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5. Process Improvement
5.3. Choosing an experimental design
5.3.3. How do you select an experimental design?

5.3.3.10.

Three-level, mixed-level and fractional factorial designs

Mixed level designs have some factors with, say, 2 levels, and some with 3 levels or 4 levels The 2k and 3k experiments are special cases of factorial designs. In a factorial design, one obtains data at every combination of the levels. The importance of factorial designs, especially 2-level factorial designs, was stated by Montgomery (1991): It is our belief that the two-level factorial and fractional factorial designs should be the cornerstone of industrial experimentation for product and process development and improvement. He went on to say: There are, however, some situations in which it is necessary to include a factor (or a few factors) that have more than two levels.

This section will look at how to add three-level factors starting with two-level designs, obtaining what is called a mixed-level design. We will also look at how to add a four-level factor to a two-level design. The section will conclude with a listing of some useful orthogonal three-level and mixed-level designs (a few of the so-called Taguchi "L" orthogonal array designs), and a brief discussion of their benefits and disadvantages.

Generating a Mixed Three-Level and Two-Level Design
Montgomery scheme for generating a mixed design Montgomery (1991) suggests how to derive a variable at three levels from a 23 design, using a rather ingenious scheme. The objective is to generate a design for one variable, A, at 2 levels and another, X, at three levels. This will be formed by combining the -1 and 1 patterns for the B and C factors to form the levels of the three-level factor X:

TABLE 3.38  Generating a Mixed Design

Two-Level Three-Level

B C X

-1 -1 x1
+1 -1 x2
-1 +1 x2
+1 +1 x3

Similar to the 3k case, we observe that X has 2 degrees of freedom, which can be broken out into a linear and a quadratic component. To illustrate how the 23 design leads to the design with one factor at two levels and one factor at three levels, consider the following table, with particular attention focused on the column labels.
Table illustrating the generation of a design with one factor at 2 levels and another at 3 levels from a 23 design
  A XL XL AXL AXL XQ AXQ TRT MNT

Run A B C AB AC BC ABC A X

1 -1 -1 -1 +1 +1 +1 -1 Low Low
2 +1 -1 -1 -1 -1 +1 +1 High Low
3 -1 +1 -1 -1 +1 -1 +1 Low Medium
4 +1 +1 -1 +1 -1 -1 -1 High Medium
5 -1 -1 +1 +1 -1 -1 +1 Low Medium
6 +1 -1 +1 -1 +1 -1 -1 High Medium
7 -1 +1 +1 -1 -1 +1 -1 Low High
8 +1 +1 +1 +1 +1 +1 +1 High High

If quadratic effect negligble, we may include a second two-level factor If we believe that the quadratic effect is negligible, we may include a second two-level factor, D, with D = ABC. In fact, we can convert the design to exclusively a main effect (resolution III) situation consisting of four two-level factors and one three-level factor. This is accomplished by equating the second two-level factor to AB, the third to AC and the fourth to ABC. Column BC cannot be used in this manner because it contains the quadratic effect of the three-level factor X.
More than one three-level factor
3-Level factors from 24 and 25 designs We have seen that in order to create one three-level factor, the starting design can be a 23 factorial. Without proof we state that a 24 can split off 1, 2 or 3 three-level factors; a 25 is able to generate 3 three-level factors and still maintain a full factorial structure. For more on this, see Montgomery (1991).
Generating a Two- and Four-Level Mixed Design
Constructing a design with one 4-level factor and two 2-level factors We may use the same principles as for the three-level factor example in creating a four-level factor. We will assume that the goal is to construct a design with one four-level and two two-level factors.

Initially we wish to estimate all main effects and interactions. It has been shown (see Montgomery, 1991) that this can be accomplished via a 24 (16 runs) design, with columns A and B used to create the four level factor X.

