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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

5.3.3.8.1.

Mirror-Image foldover designs
A foldover design is obtained from a fractional factorial design by reversing the signs of all the columns A mirror-image fold-over (or foldover, without the hyphen) design is used to augment fractional factorial designs to increase the resolution of \( 2_{III}^{3-1} \) and Plackett-Burman 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 mirror-image fold-over design runs, and this combination can then be used to estimate all main effects clear of any two-factor interaction. This is referred to as: breaking the alias link between main effects and two-factor interactions.

Before we illustrate this concept with an example, we briefly review the basic concepts involved.

Review of Fractional 2k-p Designs
A resolution III design, combined with its mirror-image 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 2-level fractional factorial is constructed as follows: Let the number of runs be 2k-p. Start by constructing the full factorial for the k-p variables. Next associate the extra factors with higher-order interaction columns. The Table shown previously details how to do this to achieve a minimal amount of confounding.

For example, consider the 25-2 design (a resolution III design). The full factorial for k = 5 requires 25 = 32 runs. The fractional factorial can be achieved in 25-2 = 8 runs, called a quarter (1/4) fractional design, by setting X4 = X1*X2 and X5 = X1*X3.

Design matrix for a 25-2 fractional factorial The design matrix for a 25-2 fractional factorial looks like:

TABLE 3.34: Design Matrix for a 25-2 Fractional Factorial
run X1 X2 X3 X4 = X1X2 X5 = X1X3
1 -1 -1 -1 +1 +1
2 +1 -1 -1 -1 -1
3 -1 +1 -1 -1 +1
4 +1 +1 -1 +1 -1
5 -1 -1 +1 +1 -1
6 +1 -1 +1 -1 +1
7 -1 +1 +1 -1 -1
8 +1 +1 +1 +1 +1
Design Generators, Defining Relation and the Mirror-Image Foldover
Increase to resolution IV design by augmenting design matrix In this design the X1X2 column was used to generate the X4 main effect and the X1X3 column was used to generate the X5 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 first-order 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 mirror-image fold-over 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
run X1 X2 X3 X4 = -X1X2 X5 = -X1X3
9 +1 +1 +1 -1 -1
10 -1 +1 +1 +1 +1
11 +1 -1 +1 +1 -1
12 -1 -1 +1 -1 +1
13 +1 +1 -1 -1 +1
14 -1 +1 -1 +1 -1
15 +1 -1 -1 +1 +1
16 -1 -1 -1 -1 -1
Mirror-image foldover design reverses all signs in original design matrix A mirror-image foldover design is the original design with all signs reversed. It breaks the alias chains between every main factor and two-factor interactionof a resolution III design. That is, we can estimate all the main effects clear of any two-factor interaction.
A 1/16 Design Generator Example
27-3 example Now we consider a more complex example.

We would like to study the effects of 7 variables. A full 2-level factorial, 27, would require 128 runs.

Assume economic reasons restrict us to 8 runs. We will build a 27-4 = 23 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 first-order and higher interaction terms. The design matrix (see the previous Table for a complete description of this fractional factorial design) is:

Design matrix for 27-3 fractional factorial
Design Matrix for a 27-3 Fractional Factorial
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
Design generators and defining relation for this example The design generators for this 1/16 fractional factorial design are:
    4 = 12, 5 = 13, 6 = 23 and 7 = 123
From these we obtain, by multiplication, the defining relation:
    I = 124 = 135 = 236 = 347 = 257 = 167 = 456 = 1237 =
    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 X1 we multiply the defining relation by 1 to obtain:
    1 = 24 = 35 = 1236 = 1347 = 1257 = 67 = 1456 = 237 = 12345 = 346 = 256 = 457 = 12467 = 13567 = 234567
In order to simplify matters, let us ignore all interactions with 3 or more factors; we then have the following 2-factor alias pattern for X1: 1 = 24 = 35 = 67 or, using the full notation, X1 = X2*X4 = X3*X5 = X6*X7.

The same procedure can be used to obtain all the other aliases for each of the main effects, generating the following list:

1 = 24 = 35 = 67
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 mirror-image foldover design The chosen design used a set of generators with all positive signs. The mirror-image 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 mirror-image foldover design runs (with all column signs reversed), we can de-alias all the main effect estimates from the 2-way interactions. The additional runs are:

Design matrix for mirror-image foldover runs
Design Matrix for the Mirror-Image Foldover Runs of the 27-3 Fractional Factorial
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 structure for augmented runs Following the same steps as before and making the same assumptions about the omission of higher-order interactions in the alias structure, we arrive at:

1 = -24 = -35 = -67
2 = -14 = -36 =- 57
3 = -15 = -26 = -47
4 = -12 = -37 = -56
5 = -13 = -27 = -46
6 = -17 = -23 = -45
7 = -16 = -25 = -34

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 mirror-image foldover design is a method to build a resolution IV design from a resolution III design. It is never used to follow-up a resolution IV design.
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