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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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[1] Chen, J.; Sun, D.X.; Wu, C.F.J., A catalogue of two-level and three-level fractional factorial designs with small runs, Internat. statist. rev., 61, 131-145, (1993) · Zbl 0768.62058
[2] Cheng, C.S.; Mukerjee, R., Regular fractional factorial designs with minimum aberration and maximum estimation capacity, Ann. statist., 26, 2289-2300, (1998) · Zbl 0927.62076
[3] Cheng, C.S.; Steinberg, D.M.; Sun, D.X., Minimum aberration and model robustness for two-level factorial designs, J. roy. statist. soc. ser. B., 61, 85-93, (1999) · Zbl 0913.62072
[4] Cheng, S.W.; Wu, C.F.J., Choice of optimal blocking schemes in two-level and three-level designs, Technometrics, 44, 269-277, (2002)
[5] Fang, K.T.; Mukerjee, R., A connection between uniformity and aberration in regular fractions of two-level factorials, Biometrika, 87, 1, 193-198, (2000) · Zbl 0974.62059
[6] Fries, A.; Hunter, W.G., Minimum aberration \(2^{n - p}\) designs, Technometrics, 40, 314-326, (1980)
[7] Hickernell, F.J.; Liu, M.Q., Uniform designs limit aliasing, Biometrika, 89, 4, 893-904, (2002) · Zbl 1036.62060
[8] Tang, B.; Ma, F.; Ingram, D.; Wang, H., Bounds on the maximum number of clear two-factor interactions for \(2^{m - p}\) designs of resolution III and IV, Canad. J. statist., 30, 127-136, (2002) · Zbl 0999.62059
[9] Wu, C.F.J.; Hamada, M., Experiments: planning, analysis, and parameter design optimization, (2000), Wiley New York · Zbl 0964.62065
[10] Wu, H.; Wu, C.F.J., Clear two-factor interactions and minimum aberration, Ann. statist., 30, 1496-1511, (2002) · Zbl 1015.62083
[11] Zhang, R.C.; Park, D.K., Optimal blocking of two-level fractional factorial designs, J. statist. plann. inference, 91, 107-121, (2000) · Zbl 0958.62072
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