Table showing design with 4-level, two 2-level factors in 16 runs
TABLE 3.39  A Single Four-level Factor and Two Two-level Factors in 16 runs
Run (A B) = X C D

1 -1 -1 x1 -1 -1
2 +1 -1 x2 -1 -1
3 -1 +1 x3 -1 -1
4 +1 +1 x4 -1 -1
5 -1 -1 x1 +1 -1
6 +1 -1 x2 +1 -1
7 -1 +1 x3 +1 -1
8 +1 +1 x4 +1 -1
9 -1 -1 x1 -1 +1
10 +1 -1 x2 -1 +1
11 -1 +1 x3 -1 +1
12 +1 +1 x4 -1 +1
13 -1 -1 x1 +1 +1
14 +1 -1 x2 +1 +1
15 -1 +1 x3 +1 +1
16 +1 +1 x4 +1 +1
Some Useful (Taguchi) Orthogonal "L" Array Designs
L9
design
L9 - A 34-2 Fractional Factorial Design 4 Factors at Three Levels (9 runs)
Run X1 X2 X3 X4
1 1 1 1 1
2 1 2 2 2
3 1 3 3 3
4 2 1
2
3
5 2 2 3 1
6 2 3 1 2
7 3 1 3 2
8 3 2 1 3
9 3 3 2 1
L18
design
L18 - A 2 x 37-5 Fractional Factorial (Mixed-Level) Design
1 Factor at Two Levels and Seven Factors at 3 Levels (18 Runs)
Run
X1
X2
X3
X4
X5
X6
X7
X8
1
1
1
1
1
1
1
1
1
2
1
1
2
2
2
2
2
2
3
1
1
3
3
3
3
3
3
4
1
2
1
1
2
2
3
3
5
1
2
2
2
3
3
1
1
6
1
2
3
3
1
1
2
2
7
1
3
1
2
1
3
2
3
8
1
3
2
3
2
1
3
1
9
1
3
3
1
3
2
1
2
10
2
1
1
3
3
2
2
1
11
2
1
2
1
1
3
3
2
12
2
1
3
2
2
1
1
3
13
2
2
1
2
3
1
3
2
14
2
2
2
3
1
2
1
3
15
2
2
3
1
2
3
2
1
16
2
3
1
3
2
3
1
2
17
2
3
2
1
3
1
2
3
18
2
3
3
2
1
2
3
1
L27
design
L27 - A 313-10 Fractional Factorial Design
Thirteen Factors at Three Levels (27 Runs)
Run
X1
X2
X3
X4
X5
X6
X7
X8
X9
X10
X11
X12
X13
1
1
2
1
3
 1 
4
2
2
2
1
1
5
6
7
8
9
1
10
11
12
13
14
15
16
2
17
18
1
19
20
21
 2
 1
 3
 2
 1
 3
 2
 1
22
23
24
3
25
26
27
L36
design
L36 - A Fractional Factorial (Mixed-Level) Design Eleven Factors at Two Levels and Twelve Factors at 3 Levels (36 Runs)
Run X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11 X12 X13 X14 X15 X16 X17 X18 X19 X20 X21 X22 X23
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
2 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2
3 1 1 1 1 1 1 1 1 1 1 1 3 3 3 3 3 3 3 3 3 3 3 3
4 1 1 1 1 1 2 2 2 2 2 2 1 1 1 1 2 2 2 2 3 3 3 3
5 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 3 3 3 3 1 1 1 1
6 1 1 1 1 1 2 2 2 2 2 2 3 3 3 3 1 1 1 1 2 2 2 2
7 1 1 2 2 2 1 1 1 2 2 2 1 1 2 3 1 2 3 3 1 2 2 3
8 1 1 2 2 2 1 1 1 2 2 2 2 2 3 1 2 3 1 1 2 3 3 1
9 1 1 2 2 2 1 1 1 2 2 2 3 3 1 2 3 1 2 2 3 1 1 2
10 1 2 1 2 2 1 2 2 1 1 2 1 1 3 2 1 3 2 3 2 1 3 2
11 1 2 1 2 2 1 2 2 1 1 2 2 2 1 3 2 1 3 1 3 2 1 3
