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

5.3.3.4.2.

Constructing the 23-1 half-fraction design

Construction of a 2^3-1 half fraction design by staring with a 2² full factorial design

First note that, mathematically, 2^3-1 = 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 2^3-1 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 2^3-1 design, we would have obtained:

TABLE 3.15: A 2^3-1 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 2^3-1 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 2^3-1 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 2^3-1 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