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Method: Fractional Factorial Designs
A fractional factorial design is an experimental layout where a full factorial design is augmented with
one or more factors (independent variables) to be analyzed without increasing the number of experimental
k−p
runs. These designs are labeled 2 , where k is the number of factors that could be evaluated in a full
k
factorial design of size 2 and p is the number of additional factors to be included. When a fourth factor
3 4−1
is to be incorporated in a 2 design of eight runs, the resulting design is a 2 fractional factorial, which
4
4−1
3
also has 2 = 8 runs. The full 2 factorial would have 16 runs. The 2 has only eight runs. It is a half-
5−2
fraction of the full four-factor design. Likewise, a 2 experimental design has eight runs; it is a quarter-
fraction of the full five-factor design.
4
To design a half-fraction of the full four-factor design, we must determine which half of the 2 = 16
experiments is to be done. To preserve the balance of the design, there must be four experiments at the
high setting of X 4 and four experiments at the low setting. Note that any combination of four high and four
low that we choose for factor 4 will correspond exactly to one of the column combinations for interactions
3
among factors X 1 , X 2 , and X 3 already used in the matrix of the 2 factorial design (Table 28.2). Which
combination should we select? Standard procedure is to choose the three-factor interaction X 1 X 2 X 3 for
setting the levels of X 4 . Having the levels of X 4 the same as the levels of X 1 X 2 X 3 means that the separate
effects of X 4 and X 1 X 2 X 3 cannot be estimated. We can only estimate their combined effect. Their individual
effects are confounded. Confounded means confused with, or tangled up with, in a way that we cannot
separate without doing more experiments.
4−1
The design matrix for a 2 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
3
is the same as the three-factor interaction column in the full 2 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 i 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 · X 1 is not an independent estimate of the main effect of
factor 1. It is the main effect of X 1 plus the three-way interaction of factors 2, 3, and 4. We say that the
main effect of X 1 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
The defining relation of the design allows us to determine all the confounding relationships in the
4−1
fractional design. In this 2 design, the defining relation is I = 1234. I indicates a vector of +1’s.
TABLE 28.3
4−1
Design Matrix for a 2 Fractional Factorial Design
Factor (Independent Variable)
Run 1 2 3 4
1 − − − −
2 + − − +
3 − + − +
4 + + − −
5 − − + +
6 + − + −
7 − + + −
8 + + + +

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