Is a fractional design a random subset of a full factorial?

Usually it is selected deliberately to obtain particular estimation properties.

Does resolution IV separate every two-factor interaction?

No. Two-factor interactions can remain aliased with one another.

Can more runs resolve ambiguity?

Often yes, if the augmentation is chosen to separate the relevant effects and other design conditions are respected.

AI Glossary / Statistics and probability / Advanced

Fractional factorial design

Definition

A fractional factorial design is an experimental design that uses a carefully selected subset of the treatment combinations in a full factorial experiment. It can study several factors with fewer runs, especially when screening for influential factors. The reduction comes with limits on which effects can be distinguished. In regular two-level fractions, some effect columns are identical or sign-reversed, creating aliasing. Useful interpretation therefore depends on the chosen fraction, its alias structure and justified assumptions about effects that may be negligible. A fraction is not simply an arbitrary collection of convenient runs.

What it means

A full experiment with k factors at two levels has 2 to the kth power combinations before replication. A fractional design uses fewer combinations while preserving particular balance and estimation properties. This can make early experimentation feasible, but it does not create information for free. If a main effect shares a column with an interaction, the data may identify their combined contribution without identifying each separately. The design should be chosen around the scientific questions and plausible interactions, rather than selecting the smallest run count and assuming all important effects remain estimable.

How it works

For a regular two-level design, code factor levels as minus one and plus one. Select generators that define the fraction, derive its defining relation and inspect the resulting aliases. Resolution summarizes aspects of that structure: in resolution III designs, main effects can be aliased with two-factor interactions; resolution IV separates those but may alias two-factor interactions with each other. Randomize runs where appropriate, account for blocking and plan replication or follow-up. Fit only an identifiable model and state the assumptions used to interpret aliased estimates. Additional runs, including suitable foldovers, can help separate effects that the initial fraction cannot distinguish.

Three practical examples

Process screening: investigate several operating factors before committing to a larger optimization experiment.
Product development: compare combinations of controllable settings when a full factorial would require too many trials.
Software experimentation: screen configuration factors under a designed run plan, while checking whether important interactions could be confounded with apparent main effects.

Think of it this way

Think of listening to several instruments through a limited number of audio channels. A careful arrangement can reveal useful patterns, but if two instruments share the same channel, their contributions cannot be separated without more information. The design tells you which effects share a channel.

Worked example

Consider three two-level factors A, B and C. A full factorial requires eight combinations. Choose a half fraction with C = AB, giving four runs: (-1,-1,+1), (-1,+1,-1), (+1,-1,-1) and (+1,+1,+1). The defining relation is I = ABC. Multiplying it by A gives A = BC, so the A column is identical to the BC interaction column. Suppose the hypothetical response is y = 10 + 2A + 3BC, with no noise. On this fraction, y becomes 10 + 5A: the two A-low responses are five and the two A-high responses are 15. The observed high-minus-low contrast is 10, corresponding to coded coefficient five. The design cannot determine from these runs alone whether that coefficient arose from A, BC or their combination. Calling it a pure A effect would require an additional assumption or additional runs.

Illustrative case study

Fictional case: Spruce Process Lab chooses an initial screening fraction after discussing which interactions are plausible. One apparently influential main-effect contrast is aliased with an interaction that engineers consider credible. Instead of immediately changing the process, the team adds a planned follow-up fraction to separate the explanations. It retains the initial alias table in the report. This invented case illustrates how an economical first stage can guide learning without pretending that a small experiment identifies every component of the response.

Common misconceptions and mistakes

Do not equate fewer runs with unchanged inferential power or identifiability. Orthogonal columns in the fitted model do not remove aliases with omitted effects. Design resolution does not guarantee that high-order interactions are negligible. A saturated design may leave no residual degrees of freedom for estimating experimental error without further assumptions. The coded regression coefficient and the high-minus-low factorial effect differ by a factor of two under usual two-level coding. Curvature across quantitative settings is not identified by two endpoints alone, and blocking can introduce additional confounding that must be examined.

Business use

Businesses can use fractional designs to screen factors efficiently when testing is costly. Document factor ranges, feasible combinations, run order, response definitions and the effects the design can estimate. Choose a fraction that protects the comparisons needed for the decision, and reserve resources for confirmation or augmentation. Analyze practical effect sizes alongside uncertainty and plausible interactions. If constraints exclude combinations, a standard regular fraction may no longer fit the problem and a different design approach may be needed. Clear communication of aliases is essential before translating a screening result into an operational change.

Check your understanding

In the fraction I = ABC, can A and BC be estimated separately without further information?

Answer: No. Their columns coincide in this fraction, so the observed contrast combines their contributions under the usual coding.

Related terms

Aliasing (factorial experiments)
Confounding
Generator
Replication (statistics)
Full factorial design
aliasing
design resolution
defining relation
interaction
foldover design
screening experiment
replication

Learn next

Learn how generators produce alias relations, then compare resolution III, IV and V designs. Study augmentation and confirmation experiments before interpreting screening contrasts as isolated causal effects.

Sources

NIST/SEMATECH, Confounding and aliasing: https://www.itl.nist.gov/div898/handbook/pri/section3/pri3343.htm NIST/SEMATECH, Design resolution: https://www.itl.nist.gov/div898/handbook/pri/section3/pri3324.htm NIST/SEMATECH, Improving fractional factorial design resolution: https://www.itl.nist.gov/div898/handbook/pri/section3/pri338.htm

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