# Statistics for Environmental Engineers

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The two-factor effects are also confounded in the design. Multiplying columns 123 gives a column identical to 45. As a consequence, the quantity that estimates the 45 interaction also includes the three-factor interaction of 123 (the 123 and 45 interactions are confounded with each other). Also, 12 = 345, 13 = 245, 14 = 235, 15 = 234, 23 = 145, 24 = 135, 25 = 134, 34 = 125, and 35 = 124. Each two-factor interaction is confounded with a three-factor interaction. If a two-factor interaction appeared to be significant, its interpretation would have to take into account the possibility that the effect may be due in part to the third-order interaction. Fortunately, the third-order interactions are usually small and can be neglected.

Using the 25-1 design instead of the full 25 saves us 16 runs, but at the cost of having the main effects confounded with four-factor interactions and having the two-factor interactions confounded with three-factor interactions. If the objective is mainly to learn about the main effects, and if the four-factor interactions are small, the design is highly efficient for identifying the most important factors. Furthermore, because each estimated main effect is the average of eight virtually independent comparisons, the precision of the estimates can be excellent.

### Case Study Solution

The measured permeabilities are plotted in Figure 29.1. Because the permeability vary over several orders of magnitude, the data are best displayed on a logarithmic plot. Fly ash A (solid circles) clearly has higher permeabilities than fly ash B (open circles).

The main effects of each variable were of primary interest. Two-factor interactions were of minor interest. Three-factor and higher-order interactions were expected to be negligible. A half-fraction, or 25-1 fractional factorial design, consisting of 16 runs was used. There are 16 data points, so it is possible to estimate 16 parameters. The “parameters” in this case are the mean, five main effects, and 10 two-factor interactions. Table 29.4 gives the model matrix in terms of the coded variables. The products X1X2, X1X3, etc. indicate two-factor interactions. Also listed are the permeability (y) and ln(y).

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