12 1 2 1 2 2 1 2 2 1 1 2 3 3 2 1 3 2 1 2 1 3 2 1
13 1 2 2 1 2 2 1 2 1 2 1 1 2 3 1 3 2 1 3 3 2 1 2
14 1 2 2 1 2 2 1 2 1 2 1 2 3 1 2 1 3 2 1 1 3 2 3
15 1 2 2 1 2 2 1 2 1 2 1 3 1 2 3 2 1 3 2 2 1 3 1
16 1 2 2 2 1 2 2 1 2 1 1 1 2 3 2 1 1 3 2 3 3 2 1
17 1 2 2 2 1 2 2 1 2 1 1 2 3 1 3 2 2 1 3 1 1 3 2
18 1 2 2 2 1 2 2 1 2 1 1 3 1 2 1 3 3 2 1 2 2 1 3
19 2 1 2 2 1 1 2 2 1 2 1 1 2 1 3 3 3 1 2 2 1 2 3
20 2 1 2 2 1 1 2 2 1 2 1 2 3 2 1 1 1 2 3 3 2 3 1
21 2 1 2 2 1 1 2 2 1 2 1 3 1 3 2 2 2 3 1 1 3 1 2
22 2 1 2 1 2 2 2 1 1 1 2 1 2 2 3 3 1 2 1 1 3 3 2
23 2 1 2 1 2 2 2 1 1 1 2 2 3 3 1 1 2 3 2 2 1 1 3
24 2 1 2 1 2 2 2 1 1 1 2 3 1 1 2 2 3 1 3 3 2 2 1
25 2 1 1 2 2 2 1 2 2 1 1 1 3 2 1 2 3 3 1 3 1 2 2
26 2 1 1 2 2 2 1 2 2 1 1 2 1 3 2 3 1 1 2 1 2 3 3
27 2 1 1 2 2 2 1 2 2 1 1 3 2 1 3 1 2 2 3 2 3 1 1
28 2 2 2 1 1 1 1 2 2 1 2 1 3 2 2 2 1 1 3 2 3 1 3
29 2 2 2 1 1 1 1 2 2 1 2 2 1 3 3 3 2 2 1 3 1 2 1
30 2 2 2 1 1 1 1 2 2 1 2 3 2 1 1 1 3 3 2 1 2 3 2
31 2 2 1 2 1 2 1 1 1 2 2 1 3 3 3 2 3 2 2 1 2 1 1
32 2 2 1 2 1 2 1 1 1 2 2 2 1 1 1 3 1 3 3 2 3 2 2
33 2 2 1 2 1 2 1 1 1 2 2 3 2 2 2 1 2 1 1 3 1 3 3
34 2 2 1 1 2 1 2 1 2 2 1 1 3 1 2 3 2 3 1 2 2 3 1
35 2 2 1 1 2 1 2 1 2 2 1 2 1 2 3 1 3 1 2 3 3 1 2
36 2 2 1 1 2 1 2 1 2 2 1 3 2 3 1 2 1 2 3 1 1 2 3
Advantages and Disadvantages of Three-Level and Mixed-Level "L" Designs
Advantages and disadvantages of three-level mixed-level designs The good features of these designs are:
  • They are orthogonal arrays. Some analysts believe this simplifies the analysis and interpretation of results while other analysts believe it does not.
  • They obtain a lot of information about the main effects in a relatively few number of runs.
  • You can test whether non-linear terms are needed in the model, at least as far as the three-level factors are concerned.
On the other hand, there are several undesirable features of these designs to consider:
  • They provide limited information about interactions.
  • They require more runs than a comparable 2k-pdesign, and a two-level design will often suffice when the factors are continuous and monotonic (many three-level designs are used when two-level designs would have been adequate).
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