The ANOVA Procedure
Example 25.4 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 25.4.1, Output 25.4.2, and Output 25.4.3:
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 25.4.1: Class Level Information
| 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 Read | 72 |
|---|---|
| Number of Observations Used | 72 |
Output 25.4.2: ANOVA Table
| Sugar Beet Varieties |
| Latin Square Split-Plot Design |
| 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 25.4.2 that the overall model is significant.
Output 25.4.3: Tests of Effects
| 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 25.4.3 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.