Lesson 4: Fractional factorial designs

A full 2ᵏ factorial answers every question a team could ask about k factors, but the run count doubles with every added factor: 5 factors means 32 runs, 7 factors means 128. Most of the time, most of that information is wasted, because most of the higher-order interactions a full factorial estimates turn out not to matter. A fractional factorial design deliberately runs only a fraction of the full factorial, trading away the ability to estimate every effect cleanly in exchange for a design that screens many factors in far fewer runs. This lesson covers how that fraction is chosen, what gets tangled together as a result, and how to tell a strong fractional design from a weak one.

2ᵏ⁻ᵖ notation and the idea of a fraction

A fractional factorial is written 2ᵏ⁻ᵖ: k is the number of factors, and p tells how many times the full 2ᵏ design has been cut in half. A 2⁵⁻¹ design studies 5 factors in 2⁵⁻¹ = 2⁴ = 16 runs, a half fraction of the 32-run full factorial. A 2⁷⁻² design studies 7 factors in 2⁵ = 32 runs, a quarter fraction of the 128-run full factorial. The larger p is relative to k, the fewer runs the design costs, and the more effects get tangled together as a result.

Design generators and the defining relation

To build the fraction, one or more high-order interaction columns are deliberately set equal to a new factor instead of being estimated. In a 2⁴⁻¹ design, the first three factors A, B, and C are run as a full 2³ (8 runs), and the fourth factor D is assigned using the generator D = ABC: wherever the ABC contrast in the 2³ is at its high level, D is run high, and wherever ABC is low, D is run low. Multiplying both sides of the generator by D gives the defining relation: D x D = ABC x D, and since any letter multiplied by itself cancels (squares to the identity, written I), this becomes I = ABCD. The defining relation is the single most important piece of information about a fractional design: everything about which effects are tangled together follows from it.

Aliasing (confounding): finding it by multiplying through the defining relation

Because D was never run independently of ABC, any effect estimated by the design is really estimating the combined effect of two or more terms at once, a relationship called aliasing or confounding. To find what any effect is aliased with, multiply it by the defining relation and cancel repeated letters. For I = ABCD:

• A x (I = ABCD) gives A = A x ABCD = BCD, so A is aliased with BCD.
• AB x (I = ABCD) gives AB = AB x ABCD = CD, so AB is aliased with CD.

Working through every effect the same way for this design gives the full alias structure: A = BCD, B = ACD, C = ABD, D = ABC, AB = CD, AC = BD, and AD = BC (shown in the figure above). A significant effect on the "A" contrast in the data cannot, by itself, tell the team whether A truly matters, or whether it is really the three-factor interaction BCD driving the result, or some combination of both; the two are statistically indistinguishable from this data alone.

Resolution: III, IV, and V

A design's resolution is defined as the length (number of letters) of the shortest word in its defining relation, and it is reported in Roman numerals as a subscript, such as 2⁴⁻¹_IV. Resolution sets a design's quality at a glance:

The 2⁴⁻¹ design above has defining relation I = ABCD, a single 4-letter word, so it is Resolution IV: exactly matching the pattern of main effects clear of 2FIs, but 2FIs aliased with each other (AB = CD, AC = BD, AD = BC). Higher resolution is always preferred when the run budget allows it, because it keeps more of the effects a team cares about (main effects, and ideally two-factor interactions) free of confounding.

Sparsity of effects and screening

Fractional designs are defensible because of the sparsity of effects principle: in most real processes, only a small number of the possible main effects and low-order interactions actually matter, and higher-order interactions (three-factor and above) are rarely significant. Because a fractional design's confounding is engineered to tangle the important lower-order effects with the unimportant higher-order ones (as Resolution IV and V designs do), the loss of information is usually acceptable. This is exactly why fractional factorials are the standard tool for screening: narrowing a long list of candidate X's down to the vital few before committing to a full factorial or response surface study on just those few.

Fold-over: de-aliasing a design

If a screening experiment turns up a significant effect that is aliased with another effect the team cannot rule out, a second fraction, called a fold-over, can resolve the ambiguity. A fold-over reverses the sign of one or more factor columns (a full fold-over reverses every factor) and runs a second set of trials. Combining the original data with the fold-over data cancels out the aliasing for the reversed factors, separating effects that were tangled together in the first fraction. The cost is essentially a second experiment the same size as the first, so fold-over is used selectively, on the specific ambiguity that matters, rather than as a routine step. At GCHS, a team screening rehab defect drivers with a Resolution III design that flags "crew size" as significant, aliased with a two-factor interaction it cannot rule out, would fold over rather than act on an ambiguous result.