# The ANOVA Procedure

### Example 26.4 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 26.4.1, Output 26.4.2, and Output 26.4.3:

```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 26.4.1: Class Level Information

 Sugar Beet Varieties Latin Square Split-Plot Design

The ANOVA Procedure

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 72

Output 26.4.2: ANOVA Table

 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 26.4.2 that the overall model is significant.

Output 26.4.3: 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 26.4.3 shows that the effects for `Rep` and `Harvest` are significant, while the `Column` effect is not. The average `Y`s for the six different `Variety`s 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.