Full vs fractional factorial designs (DOE)

Short answer: a full factorial tests every combination of factor levels. With k factors at 2 levels, that's 2^k runs, and you can estimate every main effect and interaction. A fractional factorial (2^k−p) runs only a planned fraction of those combinations. That saves runs, but some effects become aliased (confounded): the data can't tell them apart.

Run counts

Resolution in one line each

• III: main effects are aliased with two-factor interactions. Use it for screening only.
• 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 all clear of each other.

Worked example: estimating effects in a 2² design

The factors are temperature (A) and pressure (B), each at a low (−) and high (+) level, and the response is yield. The results: (−,−) = 20, (+,−) = 30, (−,+) = 24, (+,+) = 40.

What it means: temperature has the biggest effect, pressure has a smaller one, and the interaction is small. In a fractional design, each estimate like these is really the sum of the effect and all of its aliases. That's why you choose the resolution based on which interactions you can assume are negligible.

Common exam traps

• Effect vs coefficient. A regression coefficient in coded units is half the effect.
• Reading a resolution III result as proof. A big "main effect" there could really be an interaction. Follow up with a fold-over or added runs.
• Skipping randomization and replication. Without them, time trends hide in your effects and you have no estimate of pure error.
• Skipping center points. Two-level designs assume the response is linear. Center points check for curvature.

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