A note on minimum aberration and clear criteria.

*(English)*Zbl 1090.62080Summary: Minimum aberration and clear criteria are two important rules for selecting optimal fractional factorial designs, in both unblocked and blocked cases. We first show that under some given conditions, a blocked design \(D_B =(D, B)\) having blocked minimum aberration is equivalent to \(D\) having minimum aberration. Let \(m = n/4 + 1\) and \(n = 2^q\). >From the results of B. Tang et al. [Bounds on the maximum number of clear two-factor interactions for \(2^{m-p}\) designs of resolution III and IV. Can. J. Statist. 30, No. 1, 127–136 (2002; Zbl 0999.62059)] and H. Wu and C. F. J. Wu [Clear two-factor interactions and minimum aberration. Ann. Stat. 30, No. 5, 1496–1511 (2002; Zbl 1015.62083)] we know that the maximum number of clear two-factor interactions (2FIs) in \(2^{m-(m-q)}_{\text{IV}}\) designs is \(n/2-1\).

Here it is proved that the maximum number of clear 2FIs in \(2^{m-(m-q)}\) designs in \(2^l\) blocks, denoted by \(2^{m-(m-q)}_{\text{IV}}:2^l\) is also \(n/2 -1\) when \(q-l\geq 2\). Furthermore, it is shown that any \(2^{m-(m-q)}_{\text{IV}}\) design that contains the maximum number of clear 2FIs is not a minimum aberration design, and this conclusion also holds when the design is a \(2^{m-(m-q)}_{\text{IV}^-}:2^l\) design with \(q-l\geq 2\).

Here it is proved that the maximum number of clear 2FIs in \(2^{m-(m-q)}\) designs in \(2^l\) blocks, denoted by \(2^{m-(m-q)}_{\text{IV}}:2^l\) is also \(n/2 -1\) when \(q-l\geq 2\). Furthermore, it is shown that any \(2^{m-(m-q)}_{\text{IV}}\) design that contains the maximum number of clear 2FIs is not a minimum aberration design, and this conclusion also holds when the design is a \(2^{m-(m-q)}_{\text{IV}^-}:2^l\) design with \(q-l\geq 2\).

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\textit{P.-F. Li} et al., Stat. Probab. Lett. 76, No. 10, 1007--1011 (2006; Zbl 1090.62080)

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##### References:

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