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 can be an economical way to break up a selected alias pattern  The mirrorimage foldover (in which signs in all columns are reversed) is only one of the possible followup 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 dealiasing a single factor  Previously, we described how we dealias all the factors of a 2^{74} experiment. Suppose that we only want to dealias the X_{4} factor. This can be accomplished by only changing the sign of X_{4} = X_{1}X_{2} to X_{4} = X_{1}X_{2}. The resulting design is:  
Table showing design matrix of a reverse X4 foldover design 


Alias patterns and effects that can be estimated in the example design 
The twofactor alias patterns for X_{4} are:
The following effects can be estimated by combining the original \( 2_{III}^{74} \) with the "Reverse X_{4}" foldover fraction: Note: The 16 runs allow estimating the above 14 effects, with one degree of freedom left over for a possible block effect.X_{1} + X_{3}X_{5} + X_{6}X_{7} 

Advantage and disadvantage of this example design 
The advantage of this followup design is that it permits estimation of
the X_{4} effect and each of the six twofactor interaction
terms involving X_{4}.
The disadvantage is that the combined fractions still yield a resolution III design, with all main effects other than X_{4} aliased with twofactor interactions. 

Case when purpose is simply to estimate all twofactor interactions of a single factor  Reversing a single factor column to obtain dealiased twofactor interactions for that one factor works for any resolution III or IV design. When used to followup a resolution IV design, there are relatively few new effects to be estimated (as compared to \( 2_{III}^{kp} \) designs). When the original resolution IV fraction provides sufficient precision, and the purpose of the followup runs is simply to estimate all twofactor 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 twofactor 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). 