10.5 De-aliasing: the fold-over
A resolution III design is a cheap screen, but its main effects are aliased with two-factor interactions. If the results are ambiguous, a second fraction can be run to separate them.
Theorem 10.5.1. Let a second fraction be formed from a resolution III fraction by reversing the signs of every factor in every run (a full fold-over). In the combined design, every main effect is free of aliasing with two-factor interactions; the combined design has resolution at least IV.
Proof. Reversing every factor multiplies the column of an effect with \(r\) letters by \((-1)^r\). In the defining relation of the folded fraction, each word of odd length therefore changes sign and each word of even length keeps it. Combining the two fractions, a word that appears with opposite signs is satisfied by neither sign on all runs, and drops out of the combined defining relation; only words of even length remain. A design whose defining words all have even length, and none of length two, has resolution at least IV. (No word of length two can arise, because the original design had resolution III, so all its words have length at least three, and the combined relation is a subset of them.) □
In practice, if the original fraction gives the estimate \(l_i\) of \(A + BC + \dots \) and the folded fraction gives \(l_i'\) of \(A - BC - \dots \), then \(\frac 12(l_i + l_i')\) estimates \(A\) (plus three-factor interactions) and \(\frac 12(l_i - l_i')\) estimates the two-factor interactions aliased with it. Reversing the signs of a single factor (a single-factor fold-over) instead frees that factor and all its two-factor interactions.
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