1 4 1 4 1 4   0  
37.2149  0  1 4  0.4833  0.7613  0.2030  0 


 0.4833
 0.7613


 0.2030 


 0 
 37.2149 
 0.4833


 0.7613


 0.2030 
 ˆ1.  1 1 4 1 4 1 4 1 4 1 0 0 0 0 0   0  37.2149  1 4   0.4833  0.7613  0.2030  0    2.6545    34.3 
 ˆ   1 1 4 1 4 1 4 1 4 0 1 0 0 0 0   2.6545  37.2149  1 4 0.4833  0.7613  0.2030  0  3.1214   33.8331
 
 
 
 
 2.  


 36.9545 
 ˆ 6.  1 1 4 1 4 1 4 1 4 0 0 0 0 0 1   3.1214  
37.2149  1 4  0.4833  0.7613  0.2030  0   0


 0.2545
 0.9045


 1.8545


 0 
Standard
treat
y LSMEAN
Error
Pr > |t|
Early_Bloom
Full_Bloom
Full_Bloom_P
Ripening
Seedling
uninoculated
Balanced Design with missing observations:
Least squares means are estimates of
34.3000000
33.8330357
36.7000000
36.0500000
35.1000000
36.9544643
Simple Arithmetic Mean
 i  
0.6103597
0.7226477
0.6103597
0.6103597
0.6103597
0.7226477

<.0001
<.0001
<.0001
<.0001
<.0001
<.0001
Least Squares Mean
 ˆ1*  ˆ2*  ˆ3*  ˆ4* 
*
*
*
  ˆ  ˆi  ˆ
4


ˆ i.  ˆ *  ˆi*  
Pairwise differences of LSMEANS are free of block effects.
ErrorMS
1.49015568

 0.6103597, for r  4,
ri
ri
but, se(Full Bloom LSMEAN) = 0.7226  0.7048
LSMEAN Standard Error =
Tuesday September 16, 2008
0.7047826, for r  3
4
ST 524
RCBD and Missing observations
NCSU - Fall 2008
To calculate LSMEANS we can use the ESTIMATE statement in PROC GLM or proc MIXED
Label
LSMEAN
LSMEAN
LSMEAN
LSMEAN
LSMEAN
LSMEAN
T1
T2
T3
T4
T5
T6
For LSMEAN (Full BLOOM)
estimate "LSMEAN T2" intercept 4
Estimate
Estimates
Standard
Error
DF
t Value
Pr > |t|
34.3000
33.8330
36.7000
36.0500
35.1000
36.9545
0.6104
0.7226
0.6104
0.6104
0.6104
0.7226
13
13
13
13
13
13
56.20
46.82
60.13
59.06
57.51
51.14
<.0001
<.0001
<.0001
<.0001
<.0001
<.0001
treat 0 4
block 1 1 1 1/divisor=4;
Simple Mean differences includes block effects
y Early Bloom  y Full Bloom

y11  y12  y13  y14 y21  y23  y24

4
3
   1   1 e11      1   2 e12       1   3 e13       1   4 e14 
    2   1  e21       2   3  e23       1   4  e24 

3
4







e

e

e

e





e

e

e

 

3
4
4
    1  1 2
 11 12 13 14       2  1 3
 21 23 24 
4
4
3
3

 

