Fractional Factorial Design
(Confounding Pattern or Alias Structure)
Example 1:
2^(3

1) design with 3 = 12 (X3 = X1 * X2) or C = AB (textbook notation)
Design layout (or design matrix) is given in page 335 of the DOE handouts.
See page 334 for a graphic presentation of the fractional factorial design (also given in lecture
notes on 08/28/02).
1. Multiply X3 to both sides of X3 = X1 * X2.
2.
Left hand side (LHS) becomes X3 * X3, which is the same column of +1 and

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multiplied by itself.
Thus, it becomes a column of all +1’s, i.e., X3 * X3 = I = (+1,
+1, …, +1).
3. Right hand side (RHS) becomes X1 * X2 * X3.
4.
We thus obtain a “
generating relationship
”:
I = X1 * X2 * X3 (or I = 123 in short).
5.
X3 = X1 * X2 (or 3 = 12) is called the “
generator
” of the design.
6.
Because there are three letters in the RHS of the generating relationship, the
“
resolution
” of the design is 3.
7. From statement #4, if one multiplies X1 to both sides of the generating relationship,
the resulted relationship becomes
X1 = (X1 * X1) * X2 * X3 = X2 * X3;
that is
1 = 23
in short.
8. Similarly, one can obtain 2 = 13 and 3 = 12 relationships.
9. You will find that the column of X2*X3 (i.e., 23) of the design layout will be the same
as the column X1 (verify it yourself).
Similarly, X2 column is the same as X1*X3,
and X3 column and the same as X1*X2.
10.
In summary, the “
confounding pattern
” of the design in this example is
I = 123, 1 = 23, 2 = 13, and 3 = 12.
11.
Then, the “
Estimable Effects
” are
Average (for the column of one’s), X1 (which is equal to X2*X3), X2 (= X1*X3)
and X3 (= X1*X2), i.e., only 4 estimable effects from the experiment of 4 runs.
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