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A note on minimum aberration and clear criteria. (English) Zbl 1090.62080
Summary: 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 $$q-l\geq 2$$.

##### MSC:
 62K15 Factorial statistical designs 62K05 Optimal statistical designs
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##### References:
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