Applied Statistics and Probability for Engineers

14-3 FACTORIAL EXPERIMENTS 507

Table 14-1 A Factorial Experiment with

Two Factors

Factor B

FactorA Blow Bhigh

Alow 10 20

Ahigh 30 40

Table 14-2 A Factorial Experiment with

Interaction

Factor B

FactorA Blow Bhigh

Alow 10 20

Ahigh 30 40

Thus, if there are two factors Aand Bwith alevels of factor Aand blevels of factor B, each

replicate contains all abtreatment combinations.

The effect of a factor is defined as the change in response produced by a change in the level

of the factor. It is called a main effect because it refers to the primary factors in the study. For ex-

ample, consider the data in Table 14-1. This is a factorial experiment with two factors, Aand B,

each at two levels (Alow, Ahigh, and Blow, Bhigh). The main effect of factor Ais the difference between

the average response at the high level of Aand the average response at the low level of A, or

That is, changing factor Afrom the low level to the high level causes an average response in-

crease of 20 units. Similarly, the main effect of Bis

In some experiments, the difference in response between the levels of one factor is not the

same at all levels of the other factors. When this occurs, there is an interactionbetween the fac-

tors. For example, consider the data in Table 14-2. At the low level of factor B, the Aeffect is

and at the high level of factor B, the Aeffect is

Since the effect of Adepends on the level chosen for factor B, there is interaction between Aand B.

When an interaction is large, the corresponding main effects have very little practical

meaning. For example, by using the data in Table 14-2, we find the main effect of Aas

and we would be tempted to conclude that there is no factor Aeffect. However, when we ex-

amined the effects of Aat different levels of factor B, we saw that this was not the case. The

effect of factor Adepends on the levels of factor B. Thus, knowledge of the ABinteraction is

more useful than knowledge of the main effect. A significant interaction can mask the signif-

icance of main effects. Consequently, when interaction is present, the main effects of the

factors involved in the interaction may not have much meaning.

A

30  0

2



10  20

2

 0

A 0  20  20

A 30  10  20

B

20  40

2



10  30

2

 10

A

30  40

2



10  20

2

 20

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