Fractional Factorials and Design Resolution
How fractional factorial designs cut run counts, what aliasing means, and how to read design resolution III, IV and V when screening many factors.
By Covariance Labs · · 3 min read
This guide has not yet been reviewed by an independent statistician. If you spot an error, please tell us.
A full factorial gets expensive quickly: six factors at two levels is 64 runs. Yet in most processes only a few factors matter, and high-order interactions are usually negligible. A fractional factorial exploits that by running a fraction (a half, a quarter) of the full design.
How a fraction is built
Start with a full factorial in some base factors, then define the remaining factors as products of the base columns. These products are called generators.
For four factors A, B, C and D, run a full factorial in A, B and C (8 runs) and set the level of D to the product of the other three:
This is a 24−1 design: four factors in 23 = 8 runs, half the 16 of the full design.
Aliasing: the price of fewer runs
Because D is defined as ABC, the two columns are identical, so the data cannot distinguish an effect of D from an effect of the three-way interaction ABC. They are aliased. The defining relation for this design is I = ABCD, and multiplying through by any effect gives its alias:
- A is aliased with BCD
- AB is aliased with CD
- AC is aliased with BD
- AD is aliased with BC
If you see a big AB effect, you cannot tell whether it is AB, CD, or a mixture. Whether that is acceptable depends on what you believe about interactions.
Reading resolution
Resolution is the length of the shortest word in the defining relation, and summarizes how bad the aliasing is.
| Resolution | What it means |
|---|---|
| III | Main effects are aliased with two-factor interactions. Fine only if interactions are assumed negligible. |
| IV | Main effects are clear of two-factor interactions, but two-factor interactions are aliased with each other. |
| V | Main effects and two-factor interactions are clear of each other (aliased only with three-factor or higher). |
The 24−1 design above has defining relation I = ABCD, a word of length 4, so it is resolution IV.
Common fractions for 4 to 6 factors
| Design | Runs | Generators | Resolution |
|---|---|---|---|
| 24−1 | 8 | D = ABC | IV |
| 25−1 | 16 | E = ABCD | V |
| 26−2 | 16 | E = ABC, F = BCD | IV |
For the 26−2 design the defining relation is I = ABCE = BCDF = ADEF (the third word is the product of the first two), so its shortest word has length 4.
A screening strategy
- Use a fractional factorial to screen many factors and find the few that matter.
- Drop the inactive factors and run a full factorial or a response-surface design on the survivors.
- Confirm the final settings with real runs.
Explore these designs with the free design generator, and see the practical guide to DoE for how screening fits into the whole workflow.
Frequently asked questions
- What is a fractional factorial design?
- A fractional factorial runs a carefully chosen subset of the runs of a full factorial. It needs fewer runs but confounds (aliases) some effects with each other.
- What does resolution IV mean?
- In a resolution IV design, main effects are not aliased with two-factor interactions, but two-factor interactions are aliased with each other.