5. Process Improvement
5.3. Choosing an experimental design
5.3.3. How do you select an experimental design?
5.3.3.4. Fractional factorial designs

## Constructing the 23-1 half-fraction design

Construction of a 23-1 half fraction design by staring with a 22 full factorial design First note that, mathematically, 23-1 = 22. This gives us the first step, which is to start with a regular 22 full factorial design. That is, we start with the following design table.

TABLE 3.12: A Standard Order 22 Full Factorial Design Table
X1 X2
1 -1 -1
2 +1 -1
3 -1 +1
4 +1 +1
Assign the third factor to the interaction column of a 22 design This design has four runs, the right number for a half-fraction of a 23, but there is no column for factor X3. We need to add a third column to take care of this, and we do it by adding the X1*X2 interaction column. This column is, as you will recall from full factorial designs, constructed by multiplying the row entry for X1 with that of X2 to obtain the row entry for X1*X2.

TABLE 3.13: A 22 Design Table Augmented with the X1*X2 Interaction Column 'X1*X2'
X1 X2 X1*X2
1 -1 -1 +1
2 +1 -1 -1
3 -1 +1 -1
4 +1 +1 +1
Design table with X3 set to X1*X2 We may now substitute 'X3' in place of 'X1*X2' in this table.

TABLE 3.14: A 23-1 Design Table with Column X3 set to X1*X2
X1 X2 X3
1 -1 -1 +1
2 +1 -1 -1
3 -1 +1 -1
4 +1 +1 +1
Design table with X3 set to -X1*X2 Note that the rows of Table 3.14 give the dark-shaded corners of the design in Figure 3.4. If we had set X3 = -X1*X2 as the rule for generating the third column of our 23-1 design, we would have obtained:

TABLE 3.15: A 23-1 Design Table with Column X3 set to -X1*X2
X1 X2 X3
1 -1 -1 -1
2 +1 -1 +1
3 -1 +1 +1
4 +1 +1 -1
Main effect estimates from fractional factorial not as good as full factorial This design gives the light-shaded corners of the box of Figure 3.4. Both 23-1 designs that we have generated are equally good, and both save half the number of runs over the original 23 full factorial design. If c1, c2, and c3 are our estimates of the main effects for the factors X1, X2, X3 (i.e., the difference in the response due to going from "low" to "high" for an effect), then the precision of the estimates c1, c2, and c3 are not quite as good as for the full 8-run factorial because we only have four observations to construct the averages instead of eight; this is one price we have to pay for using fewer runs.
Example Example: For the 'Pressure (P), Table speed (T), and Down force (D)' design situation of the previous example, here's a replicated 23-1 in randomized run order, with five centerpoint runs ('000') interspersed among the runs. This design table was constructed using the technique discussed above, with D = P*T.
Design table for the example
TABLE 3.16: A 23-1 Design Replicated Twice, with Five Centerpoint Runs Added
Pattern P T D Center
Point
1 000 0 0 0 1
2 +-- +1 -1 -1 0
3 -+- -1 +1 -1 0
4 000 0 0 0 1
5 +++ +1 +1 +1 0
6 --+ -1 -1 +1 0
7 000 0 0 0 1
8 +-- +1 -1 -1 0
9 --+ -1 -1 +1 0
10 000 0 0 0 1
11 +++ +1 +1 +1 0
12 -+- -1 +1 -1 0
13 000 0 0 0 1