Statistics for Environmental Engineers

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column and they are identified by run number in the fractional design. (An equivalent fractional design consists of the 16 runs that are not marked with asterisks.) Notice that in each column of the full design, half the values are +1 and half are -1. This balance must be preserved when we select the half-fraction.

The 16 runs of the fractional design constructed as follows. A full 24 design was written for the four variables 1, 2, 3, and 4. The column of signs for the 1234 interaction was written and these were used to define the levels of variable 5. Thus, we made 5 = 1234.

The consequence of making 5 = 1234 is that the quantity we calculate as the main effect of factor 5 is the main effect plus the four-factor interaction of factors 1, 2, 3, and 4. The main effect of 5 is confounded with the 1234 interaction. Or we can say that 5 and 1234 are aliases of each other. There are other interactions that are confounded and we need to understand what they are.

The confounding pattern between the columns in the model matrix is determined by the defining relation I = 12345. That means that multiplying columns 1234 gives column 5; that is, 5 = 1234. It is also true that 1 = 2345, 2 = 1345, 3 = 1245, and 4 = 1235. Each of the main effects is confounded with a four-factor interaction. These high order effects are expected to be negligible, in which case the design produces independent estimates of the main effects.

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