  1 3 2   3   4
  1   2  
  e1.  e2. 
12

LSMEANS differences is free of block effects


 ˆ1* 2 ˆ2*  ˆ3*  ˆ4* 
 ˆ *  ˆ2*  ˆ3*  ˆ4*  * *
Y  Xβ

e
ˆ e , eij ˆ~* iidN
ˆ* 0, 
ˆ *  ˆ2*   1
ˆ Early



  ˆ1  ˆ2
Bloom   Full Bloom     1  




4



L '  X ' X  MS Error  L


L '  1 1 4 1 4 1 4 1 4 0 1 0 0 0 0 
Var(LSMEAN for Full Bloom) =
X 'X 

0.375
-0.166667
-0.166667
-0.166667
0
-0.25
-0.25
-0.25
-0.25
-0.25
0
4

where, for Treat = 2,
=
-0.166667
0.3333333
0.1666667
0.1666667
0
0
0
0
0
0
0
-0.166667
0.1666667
0.3333333
0.1666667
0
0
0
0
0
0
0
-0.166667
0.1666667
0.1666667
0.3333333
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
-0.25
0
0
0
0
0.5
0.25
0.25
0.25
0.25
0
-0.25
0
0
0
0
0.25
0.5
0.25
0.25
0.25
0
-0.25
0
0
0
0
0.25
0.25
0.5
0.25
0.25
0
-0.25
0
0
0
0
0.25
0.25
0.25
0.5
0.25
0
-0.25
0
0
0
0
0.25
0.25
0.25
0.25
0.5
0
0
0
0
0
0
0
0
0
0
0
0
SAS MANUAL (PROC GLM)
LS-means are, in effect, within-group means appropriately adjusted for the other effects in the model. More
precisely, they estimate the marginal means for a balanced population (as opposed to the unbalanced design). For
this reason, they are also called estimated population marginal means (Searle, 1980). In the same way that the Type
I F-test assesses differences between the arithmetic treatment means (when the treatment effect comes first in
the model), the Type III F-test assesses differences between the LS-means. Accordingly, for the unbalanced twoway design, the discrepancy between the Type I and Type III tests is reflected in the arithmetic treatment means
and treatment LS-means. Note that, while the arithmetic means are always uncorrelated (under the usual
assumptions for analysis of variance), the LS-means may not be. This fact complicates the problem of multiple
comparisons for LS-means.
Estimable functions
Type I hypothesis
Tuesday September 16, 2008
5
ST 524
RCBD and Missing observations
NCSU - Fall 2008
Sum to zero parameterization
Last level is 0 parameterization, default parameterization in sas
Tuesday September 16, 2008
6
ST 524
RCBD and Missing observations
NCSU - Fall 2008
Randomized Complete Block Design with Subsampling
Example
Data from a study in potato Late Blight resistance conducted by researcher Luis Rivera at the
International Potato Center in Lima, Peru
n
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
family
fam01
fam01
fam01
fam01
fam01
fam01
fam01
fam01
fam01
fam01
fam24
fam24
fam24
fam24
fam24
fam24
fam24
fam24
fam24
fam24
fam25
fam25
fam25
fam25
fam25
fam25
fam25
fam25
fam25
fam25
fam31
fam31
fam31
fam31
fam31
fam31
fam31
fam31
fam31
fam31
fam32
fam32
fam32
fam32
fam32
fam32
fam32
fam32
fam32
fam32
fam34
fam34
fam34
fam34
fam34
fam34
fam34
fam34
fam34
fam34
Block
Yield
n
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
150
50
300
350
650
500
450
500
250
350
150
150
150
100
150
300
100
100
300
60
500
100
250
350
500
300
400
350
400
250
1300
100
350
200
200
300
100
300
300
80
150
500
500
200
270
220
900
200
340
900
390
530
120
50
150
150
420
490
140
300
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
Tuesday September 16, 2008
family
fam01
fam01
fam01
fam01
fam01
fam01
fam01
fam01
fam01
fam01
fam24
fam24
fam24
fam24
fam24
fam24
fam24
fam24
fam24
fam24
fam25
fam25
fam25
fam25
fam25
fam25
fam25
fam25
fam25
fam25
fam31
fam31
fam31
fam31
fam31
fam31
fam31
fam31
fam31
fam31
fam32
fam32
fam32
fam32
fam32
fam32
fam32
fam32
fam32
fam32
fam34
fam34
fam34
fam34
fam34
fam34
fam34
fam34
fam34
fam34
Block
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
Yield
n
700
900
300
150
450
700
350
400
200
350
300
500
50
300
50
300
200
750
550
300
60
70
60
110
150
10
190
150
140
150
350
100
100
250
100
100
150
500
650
200
200
620
300
130
240
100
250
230
300
150
400
500
250
900
750
50
400
250
100
400
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
family
fam01
fam01
fam01
fam01
fam01
fam01
fam01
fam01
fam01
fam01
fam24
fam24
fam24
fam24
fam24
fam24
fam24
fam24
fam24
fam24
fam25
fam25
fam25
fam25
fam25
fam25
fam25
fam25
fam25
fam25
fam31
fam31
fam31
fam31
fam31
fam31
fam31
fam31
fam31
fam31
fam32
fam32
fam32
fam32
fam32
fam32
fam32
fam32
fam32
fam32
fam34
fam34
fam34
fam34
fam34
fam34
fam34
fam34
fam34
fam34
Block
Yield
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
400
150
250
100
750
250
750
200
250
800
20
370
400
100
300
300
600
80
250
200
230
140
370
30
20
320
120
130
410
350
320
20
110
280
160
110
60
80
120
100
250
400
180
220
120
250
450
250
450
130
600
200
150
300
300
300
350
650
200
450
7
ST 524
RCBD and Missing observations
NCSU - Fall 2008
Linear Model : Yij     i   j   ij   ijk
t


