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When several factors are of interest in an experiment, a factorial experimental design should be used. As noted previously, in these experiments factors are varied together. De nition By a factorial experiment we mean that in each complete trial or replicate of the experiment all possible combinations of the levels of the factors are investigated.
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Thus, if there are two factors A and B with a levels of factor A and b levels of factor B, each replicate contains all ab treatment combinations. The effect of a factor is de ned 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 example, consider the data in Table 14-1. This is a factorial experiment with two factors, A and B, each at two levels (Alow, Ahigh, and Blow, Bhigh). The main effect of factor A is the difference between the average response at the high level of A and the average response at the low level of A, or A 30 2 40 10 2 20 20
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That is, changing factor A from the low level to the high level causes an average response increase of 20 units. Similarly, the main effect of B is B 20 2 40 10 2 30 10
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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 interaction between the factors. For example, consider the data in Table 14-2. At the low level of factor B, the A effect is A 30 10 20
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and at the high level of factor B, the A effect is A 0 20 20
Since the effect of A depends on the level chosen for factor B, there is interaction between A and 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 nd the main effect of A as A 30 2 0 10 2 20 0
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and we would be tempted to conclude that there is no factor A effect. However, when we examined the effects of A at different levels of factor B, we saw that this was not the case. The effect of factor A depends on the levels of factor B. Thus, knowledge of the AB interaction is more useful than knowledge of the main effect. A signi cant interaction can mask the significance of main effects. Consequently, when interaction is present, the main effects of the factors involved in the interaction may not have much meaning.
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Table 14-1 A Factorial Experiment with Two Factors Factor B Factor A Alow Ahigh B low 10 30 B high 20 40
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Table 14-2 A Factorial Experiment with Interaction Factor B Factor A Alow Ahigh B low 10 30 B high 20 40
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It is easy to estimate the interaction effect in factorial experiments such as those illustrated in Tables 14-1 and 14-2. In this type of experiment, when both factors have two levels, the AB interaction effect is the difference in the diagonal averages. This represents one-half the difference between the A effects at the two levels of B. For example, in Table 14-1, we nd the AB interaction effect to be AB 20 2 30 10 2 40 0
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Thus, there is no interaction between A and B. In Table 14-2, the AB interaction effect is AB 20 2 30 10 2 0 20
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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 A for both levels of B. Note that the Blow and Bhigh lines are approximately parallel, indicating that factors A and B do not interact signi cantly. Figure 14-2 presents a similar plot for the data in Table 14-2. In this graph, the Blow and Bhigh lines are not parallel, indicating the interaction between factors A and B. Such graphical displays are called two-factor interaction plots. They are often useful in presenting the results of experiments, and many computer software programs used for analyzing data from designed 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 plot of the data from Table 14-1. These data contain no interaction, and the surface plot is a plane lying above the A-B space. The slope of the plane in the A and B directions is proportional to the main effects of factors A and 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 time rather than to vary them simultaneously. To illustrate this one-factorat-a-time procedure, suppose that an engineer is interested in nding the values of temperature and pressure that maximize yield in a chemical process. Suppose that we x temperature at 155 F (the current operating level) and perform ve runs at different levels of time, say,
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