8.6 Two-Level Fractional Factorial Experiments and Confounding

Why fractionate

A full two-level factorial in $k$ factors needs $2^k$ runs, which grows impossibly fast.

Ten factors would consume 1,024 runs to estimate 1,023 effects, of which only 10 are main effects and 45 are two-factor interactions. The remaining 968 are three-factor and higher interactions that are almost always negligible. A fractional design spends the run budget on the effects that matter.

Three principles justify this:

• Sparsity of effects. Most processes are driven by a few main effects and a small number of two-factor interactions.
• Hierarchy. Lower-order effects are more likely to be important than higher-order ones, and are usually larger.
• Heredity. An interaction is unlikely to be active unless at least one of its parent main effects is active.

Notation

A fractional factorial is written $2^{k-p}_{R}$:

• $k$ = number of factors
• $p$ = degree of fractionation ($1/2^p$ of the full design)
• $R$ = resolution, in Roman numerals

$2^{5-1}_{V}$ means five factors in $2^4 = 16$ runs, a half fraction, at Resolution V.

Generators, defining relations, and aliases

Build a $2^{5-1}$ by writing the full $2^4$ design in factors A, B, C, D, then assigning the fifth factor to the highest-order interaction column:

$$E = ABCD \quad \text{(the generator)}$$

Multiply both sides by $E$ and use the rule that any column times itself gives the identity column $I$:

$$I = ABCDE \quad \text{(the defining relation)}$$

Every effect's alias is found by multiplying it by the defining relation:

$$A \times ABCDE = BCDE, \qquad AB \times ABCDE = CDE$$

So $A$ is aliased with the four-factor interaction $BCDE$, and $AB$ is aliased with the three-factor interaction $CDE$. Since four-factor and three-factor interactions are almost always negligible, both main effects and two-factor interactions can be interpreted cleanly. That is Resolution V.

Contrast with a $2^{7-4}$ in 8 runs, whose defining relation contains several words of length 3, so main effects are aliased with two-factor interactions: Resolution III.

Resolution from the defining relation

Resolution equals the length of the shortest word in the defining relation. An effect of order $p$ is aliased with effects of order $R - p$.

Interpreting a confounded result

Suppose a $2^{6-2}_{IV}$ screening design shows a large effect on the alias chain $AB + CE$. The experiment cannot tell you which of the two it is. Four ways forward:

1. Process knowledge. If C and E are unrelated physically while A and B are known to interact, AB is the plausible attribution. This is a judgment, and it should be recorded as one.
2. Heredity. If A and B are both active main effects while C and E are inactive, AB is far more likely.
3. Fold over. Run a second fraction with the signs of one or all factors reversed. Combining the two fractions breaks the alias chain.
4. Run the confirming pair. A small follow-up design over just those factors resolves it directly.

Fold-over designs

A fold-over doubles the run count, so the sequential strategy of a small Resolution III screen followed by a fold-over only where needed is often cheaper than starting with a larger design.

Worked example

A team screens seven factors in a $2^{7-4}_{III}$ design, 8 runs. Effects on the alias chains show three large values: A, D, and the chain $BD + CE + AF$.

• Main effects A and D are aliased with two-factor interactions, so even those attributions are not certain at Resolution III.
• The three-way chain is uninterpretable as it stands.

The team runs a full fold-over, 8 more runs, giving a 16-run Resolution IV design. Now all seven main effects are clear of two-factor interactions, and A and D are confirmed as genuinely active. Two-factor interactions remain aliased in pairs, but with A and D active and the other five factors inactive, the heredity principle points at $AD$, which a small follow-up factorial in A and D confirms.

Total cost: 16 screening runs plus a small confirmation, against 128 for the full factorial.

Plackett-Burman designs

Plackett-Burman designs screen up to $N-1$ factors in $N$ runs, where $N$ is a multiple of 4 (12, 20, 24, 28). They are Resolution III but with partial (complex) aliasing: each main effect is aliased with a fraction of many two-factor interactions rather than fully with one.

• Advantage: extremely efficient. Eleven factors in 12 runs.
• Disadvantage: interactions are smeared across main effects, and the alias structure is not clean, so interpretation is riskier and they cannot be folded over as simply.

They are appropriate as a first pass when the factor list is long and interactions are believed weak.

Choosing the design

The standard sequential strategy is to spend no more than about a quarter of the experimental budget on the first screen, then commit the rest based on what it finds.