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 The ANOVA Procedure

## 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

 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 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 anError 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.

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