Applied Statistics and Probability for Engineers

532 CHAPTER 14 DESIGN OF EXPERIMENTS WITH SEVERAL FACTORS

Since the ABinteraction is one-half of this difference,

We could write Equation 14-18 as follows:

In this form, the ABinteraction is easily seen to be the difference in averages between runs

on two diagonal planes in the cube in Fig. 14-19(b). Using similar logic and referring to

Fig. 14-19(b), we find that the ACand BCinteractions are

AB

abcabc 112

4 n



bcbaca

4 n

The ABCinteraction is defined as the average difference between the ABinteraction for

the two different levels of C. Thus,

or

ABC

1

4 n

53 abcbc 4  3 acc 4  3 abb 4  3 a 11246

As before, we can think of the ABCinteraction as the difference in two averages. If the runs

in the two averages are isolated, they define the vertices of the two tetrahedra that comprise

the cube in Fig. 14-19(c).

In Equations 14-15 through 14-21, the quantities in brackets are contrastsin the treat-

ment combinations. A table of plus and minus signs can be developed from the contrasts

and is shown in Table 14-15. Signs for the main effects are determined directly from the test

matrix in Figure 14-18(b). Once the signs for the main effect columns have been estab-

lished, the signs for the remaining columns can be obtained by multiplying the appropriate

AB (14-18)

1

4 n

3 abcbcabbacca 1124

(14-19)

BC (14-20)

1

4 n

3112 ababcacbcabc 4

AC

1

4 n

3112 ababcacbcabc 4

ABC (14-21)

1

4 n

3 abcbcaccabba 1124

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