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
|
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
|
|
