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TABLE 19.4 A 24–1 Experimental Design

Experiment No. A B C AB AC BC D

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

19.2 CONFOUNDING

One problem with fractional factorial experiments is that some of the effects cannot be determined. Only the combined influence of two or more effects can be computed. This problem is known as confounding and the effects whose influence cannot be separated are said to be confounded. The following example makes the concept clear.

Example 19.4 Let us now analyze the design prepared in the previous example of 24–1 design. If yi represents the observed y-value in the ith experiment, then the effect of A can be obtained by taking the inner product of column A and column y and dividing the sum by 8. This gives


Similarly, the effect of D is given by


The effect of the interaction ABC is obtained by multiplying the respective elements of columns A, B, C, and y. This gives


Notice that the expression for qABC is identical to that for qD. In fact, the expression is neither qD nor qABC; it is the sum of the two:


Without a full factorial design it is not possible to get separate estimates of D and ABC effects. In statistical terms the effects of D and ABC are confounded. This does not cause any serious problem, particularly if it is known prior to the experiments that the combined interaction of the factors A, B, and C is small compared to the effect of D. If that is the case, the preceding expression represents mostly qD.

Thus, we see that if two or more effects are confounded, their computation uses the same linear combination of responses, with the possible exception of the sign.

For reasons that will become apparent soon, the confounding in Example 19.4 can be denoted as

D = ABC

In this design, D and ABC are not the only effects that are confounded. It is easy to see that the A and BCD effects are also confounded.

that is,

A = BCD

In fact, every column in the design represents a sum of 2 effects. With four variables, each at two levels, there are 16 effects (including the I column of mean responses). In a 24–1 design, only eight quantities can be computed. Each quantity therefore represents a sum of 2 effects. The complete list of confoundings in this design is as follows:

A = BCD, B = ACD, C = ABD, AB = CD

AC = BD, BC = AD, ABC = D, I = ABCD

where I = ABCD is used to denote the confounding of ABCD with the mean.

A fractional factorial design is not unique. For the same number of factors k and the same number of experiments 2k–p, there are 2p possible different fractional factorial designs. One 24–1 design was presented in Example 19.3. Another 24–1 design is shown in Table 19.5. This design has the following confoundings:

I = ABD, A = BD, B = AD, C = ABCD

D = AB, AC = BCD, BC = ACD, ABC = CD.

This design is generally not considered as good as the previous design. In the previous design, the mean (I) was confounded with the fourth-order interaction, and the main effects were confounded with third-order interactions.

TABLE 19.5 Another 24–1 Experimental Design

Experiment
No.
A B C D 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


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