Changing a resolution III design to resolution IV
Screening designs are often designed to use the minimum number of runs that allow all main effects to be independently estimated.
Number of runs | Type of design | Max factors |
---|---|---|
4 | 23-1 | 3 |
8 | 27-4 | 7 |
12 | Plackett-Burman | 11 |
16 | 215-11 | 15 |
20 | Plackett-Burman | 19 |
After conducting such an experiment, it is may be suspected that some 2-factor interactions are large. Since the main effects in a resolution III design are confounded with 2-factor interactions, any 2-factor interactions could affect the apparent size of the main effects of other factors.
It may therefore be decided to conduct further runs of the experiment to separate the 2-factor interactions from the main effects — i.e. to change the design from resolution III to resolution IV.
Foldover designs
A simple way to create a resolution IV design from a resolution III design is to repeat all runs in the resolution III design with all factor levels swapped (high -> low and low -> high). This is called a foldover design.
Foldover of Plackett-Burman design
The first 12 runs below comprise a Plackett-Burman design of resolution III that can be used to estimate the main effects of up to 11 factors. Runs 13 to 24 repeat the first 12 runs but interchange the high and low levels of all factors.
Factor | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
Run | A | B | C | D | E | F | G | H | I | J | K |
1 | +1 | +1 | -1 | +1 | +1 | +1 | -1 | -1 | -1 | +1 | -1 |
2 | -1 | +1 | +1 | -1 | +1 | +1 | +1 | -1 | -1 | -1 | +1 |
3 | +1 | -1 | +1 | +1 | -1 | +1 | +1 | +1 | -1 | -1 | -1 |
4 | -1 | +1 | -1 | +1 | +1 | -1 | +1 | +1 | +1 | -1 | -1 |
5 | -1 | -1 | +1 | -1 | +1 | +1 | -1 | +1 | +1 | +1 | -1 |
6 | -1 | -1 | -1 | +1 | -1 | +1 | +1 | -1 | +1 | +1 | +1 |
7 | +1 | -1 | -1 | -1 | +1 | -1 | +1 | +1 | -1 | +1 | +1 |
8 | +1 | +1 | -1 | -1 | -1 | +1 | -1 | +1 | +1 | -1 | +1 |
9 | +1 | +1 | +1 | -1 | -1 | -1 | +1 | -1 | +1 | +1 | -1 |
10 | -1 | +1 | +1 | +1 | -1 | -1 | -1 | +1 | -1 | +1 | +1 |
11 | +1 | -1 | +1 | +1 | +1 | -1 | -1 | -1 | +1 | -1 | +1 |
12 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 |
13 | -1 | -1 | +1 | -1 | -1 | -1 | +1 | +1 | +1 | -1 | +1 |
14 | +1 | -1 | -1 | +1 | -1 | -1 | -1 | +1 | +1 | +1 | -1 |
15 | -1 | +1 | -1 | -1 | +1 | -1 | -1 | -1 | +1 | +1 | +1 |
16 | +1 | -1 | +1 | -1 | -1 | +1 | -1 | -1 | -1 | +1 | +1 |
17 | +1 | +1 | -1 | +1 | -1 | -1 | +1 | -1 | -1 | -1 | +1 |
18 | +1 | +1 | +1 | -1 | +1 | -1 | -1 | +1 | -1 | -1 | -1 |
19 | -1 | +1 | +1 | +1 | -1 | +1 | -1 | -1 | +1 | -1 | -1 |
20 | -1 | -1 | +1 | +1 | +1 | -1 | +1 | -1 | -1 | +1 | -1 |
21 | -1 | -1 | -1 | +1 | +1 | +1 | -1 | +1 | -1 | -1 | +1 |
22 | +1 | -1 | -1 | -1 | +1 | +1 | +1 | -1 | +1 | -1 | -1 |
23 | -1 | +1 | -1 | -1 | -1 | +1 | +1 | +1 | -1 | +1 | -1 |
24 | +1 | +1 | +1 | +1 | +1 | +1 | +1 | +1 | +1 | +1 | +1 |
The 24 runs comprise a resolution IV design in which the main effects are orthogonal to (not confounded with) the 2-factor interactions. With this design, we would be more confident that our estimates of the main effects are not affected by interactions between different factors.
Foldover of fractional factorial design
The table below shows the 8 runs of a 25-2 fractional factorial design defined by the confounding pattern
D = ABC
E = AC
The subsequent 8 foldover runs swap the high and low levels of all factors. Note that these 8 runs conform to the confounding pattern
D = ABC
E = -AC
Factors | |||||
---|---|---|---|---|---|
Run | A | B | C | D = ABC | E = AC |
1 | -1 | -1 | -1 | -1 | +1 |
2 | -1 | -1 | +1 | +1 | -1 |
3 | -1 | +1 | -1 | +1 | +1 |
4 | -1 | +1 | +1 | -1 | -1 |
5 | +1 | -1 | -1 | +1 | -1 |
6 | +1 | -1 | +1 | -1 | +1 |
7 | +1 | +1 | -1 | -1 | -1 |
8 | +1 | +1 | +1 | +1 | +1 |
D = ABC | E = -AC | ||||
9 | +1 | +1 | +1 | +1 | -1 |
10 | +1 | +1 | -1 | -1 | +1 |
11 | +1 | -1 | +1 | -1 | -1 |
12 | +1 | -1 | -1 | +1 | +1 |
13 | -1 | +1 | +1 | -1 | +1 |
14 | -1 | +1 | -1 | +1 | -1 |
15 | -1 | -1 | +1 | +1 | +1 |
16 | -1 | -1 | -1 | -1 | -1 |
The resulting 16 runs are also a fractional factorial design — now a 25-1 design. It can also be generated as a complete factorial design for A, B, C and E, with D then defined by:
D = ABC
A 25-1 design with this confounding pattern is resolution IV.
Note that it is possible to generate a resolution V design with 16 runs and this would also allow separate estimation of all 2-factor interactions. However a resolution V design cannot be generated sequentially by adding 8 runs to a previous 8-run resolution III design.
Uses of foldover designs
Foldover designs are useful as sequential designs that augment a resolution III design with additional runs to turn it into resolution IV.
Since foldover of a fractional factorial design is also a fractional factorial design, there is little to be gained by initially designing an experiment as a foldover of a fractional factorial.
However foldover of some Plackett-Burman designs are resolution IV designs with fewer runs than would be possible with any fractional factorial. For example, the above foldover of a 12-run Plackett-Burman is a resolution IV design for 11 factors in 24 runs. A fractional factorial resolution IV design for 11 factors would require at least 32 runs.