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 The ANOVA Procedure

## Example 17.3: Split Plot

In some experiments, treatments can be applied only to groups of experimental observations rather than separately to each observation. When there are two nested groupings of the observations on the basis of treatment application, this is known as a split plot design. For example, in integrated circuit fabrication it is of interest to see how different manufacturing methods affect the characteristics of individual chips. However, much of the manufacturing process is applied to a relatively large wafer of material, from which many chips are made. Additionally, a chip's position within a wafer may also affect chip performance. These two groupings of chips -by wafer and by position-within-wafer -might form the whole plots and the subplots, respectively, of a split plot design for integrated circuits.

The following statements produce an analysis for a split-plot design. The CLASS statement includes the variables Block, A, and B, where B defines subplots within BLOCK*A whole plots. The MODEL statement includes the independent effects Block, A, Block*A, B, and A*B. The TEST statement asks for an F test of the A effect, using the Block*A effect as the error term. The following statements produce Output 17.3.1 and Output 17.3.2:

```   title 'Split Plot Design';
data Split;
input Block 1 A 2 B 3 Response;
datalines;
142 40.0
141 39.5
112 37.9
111 35.4
121 36.7
122 38.2
132 36.4
131 34.8
221 42.7
222 41.6
212 40.3
211 41.6
241 44.5
242 47.6
231 43.6
232 42.8
;

proc anova;
class Block A B;
model Response = Block A Block*A B A*B;
test h=A e=Block*A;
run;
```

Output 17.3.1: Class Level Information and ANOVA Table

 Split Plot Design

 The ANOVA Procedure

 Class Level Information Class Levels Values Block 2 1 2 A 4 1 2 3 4 B 2 1 2

 Number of observations 16

 Split Plot Design

 The ANOVA Procedure Dependent Variable: Response

 Source DF Sum of Squares Mean Square F Value Pr > F Model 11 182.0200000 16.5472727 7.85 0.0306 Error 4 8.4300000 2.1075000 Corrected Total 15 190.4500000

 R-Square Coeff Var Root MSE Response Mean 0.955736 3.609007 1.451723 40.22500

First, notice that the overall F test for the model is significant.

Output 17.3.2: Tests of Effects

 Split Plot Design

 The ANOVA Procedure Dependent Variable: Response

 Source DF Anova SS Mean Square F Value Pr > F Block 1 131.1025000 131.1025000 62.21 0.0014 A 3 40.1900000 13.3966667 6.36 0.0530 Block*A 3 6.9275000 2.3091667 1.10 0.4476 B 1 2.2500000 2.2500000 1.07 0.3599 A*B 3 1.5500000 0.5166667 0.25 0.8612

 Tests of Hypotheses Using the Anova MS for Block*A as an Error Term Source DF Anova SS Mean Square F Value Pr > F A 3 40.19000000 13.39666667 5.80 0.0914

The effect of Block is significant. The effect of A is not significant: look at the F test produced by the TEST statement, not at the F test produced by default. Neither the B nor A*B effects are significant. The test for Block*A is irrelevant, as this is simply the main-plot error.

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