# Statistics for Environmental Engineers

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Note: Defining relation: I = 1234 (or I = SDLF).

Therefore, I = 1234 means that multiplying the column vectors for factors 1, 2, 3, and 4, which consists of +1’s and —1’s, gives a vector that consists of +1 values. It also means that multiplying the column vectors of factors 2, 3, and 4 gives the column vector for factor 1. This means that the effect calculated using the column of +1 and —1 values for factor 1 is the same as the value that is calculated using the column vector of the X2X3X4 interaction. Thus, the main effect of factor 1 is confounded with the three-factor interaction of factors 2, 3, and 4. Also, multiplying the column vectors of factors 1, 3, and 4 gives the column vector for factor 2, etc.

Having the main effects confounded with three-factor interactions is part of the price we pay we to investigate four factors in eight runs. Another price, which can be seen in the defining relation I = 1234, is that the two-factor interactions are confounded with each other:

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