Treatment is a Fixed Effect
i 1
0
i
r


Block is a Fixed Effect
j 1
j
0

Plot is a Random effect , nested on treatments

Plant is a Random effect, nested on plots
 ij

and
 ijk
 ij ~ iidN  0,  2 
 ijk ~ iidN  0,  2  ,
are independent random effects
Analysis of Variance - Expected Mean Squares
Sum of Squares Decomposition
t
r
s

i 1 j 1 k 1
t
r
t
i 1
j 1
r
s
(Yijk  Y ... )2 rs (Y i..  Y ... )2  ts (Y . j.  Y ... )2  s (Y ij.  Y i..  Y . j. )2   (Yijk  Y ij. )2
i 1 j 1
i
j
k 1
SS due to Treatment Effects
SS due to Block Effects
SS due to Sampling Error
Variation among samples within pots
SS due to Experimental Error
Variation among pots within treatments
Analysis of Variance Table
Sources
df
Sum of Squares
r
Block
r-1
ts (Y . j.  Y ... )
2
j 1
t
Treatment
t-1
rs  (Y i..  Y ... )
2
i 1
Experimental
Error
Sampling Error
t 1   r 1
r
i 1 j 1
t  r   s  1
Plants(Plot)
Corrected Total
t
s (Y ij .  Y i..  Y . j . )2
t  r  s 1
s
 (Y
i
j
k 1
t
r
s

i 1 j 1 k 1
Tuesday September 16, 2008
ijk
 Y ij . )2
Mean Square
Block SS
r 1
E(MS)
 2  s e2   t  s 
Treatment SS
t 1
 2  s e2   r  s 
Exp Error SS
 r  1 t  1
 2  s 2
Sampling Error SS
rt  s  1
 2

2
j
j
 r  1

2
i
i
 t  1
(Yijk  Y ... )2
8
ST 524
RCBD and Missing observations
NCSU - Fall 2008
SAS Output
Dependent Variable: Yield
Source
DF
Sum of Squares
Mean Square
F Value
Pr > F
Model
17
1619185.000
95246.176
2.43
0.0021
Error
162
6351310.000
39205.617
Corrected Total
179
7970495.000
R-Square
Coeff Var
Root MSE
Yield Mean
0.203147
67.53977
198.0041
293.1667
Source
DF
Type I SS
Mean Square
F Value
Pr > F
Block
2
52990.0000
26495.0000
0.68
0.5102
family
5
727285.0000
145457.0000
3.71
0.0033
family*Block
10
838910.0000
83891.0000
2.14
0.0242
Source
DF
Type III SS
Mean Square
F Value
Pr > F
Block
2
52990.0000
26495.0000
0.68
0.5102
family
5
727285.0000
145457.0000
3.71
0.0033
family*Block
10
838910.0000
83891.0000
2.14
0.0242
Expected Mean Squares table
The GLM Procedure
Source
Type III Expected Mean Square
Block
Var(Error) + 10 Var(family*Block) + Q(Block)
family
Var(Error) + 10 Var(family*Block) + Q(family)
family*Block
Var(Error) + 10 Var(family*Block)
Test of Hypothesis
Source
DF
Type III SS
Mean Square
F Value
Pr > F
Block
2
52990
26495
0.32
0.7362
family
5
727285
145457
1.73
0.2145
Error
10
838910
83891
Error: MS(family*Block)
Tuesday September 16, 2008
9
ST 524
RCBD and Missing observations
NCSU - Fall 2008
Source
DF
Type III SS
Mean Square
F Value
Pr > F
family*Block
10
838910
83891
2.14
0.0242
162
6351310
39206
Error: MS(Error)

