10.3 Smaller fractions: the \(2^{k-p}\) design
A \(2^{k-p}\) design, a \(1/2^p\) fraction, is constructed by writing down a full factorial in \(k - p\) basic factors and defining each of the remaining \(p\) factors as an interaction of the basic ones, through \(p\) generators. The defining relation then contains the \(p\) generator words and all their generalised interactions: \(2^p - 1\) words in all, which, with \(I\), form a group under multiplication. Each effect has \(2^p - 1\) aliases, obtained by multiplying it by each word, and the resolution is again the length of the shortest word.
Example 10.3.1. Consider the \(2^{5-2}\) design with generators \(I = ABC\) and \(I = BDE\).
Defining relation. The generalised interaction is \(ABC \cdot BDE = AB^2CDE = ACDE\), so \[ I = ABC = BDE = ACDE . \] Resolution. The shortest words have three letters: the design is \(2^{5-2}_{\mathrm {III}}\).
Alias structure. Multiplying each effect by the three words, and ignoring repeats,
| \(A = BC = ABDE = CDE\) |
| \(B = AC = DE = ABCDE\) |
| \(C = AB = BCDE = ADE\) |
| \(D = ABCD = BE = ACE\) |
| \(E = ABCE = BD = ACD\) |
| \(AD = BCD = ABE = CE\) |
| \(AE = BCE = ABD = CD\) |
These seven alias sets account for the seven degrees of freedom among eight runs. Every main effect is aliased with at least one two-factor interaction.
A quarter fraction. Take \(A\), \(B\), \(D\) as the basic factors and set \(C = AB\), \(E = BD\):
| \(A\) | \(B\) | \(D\) | \(C = AB\) | \(E = BD\) | run |
| \(-\) | \(-\) | \(-\) | \(+\) | \(+\) | \(ce\) |
| \(+\) | \(-\) | \(-\) | \(-\) | \(+\) | \(ae\) |
| \(-\) | \(+\) | \(-\) | \(-\) | \(-\) | \(b\) |
| \(+\) | \(+\) | \(-\) | \(+\) | \(-\) | \(abc\) |
| \(-\) | \(-\) | \(+\) | \(+\) | \(-\) | \(cd\) |
| \(+\) | \(-\) | \(+\) | \(-\) | \(-\) | \(ad\) |
| \(-\) | \(+\) | \(+\) | \(-\) | \(+\) | \(bde\) |
| \(+\) | \(+\) | \(+\) | \(+\) | \(+\) | \(abcde\) |
Every run has \(ABC = +\) and \(BDE = +\); this is the principal fraction. Changing the signs of the generators (\(C = -AB\) and/or \(E = -BD\)) gives the other three quarter fractions, and the four together make up the full \(2^5\).
When several designs of the same resolution are available, the one with the fewest words of minimum length is preferred, since it aliases the fewest pairs of low-order effects. This is the criterion of minimum aberration, and tables of recommended generators are constructed with it.
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