Fractional Factorial with Aliased Two-Factor Interactions
A 2^(4−1) fractional factorial experiment studies conversion yield. Factors: A temperature, B pressure, C catalyst, D residence time. The generator is D=ABC; eight runs are completed in random order. Estimated effects from the run data are A=5.1, B=−2.3, C=1.4, D=3.8, and the alias sums AB+CD=4.6, AC+BD=−0.9, AD+BC=1.2. The team wants to know whether the AB interaction is real, because if it is, they will change pressure settings. There is no replication, and the project has budget for eight additional runs. A statistician says the design is resolution IV and that the AB effect cannot be separated from CD; the team proposes simply adding eight runs at the current factor settings with the signs of A reversed.
1. State the resolution and alias structure of this design, and explain which effects are estimable.
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With generator D=ABC, the defining relation is I=ABCD. The design is resolution IV: main effects are clear of two-factor interactions, but two-factor interactions are aliased in pairs: AB=CD, AC=BD, AD=BC. Thus A, B, C, D are estimable if three-factor and higher interactions are negligible. The observed alias sum AB+CD=4.6 cannot be attributed to AB alone.
2. Evaluate the team's proposed eight additional runs. What design should be used instead?
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The proposed extra runs are not a proper foldover. Holding B, C and D at their original settings while reversing only A violates the generator D=ABC, so the combined set is not the complementary fraction and the AB/CD alias is not cleanly broken. A proper foldover runs the second fraction with D=-ABC, i.e. the eight treatment combinations not already run. That yields the full 2^4 factorial in 16 runs and separates all two-factor interactions. Use the eight-run budget for the complementary fraction, not an A-only flip.
3. Given AB+CD=4.6, what can be concluded, and what follow-up resolves the ambiguity?
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Because AB+CD=4.6, either AB or CD (or both) is nonzero. The team cannot decide to change pressure settings on this evidence. After the full foldover, estimate AB and CD separately; if AB remains large and CD is small, the interaction is supported. If both are moderate, the original eight runs cannot resolve them. Also check whether the alias sum is consistent with effect sparsity and whether any run had unusual yield; with no replication, the original design has no pure-error estimate.