# Statistics for Environmental Engineers

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The design matrix for a 24—1 design is shown in Table 28.3. The signs of the factor 4 column vector of levels are determined by the product of column vectors for the column 1, 2, and 3 factors. (Also, it is the same as the three-factor interaction column in the full 23 design.) For example, the signs for run 4 (row 4) are (+) (+)    (—) (—),    where the last    (—)    comes from the product    (+)    (+)    (—)    =    (—).

The model matrix is given in Table 28.4. The eight experimental runs allow estimation of eight effects, which are computed as the product of a column vector X; and the y vector just as was explained for the full factorial experiment discussed in Chapter 27. The other effects also are computed as for the full factorial experiment but they have a different interpretation, which will be explained now.

To evaluate four factors with only eight runs, we give up the ability to estimate independent main effects. Notice in the design matrix that column vector 1 is identical to the product of column vectors 2, 3, and 4. The effect that is computed as y ■ X1 is not an independent estimate of the main effect of factor 1. It is the main effect of X1plus the three-way interaction of factors 2, 3, and 4. We say that the main effect of X1 is confounded with the three-factor interaction of X 2, X 3, and X 4. Furthermore, each main effect is confounded with a three-factor interaction, as follows:

1 + 234    2 + 134    3 + 124    4 + 123

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