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Example 19.1 Consider the data shown in Table 19.2 for a seven-factor study.
TABLE 19.2 Data for a Seven-Factor Experimental Design

I A B C D E F G y

1 –1 –1 –1 1 1 1 –1 20
1 1 –1 –1 –1 –1 1 1 35
1 –1 1 –1 –1 1 –1 1 7
1 1 1 –1 1 –1 –1 –1 42
1 –1 –1 1 1 –1 1 36
1 1 –1 1 –1 1 –1 –1 50
1 –1 1 1 –1 –1 1 –1 45
1 1 1 1 1 1 1 1 82
317 101 35 109 43 1 47 3 Total
39.62 12.62 4.37 13.62 537 0.125 5.87 0.37 Total/8


The effects of factors A through G are given in the last line of the table. The percentage of variation explained by each factor is proportional to the square of the corresponding effects. Thus, factors A through G explain 37.26, 4.74, 43.40, 6.75, 0, 8.06, and 0.03% variation, respectively. It is clear that further experimentation, if necessary, should be done using only factors C and A. Other factors explain very little variation.

Notice that in Example 19.1, we could obtain all main effects but we could not obtain any interactions. Thus, if the interactions are small or negligible, this design would save us a lot of experimental work.

19.1 PREPARING THE SIGN TABLE FOR A 2k–p DESIGN

The next step in understanding 2k–p designs is to learn to prepare tables with orthogonal columns. The procedure consists of the following two steps:

1.  Choose kp factors and prepare a complete sign table for a full factorial design with kp factors. This will result in a table of 2k–p rows and 2k–p columns. The first column will be marked I and consists of all I’s. The next kp columns will be marked with the kp factors that were chosen. The remaining columns are simply products of these factors.
2.  Of the 2k–pk + p – 1 columns on the right, choose p columns and mark them with the p factors that were not chosen in step 1.
Example 19.2 To prepare a 27–4 table, we start with a 23 design, shown in Table 19.3.
TABLE 19.3 A 23 Experimental Design

Experiment
No.
A B C AB AC BC ABC

1 –1 –1 –1 1 1 1 –1
2 1 –1 –1 –1 –1 1 1
3 –1 1 –1 –1 1 –1 1
4 1 1 –1 1 –1 –1 –1
5 –1 –1 1 1 –1 –1 1
6 1 –1 1 –1 1 –1 –1
7 –1 1 1 –1 –1 1 –1
8 1 1 1 1 1 1 1


Now we mark the rightmost four columns with D, E, F, and G instead of AB, AC, BC, and ABC, respectively. This gives us the 27–4 design shown in Table 19.1 earlier.
Example 19.3 Let us prepare a 24–1 design. Again we start with the 23 design shown in Table 19.3. From the four columns on the right, we arbitrarily pick the rightmost column and mark it D. This gives us the design shown in Table 19.4. This design will allow us to compute the main effects qA, qB, qC, and qD along with the interactions qAB, qAC, and qBC.


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