Third factor
The column of ±1 values for the interaction on the previous page can be used to define the levels for a third factor, C
Factor | ||||
---|---|---|---|---|
Run | A | B | C = AB | Response |
1 | -1 | -1 | +1 | y—+ |
2 | -1 | +1 | -1 | y-+- |
3 | +1 | -1 | -1 | y+— |
4 | +1 | +1 | +1 | y+++ |
When the levels for C are defined in this way, all three factors are orthogonal — the columns of ±1 are uncorrelated — so the main effects for the factors can be independently estimated
main effect for A = (y+++ + y+— - y-+- - y—+) / 2
main effect for B = (y+++ + y-+- - y+— - y—+) / 2
main effect for C = (y+++ + y—+ - y+— - y-+-) / 2
Since this design uses half the number of runs that would be required for a full factorial design for 3 factors, it is called a fractional factorial design — a 23-1 design.
Interactions
The problem with this design is that the estimates of the main effect for each factor is intertwined with the interaction between the other two factors. Because the levels for C were defined from the AB interaction,
C = AB
the estimated main effect for C is identical to the estimated AB interaction effect that was shown on the previous page. The main effect for C is said to be confounded with the AB interaction.
estimate of main effect of C = (true main effect of C) + (AB interaction effect)
This has the potential to mislead if there are interactions.
If A and B interact in their effect on the response, their interaction could incorrectly make C appear to have a main effect when it does not, or make C appear to have no effect when it does.
In a similar way, the main effects for A and B are confounded with 2-factor interactions,
A = BC
B = AC
The design is called a resolution III design since the main effects are orthogonal with each other but are confounded with 2-factor interactions.
3-factor interaction
If the three columns of ±1 for A, B and C are multiplied, the resulting value is +1 for all runs of the experiment. This means that the experiment can give no information about the ABC interaction.
There are six possible ways to assign two high and two low levels for the factor C to the four runs of the experiment. In the diagram below, you can select three of these options — the other three have the high and low levels of C interchanged and are essentially equivalent.
Initially the levels of C are identical to those of A — C is at a high level in the same runs as A. The main effects of A and C are therefore confounded, so it is impossible to tell whether the estimated effect results from A or C. This is clearly a bad design since it can give no information about which of A or C is more important.
Click the red heading row above B to use the relationship
C = B
to define the levels of C. The factors B and C are now confounded and their effects cannot be separated.
Finally click the red heading row above AB to define the levels of C with
C = AB
The table shows the resulting interactions that are confounded with the main effects.