# Statistics for Environmental Engineers

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(a) Main effect X1    (b) Main effect X2    (c) Main effect X3

(d) Interaction X1 & X2

(e) Interaction X1 & X2    (f) Interaction X2 & X3

FIGURE 27.2 Cube plots showing the main effects and two-factor interactions of a 23 factorial experimental design. The main effects and interactions are estimated by subtracting the average of the four values indicated with small dots from the average of the four values indicated by large dots.

The interactions measure the non-additivity of the effects of two or more factors. A significant two-factor interaction indicates antagonism or synergism between two factors; their combined effect is not the sum of their separate contributions. The interaction between factors 1 and 2 (panel d) is the average difference between the effect of factor 1 at the high setting of factor 2 and the effect of factor 1 at the low setting of factor 2. Equivalently, it is the effect of factor 2 at the high setting of factor 1 minus the effect of factor 2 at the low setting of factor 1. This interpretation holds for the two-factor interactions between factors 1 and 3 (panel e) and factors 2 and 3 (panel f). This is equivalent to subtracting the average of the four corners with small dots from the average of the four corners with large dots.

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