# Statistics for Environmental Engineers

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12 + 34    13 + 24    23 + 14

The two-way interaction of factors 1 and 2 is confounded with the two-way interaction of factors 3 and 4, etc.

The consequence of this intentional confounding is that the estimated main effects are biased unless the three-factor interactions are negligible. Fortunately, three-way interactions are often small and can be ignored. There is no safe basis for ignoring any of the two-factor interactions, so the effects calculated as two-factor interactions must be interpreted with caution.

Understanding how confounding is identified by the defining relation reveals how the fractional design was created. Any fractional design will involve some confounding. The experimental designer wants to make this as painless as possible. The best we can do is to hope that the three-factor interactions are unimportant and arrange for the main effects to be confounded with three-factor interactions. Intentionally confounding factor 4 with the three-factor interaction of factors 1, 2, and 3 accomplishes that. By convention, we write the design matrix in the usual form for the first three factors. The fourth column becomes the product of the first three columns. Then we multiply pairs of columns to get the columns for the two-factor interactions, as shown in Table 28.4.

### Case Study Solution

The average response at each experimental setting is shown in Figure 28.1. The small boxes identify the four tests that were conducted at the high flow rate (X 4); the low flow rate tests are the four unboxed values. Calculation of the effects was explained in Chapter 27 and are not repeated here. The estimated effects are given in Table 28.5.

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