Example 17.4: 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 17.4.1 and Output 17.4.2:
title '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;
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 17.4.1: Class Level Information and ANOVA Table
|
| Sugar Beet Varieties |
| Latin Square Split-Plot Design |
| Class Level Information |
| Class |
Levels |
Values |
| Column |
6 |
1 2 3 4 5 6 |
| Rep |
6 |
1 2 3 4 5 6 |
| Variety |
6 |
1 2 3 4 5 6 |
| Harvest |
2 |
1 2 |
| Number of observations |
72 |
|
|
| Sugar Beet Varieties |
| Latin Square Split-Plot Design |
| The ANOVA Procedure |
| Dependent Variable: Y |
| Source |
DF |
Sum of Squares |
Mean Square |
F Value |
Pr > F |
| Model |
46 |
98.9147222 |
2.1503200 |
7.22 |
<.0001 |
| Error |
25 |
7.4484722 |
0.2979389 |
|
|
| Corrected Total |
71 |
106.3631944 |
|
|
|
| R-Square |
Coeff Var |
Root MSE |
Y Mean |
| 0.929971 |
3.085524 |
0.545838 |
17.69028 |
| Source |
DF |
Anova SS |
Mean Square |
F Value |
Pr > F |
| Rep |
5 |
4.32069444 |
0.86413889 |
2.90 |
0.0337 |
| Column |
5 |
1.57402778 |
0.31480556 |
1.06 |
0.4075 |
| Variety |
5 |
20.61902778 |
4.12380556 |
13.84 |
<.0001 |
| Column*Rep*Variety |
20 |
3.25444444 |
0.16272222 |
0.55 |
0.9144 |
| Harvest |
1 |
60.68347222 |
60.68347222 |
203.68 |
<.0001 |
| Rep*Harvest |
5 |
7.71736111 |
1.54347222 |
5.18 |
0.0021 |
| Variety*Harvest |
5 |
0.74569444 |
0.14913889 |
0.50 |
0.7729 |
|
First, note from Output 17.4.1 that the overall model is
significant.
Output 17.4.2: Tests of Effects
|
| Sugar Beet Varieties |
| Latin Square Split-Plot Design |
| The ANOVA Procedure |
| Dependent Variable: Y |
Tests of Hypotheses Using the Anova MS for Column*Rep*Variety as an
Error Term |
| Source |
DF |
Anova SS |
Mean Square |
F Value |
Pr > F |
| Rep |
5 |
4.32069444 |
0.86413889 |
5.31 |
0.0029 |
| Column |
5 |
1.57402778 |
0.31480556 |
1.93 |
0.1333 |
| Variety |
5 |
20.61902778 |
4.12380556 |
25.34 |
<.0001 |
| Tests of Hypotheses Using the Anova MS for Rep*Harvest as an Error Term |
| Source |
DF |
Anova SS |
Mean Square |
F Value |
Pr > F |
| Harvest |
1 |
60.68347222 |
60.68347222 |
39.32 |
0.0015 |
|
Output 17.4.2 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
using the results from the TEST statements or are irrelevant as
they are only error terms for portions of the model.
Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.
