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

5.3.3.8.2.

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 2^7-4 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).