Treatments
H o : 1   2   3   4   5   6  0
H1 : At least one  i  0 , i  1, 2,
Fcalc 

145457.0
 1.73
83891.0
,6
p-value =0.2145
Do not Reject Ho
Experimental error
H o :  2  0
H1 :  2  0
Fcalc 
83891
 2.14
39206
p-value = 0.0242
Reject Ho
Variance Components Estimation
Var(among subsamples units within plot and family):

 2 
Var(among experimental units):

2

=39206
83891  39206
 4468.5
10
Variance among plants within plots
Variance among experimental units
An Observation, Yij     i   j   ij   ijk
Variance of an observation
Var Yijk    2   2
estimated by 4468.5 + 39206 = 43674.5
Plot mean
Y ij.     i   j   ij 

ijk
k
s
    i   j   ij 
 ij.
s
Variance of a plot mean
 


 2
, estimated by 4468.5  39206  8389.1
s
10
Note that estimate value for the variance of a plot mean is given by
Experimental Error MS 83891
var Y ij . 

 8389.1
s
10
Var Y ij .  Var    i   j   ij   ij .   2 
 
Treatment mean
  
j
Y i..     i 
jk
rs

jk
rs
ij


ijk
jk
rs
Variance of a treatment mean
Tuesday September 16, 2008
10
ST 524
RCBD and Missing observations
 
Var Y i..
NCSU - Fall 2008


j  j
2
2
4468.5 39206

 

 2796.3667
 Var     i 
  i.   i..      , estimated by
3
3 10
r
sr

 r


Note that the estimated value of variance of a treatment mean is given by
Experimental Error MS 83891
var Y i.. 

 2796.3667
rs
3 10
 
 
Var Y i...  2796.6667  52.8807
Standard error of a treatment mean =
PROC GLM
Least Squares Means
Standard Errors and Probabilities Calculated Using the Type III MS for family*Block as an
Error Term
family
Yield LSMEAN
Standard Error
Pr > |t|
fam01
398.333333
52.880683
<.0001
fam24
249.333333
52.880683
0.0008
fam25
220.333333
52.880683
0.0019
fam31
236.333333
52.880683
0.0012
fam32
313.333333
52.880683
0.0001
fam34
341.333333
52.880683
<.0001
PROC MIXED OUTPUT
Covariance Parameter Estimates
Cov Parm
Subject
Estimate
Intercept
family*Block
Residual
4468.54
39206
Wald’s test for Covariance paramters
Asymptotically Normal Wald’s Z test
Covariance Parameter Estimates
Cov Parm
Subject
Intercept
family*Block
Residual
Estimate
Standard Error
Z Value
Pr Z
4468.54
3776.93
1.18
0.1184
39206
4356.18
9.00
<.0001
H o :  2  0
H1 :  2  0
H o :  2  0
H1 :  2  0
Type 3 Tests of Fixed Effects
Tuesday September 16, 2008
Effect
Num DF
Den DF
F Value
Pr > F
Block
2
10
0.32
0.7362
family
5
10
1.73
0.2145
11
ST 524
RCBD and Missing observations
NCSU - Fall 2008
Least Squares Means
Effect
family
Estimate
Standard Error
DF
t Value
Pr > |t|
family
fam01
398.33
52.8807
10
7.53
<.0001
family
fam24
249.33
52.8807
10
4.72
0.0008
family
fam25
220.33
52.8807
10
4.17
0.0019
family
fam31
236.33
52.8807
10
4.47
0.0012
family
fam32
313.33
52.8807
10
5.93
0.0001
family
fam34
341.33
52.8807
10
6.45
<.0001
Same results when working with average over plants within each plot
Analyzing Block*Family interaction
Test of hypothesis for family is valid when no block*family interaction is present, families should behave
similarly within each block.
Tuesday September 16, 2008
12
ST 524
RCBD and Missing observations
Tuesday September 16, 2008
NCSU - Fall 2008
13