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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 2 k 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 is it 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 fractional 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 (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

Similarly to the 3k case, we observe that X has 2 degrees of freedom, which can be broken out in a linear and quadratic component. To illustrate how the 23 design gives birth 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. 
If we think that the quadratic effect is negligible, we may include a second two level factor, D, where D = ABC. In fact, we can convert the design to exclusively a main factor (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

We have seen that in order to create one three level factor, the starting design is a 23  factorial. Without proof we state that a 24 can split off 1, 2  or 3 three-levels 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

We may use the same principles as for the three-level factor example in creating a four-level factor. It is desired 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, where columns A and B are used to create the four level factor X. 
 
 

TABLE 3.39  A Single Four-level Factor and Two Two-level Factors in 16 runs

The "+" and "-" are substituted for +1 and -1, to conserve space on the page. 
 

Some Useful (Taguchi) Orthogonal "L" Array Designs
 

L9 - A 34-2 Fractional Factorial (Three Level ) Design
4 Factors at Three Level (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 - A 21 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 - A 313-10 Fractional Factorial (Three Level ) Design
Thirteen Factors at Three Levels (27 Runs) 

 
Run 
#
X1
X2
X3
X4
X5
X6
X7
X8
X9
X10
X11
X12
X13
1
1
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
2
1
1 
1 
1 
2 
2 
2 
2 
2 
2 
2 
2 
2 
3
 1 
1 
1 
1 
3 
3 
3 
3 
3 
3 
3 
3 
3 
4
1 
2
2
 2
1
1
1 
2 
2 
2 
3 
3 
3 
5
1 
2 
2 
2 
2 
2 
2 
3 
3 
3 
1 
1 
1 
6
1 
2 
2 
2 
3 
3 
3 
1 
1 
1 
2 
2 
2 
7
1 
3 
3 
3 
1 
1 
1 
3 
3 
3 
2 
2 
2 
8
1 
3 
3 
3 
2 
2 
2 
1 
1 
1 
3 
3 
3 
9
1 
3 
3 
3 
3 
3 
3 
2 
2 
2 
1
1 
1 
10
2 
1 
2 
3 
1 
2 
3 
1 
2 
3 
1 
2 
3 
11
2 
1 
2 
3 
2 
3 
1 
2 
3 
1 
2 
3 
1 
12
2 
1 
2 
3 
3 
1 
2 
3 
1 
2 
3 
1 
2 
13
2 
2 
3 
1 
1 
2 
3 
2 
3 
1 
3 
1 
2 
14
2 
2 
3 
1 
2 
3 
1 
3 
1 
2 
1 
2 
3 
15
2 
2 
3 
1 
3 
1 
2 
1 
2 
3 
2 
3 
1 
16
2 
3 
1 
2 
1 
2 
3 
3 
1 
2 
2
3 
1 
17
2 
3 
1 
2 
2 
3 
1 
1 
2 
3 
3 
1 
2 
18
2 
3 
1 
2 
3 
1 
2 
2 
3 
1 
1
2 
3 
19
3 
1 
3 
2 
1 
3 
2 
1 
3 
2 
1 
3 
2 
20
3 
1 
3 
2 
2 
1 
3 
2 
1 
3 
2 
1 
3 
21
3 
1 
3 
2 
3 
 2
 1
 3
 2
 1
 3
 2
 1
22
3 
2 
1 
3 
1 
3 
2 
2 
1 
3 
3 
2 
1 
23
3 
2 
1 
3 
2 
1 
3 
3 
2 
1 
1 
3 
2 
24
3
2 
1 
3 
3 
2 
1 
1 
3 
2 
2 
1 
3 
25
3 
3 
2 
1 
1 
3 
2 
3 
2 
1 
2 
1 
3 
26
3 
3 
2 
1 
2 
1 
3 
1 
3 
2 
3 
2 
1 
27
3 
3 
2 
1 
3 
2 
1 
2 
1 
3 
1 
3 
2 

 

L36 - A Fractional Factorial (Mixed Level ) Design  Eleven Factors at Two Levels and Twelve Factors at 3 Levels (36 Runs)


Advantages and Disadvantages of Three Level and Mixed Level  "L" Designs

The good features of these "L" desgns are: 

  • They are orthogonal arrays, which simplifies the analysis and interpretation of  results 
  • 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-p design, and a two level design will often suffice when the factors are continuous and monotonic (many three level experiments are run where two level designs would have been adequate) 
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