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


A foldover design is obtained from a fractional factorial design by reversing the signs of all the columns 
A mirrorimage foldover (or foldover, without the hyphen) design is
used to augment fractional factorial designs to
increase the resolution of
\( 2_{III}^{31} \)
and PlackettBurman designs. It is obtained by reversing the signs of
all the columns of the original design matrix. The original design
runs are combined with the mirrorimage foldover design runs, and this
combination can then be used to estimate all main effects clear of any
twofactor interaction. This is referred to as: breaking the alias
link between main effects and twofactor interactions.
Before we illustrate this concept with an example, we briefly review the basic concepts involved. 

Review of Fractional 2^{kp} Designs  
A resolution III design, combined with its mirrorimage foldover, becomes resolution IV 
In general, a design type that uses a specified fraction of the runs
from a full factorial and is balanced and orthogonal is called a
fractional factorial.
A 2level fractional factorial is constructed as follows: Let the number of runs be 2^{kp}. Start by constructing the full factorial for the kp variables. Next associate the extra factors with higherorder interaction columns. The Table shown previously details how to do this to achieve a minimal amount of confounding. For example, consider the 2^{52} design (a resolution III design). The full factorial for k = 5 requires 2^{5} = 32 runs. The fractional factorial can be achieved in 2^{52} = 8 runs, called a quarter (1/4) fractional design, by setting X_{4} = X_{1}*X_{2} and X_{5} = X_{1}*X_{3}. 

Design matrix for a 2^{52} fractional factorial 
The design matrix for a 2^{52 }fractional factorial looks like:


Design Generators, Defining Relation and the MirrorImage Foldover  
Increase to resolution IV design by augmenting design matrix 
In this design the X_{1}X_{2} column was
used to generate the X4 main effect and the
X_{1}X_{3} column was used to generate the
X_{5} main effect. The design generators are:
4 = 12 and 5 = 13 and the defining relation is I = 124 = 135 = 2345.
Every main effect is confounded (aliased) with at least one firstorder
interaction (see the confounding structure
for this design).
We can increase the resolution of this design to IV if we augment the 8 original runs, adding on the 8 runs from the mirrorimage foldover design. These runs make up another 1/4 fraction design with design generators 4 = 12 and 5 = 13 and defining relation I = 124 = 135 = 2345. The augmented runs are: 

Augmented runs for the design matrix 


Mirrorimage foldover design reverses all signs in original design matrix  A mirrorimage foldover design is the original design with all signs reversed. It breaks the alias chains between every main factor and twofactor interactionof a resolution III design. That is, we can estimate all the main effects clear of any twofactor interaction.  
A 1/16 Design Generator Example  
2^{73} example 
Now we consider a more complex example.
We would like to study the effects of 7 variables. A full 2level factorial, 2^{7}, would require 128 runs. Assume economic reasons restrict us to 8 runs. We will build a 2^{74} = 2^{3} full factorial and assign certain products of columns to the X4, X5, X6 and X7 variables. This will generate a resolution III design in which all of the main effects are aliased with firstorder and higher interaction terms. The design matrix (see the previous Table for a complete description of this fractional factorial design) is: 

Design matrix for 2^{73} fractional factorial 


Design generators and defining relation for this example 
The design generators for this 1/16 fractional factorial design are:
2345 = 1346 = 1256 = 1457 = 2467 = 3567 = 1234567. 

Computing alias structure for complete design 
Using this defining relation, we can easily compute the alias structure
for the complete design, as shown previously in the
link to the fractional design Table given
earlier. For example, to figure out which
effects are aliased (confounded) with factor X_{1} we
multiply the defining relation by 1 to obtain:
The same procedure can be used to obtain all the other aliases for each of the main effects, generating the following list:
2 = 14 = 36 = 57 3 = 15 = 26 = 47 4 = 12 = 37 = 56 5 = 13 = 27 = 46 6 = 17 = 23 = 45 7 = 16 = 25 = 34 

Signs in every column of original design matrix reversed for mirrorimage foldover design 
The chosen design used a set of generators with all positive signs.
The mirrorimage foldover design uses generators with negative signs for
terms with an even number of factors or, 4 = 12, 5 = 13, 6 = 23 and
7 = 123. This generates a design matrix that is equal to the original
design matrix with every sign in every column reversed.
If we augment the initial 8 runs with the 8 mirrorimage foldover design runs (with all column signs reversed), we can dealias all the main effect estimates from the 2way interactions. The additional runs are: 

Design matrix for mirrorimage foldover runs 


Alias structure for augmented runs 
Following the same steps as before and making the same assumptions about
the omission of higherorder interactions in the alias structure, we
arrive at:
1 = 24 = 35 = 67
With both sets of runs, we can now estimate all the main effects free from two factor interactions. 

Build a resolution IV design from a resolution III design  Note: In general, a mirrorimage foldover design is a method to build a resolution IV design from a resolution III design. It is never used to followup a resolution IV design. 