Applied Statistics and Probability for Engineers

508 CHAPTER 14 DESIGN OF EXPERIMENTS WITH SEVERAL FACTORS

It is easy to estimate the interaction effect in factorial experiments such as those illus-

trated in Tables 14-1 and 14-2. In this type of experiment, when both factors have two

levels, the ABinteraction effect is the difference in the diagonal averages. This represents

one-half the difference between the Aeffects at the two levels of B. For example, in

Table 14-1, we find the ABinteraction effect to be

Thus, there is no interaction between Aand B. In Table 14-2, the ABinteraction effect is

As we noted before, the interaction effect in these data is very large.

The concept of interaction can be illustrated graphically in several ways. Figure 14-1

plots the data in Table 14-1 against the levels of Afor both levels of B. Note that the Blowand

Bhighlines are approximately parallel, indicating that factors Aand Bdo not interact signifi-

cantly. Figure 14-2 presents a similar plot for the data in Table 14-2. In this graph, the Blowand

Bhighlines are not parallel, indicating the interaction between factors Aand B. Such graphical

displays are called two-factor interaction plots.They are often useful in presenting the re-

sults of experiments, and many computer software programs used for analyzing data from de-

signed experiments will construct these graphs automatically.

Figures 14-3 and 14-4 present another graphical illustration of the data from Tables 14-1

and 14-2. In Fig. 14-3 we have shown a three-dimensional surface plotof the data from

Table 14-1. These data contain no interaction, and the surface plot is a plane lying above the

A-Bspace. The slope of the plane in the Aand Bdirections is proportional to the main effects

of factors Aand B, respectively. Figure 14-4 is a surface plot of the data from Table 14-2.

Notice that the effect of the interaction in these data is to “twist” the plane, so that there is

curvature in the response function. Factorial experiments are the only way to discover

interactions between variables.

An alternative to the factorial design that is (unfortunately) used in practice is to change

the factors one at a timerather than to vary them simultaneously. To illustrate this one-factor-

at-a-time procedure, suppose that an engineer is interested in finding the values of temperature

and pressure that maximize yield in a chemical process. Suppose that we fix temperature

at 155F (the current operating level) and perform five runs at different levels of time, say,

AB

20  30

2



10  0

2

 20

AB

20  30

2



10  40

2

 0

Alow

0

Ahigh

10

20

30

40

50

Observation

Factor A

Bhigh

Blow

Alow

0

Ahigh

10

20

30

40

50

Observation

Factor A

Bhigh

Blow

Figure 14-1 Factorial experiment, no

interaction.

Figure 14-2 Factorial experiment, with

interaction.

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