Example 24.4 Latin Square Split Plot
The data for this example is taken from Smith (1951). A Latin square design is used to evaluate six different sugar beet varieties arranged in a six-row (Rep) by six-column (Column) square. The data are collected over two harvests. The variable Harvest then becomes a split plot on the original Latin square design for whole plots. The following statements produce Output 24.4.1, Output 24.4.2, and Output 24.4.3:
title1 'Sugar Beet Varieties';
title3 'Latin Square Split-Plot Design';
data Beets;
do Harvest=1 to 2;
do Rep=1 to 6;
do Column=1 to 6;
input Variety Y @;
output;
end;
end;
end;
datalines;
3 19.1 6 18.3 5 19.6 1 18.6 2 18.2 4 18.5
6 18.1 2 19.5 4 17.6 3 18.7 1 18.7 5 19.9
1 18.1 5 20.2 6 18.5 4 20.1 3 18.6 2 19.2
2 19.1 3 18.8 1 18.7 5 20.2 4 18.6 6 18.5
4 17.5 1 18.1 2 18.7 6 18.2 5 20.4 3 18.5
5 17.7 4 17.8 3 17.4 2 17.0 6 17.6 1 17.6
3 16.2 6 17.0 5 18.1 1 16.6 2 17.7 4 16.3
6 16.0 2 15.3 4 16.0 3 17.1 1 16.5 5 17.6
1 16.5 5 18.1 6 16.7 4 16.2 3 16.7 2 17.3
2 17.5 3 16.0 1 16.4 5 18.0 4 16.6 6 16.1
4 15.7 1 16.1 2 16.7 6 16.3 5 17.8 3 16.2
5 18.3 4 16.6 3 16.4 2 17.6 6 17.1 1 16.5
;
proc anova data=Beets;
class Column Rep Variety Harvest;
model Y=Rep Column Variety Rep*Column*Variety
Harvest Harvest*Rep
Harvest*Variety;
test h=Rep Column Variety e=Rep*Column*Variety;
test h=Harvest e=Harvest*Rep;
run;
Output 24.4.1
Class Level Information
The ANOVA Procedure
6 |
1 2 3 4 5 6 |
6 |
1 2 3 4 5 6 |
6 |
1 2 3 4 5 6 |
2 |
1 2 |
Output 24.4.2
ANOVA Table
The ANOVA Procedure
Dependent Variable: Y
46 |
98.9147222 |
2.1503200 |
7.22 |
<.0001 |
25 |
7.4484722 |
0.2979389 |
|
|
71 |
106.3631944 |
|
|
|
0.929971 |
3.085524 |
0.545838 |
17.69028 |
5 |
4.32069444 |
0.86413889 |
2.90 |
0.0337 |
5 |
1.57402778 |
0.31480556 |
1.06 |
0.4075 |
5 |
20.61902778 |
4.12380556 |
13.84 |
<.0001 |
20 |
3.25444444 |
0.16272222 |
0.55 |
0.9144 |
1 |
60.68347222 |
60.68347222 |
203.68 |
<.0001 |
5 |
7.71736111 |
1.54347222 |
5.18 |
0.0021 |
5 |
0.74569444 |
0.14913889 |
0.50 |
0.7729 |
First, note from Output 24.4.2 that the overall model is significant.
Output 24.4.3
Tests of Effects
5 |
4.32069444 |
0.86413889 |
5.31 |
0.0029 |
5 |
1.57402778 |
0.31480556 |
1.93 |
0.1333 |
5 |
20.61902778 |
4.12380556 |
25.34 |
<.0001 |
1 |
60.68347222 |
60.68347222 |
39.32 |
0.0015 |
Output 24.4.3 shows that the effects for Rep and Harvest are significant, while the Column effect is not. The average Ys for the six different Varietys are significantly different. For these four tests, look at the output produced by the two TEST statements, not at the usual ANOVA procedure output. The Variety*Harvest interaction is not significant. All other effects in the default output should either be tested by using the results from the TEST statements or are irrelevant as they are only error terms for portions of the model.
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