5. Process Improvement
5.3. Choosing an experimental design
5.3.3. How do you select an experimental design?
5.3.3.8. Improving fractional factorial design resolution

Alternative foldover designs

Alternative foldover designs can be an economical way to break up a selected alias pattern The mirror-image foldover (in which signs in all columns are reversed) is only one of the possible follow-up fractions that can be run to augment a fractional factorial design. It is the most common choice when the original fraction is resolution III. However, alternative foldover designs with fewer runs can often be utilized to break up selected alias patterns. We illustrate this by looking at what happens when the signs of a single factor column are reversed.
Example of de-aliasing a single factor Previously, we described how we de-alias all the factors of a 27-4 experiment. Suppose that we only want to de-alias the X4 factor. This can be accomplished by only changing the sign of X4 = X1X2 to X4 = -X1X2. The resulting design is:
Table showing design matrix of a reverse X4 foldover design
TABLE 3.36: A "Reverse X4" Foldover Design
run X1 X2 X3 X4 = -X1X2 X5 = -X1X3 X6 = X2X3 X7 = X1X2X3

1 -1 -1 -1 -1 +1 +1 -1
2 +1 -1 -1 +1 -1 +1 +1
3 -1 +1 -1 +1 +1 -1 +1
4 +1 +1 -1 -1 -1 -1 -1
5 -1 -1 +1 -1 -1 -1 +1
6 +1 -1 +1 +1 +1 -1 -1
7 -1 +1 +1 +1 -1 +1 -1
8 +1 +1 +1 -1 +1 +1 +1
Alias patterns and effects that can be estimated in the example design The two-factor alias patterns for X4 are:
• Original experiment: X4 = X1X2 = X3X7 = X5X6;
• "Reverse X4" foldover experiment: X4 = -X1X2 = -X3X7 = -X5X6.

The following effects can be estimated by combining the original $$2_{III}^{7-4}$$ with the "Reverse X4" foldover fraction:

X1 + X3X5 + X6X7
X2 + X3X6 + X5X7
X3 + X1X5 + X2X6
X4
X5 + X1X3 + X2X7
X6 + X2X3 + X1X7
X7 + X2X5 + X1X6
X1X4
X2X4
X3X4
X4X5
X4X6
X4X7
X1X2 + X3X7 + X5X6
Note: The 16 runs allow estimating the above 14 effects, with one degree of freedom left over for a possible block effect.
Advantage and disadvantage of this example design The advantage of this follow-up design is that it permits estimation of the X4 effect and each of the six two-factor interaction terms involving X4.

The disadvantage is that the combined fractions still yield a resolution III design, with all main effects other than X4 aliased with two-factor interactions.

Case when purpose is simply to estimate all two-factor interactions of a single factor Reversing a single factor column to obtain de-aliased two-factor interactions for that one factor works for any resolution III or IV design. When used to follow-up a resolution IV design, there are relatively few new effects to be estimated (as compared to $$2_{III}^{k-p}$$ designs). When the original resolution IV fraction provides sufficient precision, and the purpose of the follow-up runs is simply to estimate all two-factor interactions for one factor, the semifolding option should be considered.
Semifolding
Number of runs can be reduced for resolution IV designs For resolution IV fractions, it is possible to economize on the number of runs that are needed to break the alias chains for all two-factor interactions of a single factor. In the above case we needed 8 additional runs, which is the same number of runs that were used in the original experiment. This can be improved upon.
Additional information on John's 3/4 designs We can repeat only the points that were set at the high levels of the factor of choice and then run them at their low settings in the next experiment. For the given example, this means an additional 4 runs instead 8. We mention this technique only in passing, more details may be found in the references (or see John's 3/4 designs).
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