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Some results on $$4^m 2^n$$ designs with clear two-factor interaction components. (English) Zbl 1143.62043
Summary: The clear effects criterion is one of the important rules for selecting optimal fractional factorial designs, and has become an active research issue in recent years. B. Tang et al. [Can. J. Stat. 30, No. 1, 127–136 (2002; Zbl 0999.62059)] derived upper and lower bounds on the maximum number of clear two-factor interactions (2fi’s) in $$2^{n - (n - k)}$$ fractional factorial designs of resolutions III and IV by constructing a $$2^{n - (n - k)}$$ design for given $$k$$, which are only restricted for the symmetrical case.
This paper proposes and studies the clear effects problem for the asymmetrical case. It improves the construction method of Tang et al. for $$2^{n - (n - k)}$$ designs with resolution III and derives the upper and lower bounds on the maximum number of clear two-factor interaction components (2fic’s) in $$4^m 2^n$$ designs with resolutions III and IV. The lower bounds are achieved by constructing specific designs. Comparisons show that the number of clear 2fic’s in the resulting design attains its maximum number in many cases, which reveals that the construction methods are satisfactory when they are used to construct $$4^m 2^n$$ designs under the clear effects criterion.

##### MSC:
 62K15 Factorial statistical designs
##### Keywords:
mixed levels; resolution
Full Text:
##### References:
 [1] Addelman S. Orthogonal main-effect plans for asymmetrical experiments. Technometrics, 4: 21–46 (1962) · Zbl 0116.36704 · doi:10.2307/1266170 [2] Wu C F J. Construction of 2m4n designs via a grouping scheme. Ann Statist, 17: 1880–1885 (1989) · Zbl 0695.62198 · doi:10.1214/aos/1176347399 [3] Wu C F J, Zhang R C, Wang R. Construction of asymmetrical orthogonal arrays of the type $$OA(s^k ,(s^{r_1 } )^{n_1 } )(s^{r_t } )^{n_t } )$$ . Statist Sinica, 2: 203–219 (1992) · Zbl 0822.62063 [4] Box G E P, Hunter J S. The 2k fractional factorial designs. Technometrics, 3: 311–351, 449–458 (1961) · Zbl 0100.14406 · doi:10.2307/1266725 [5] Fries A, Hunter W G. Minimum aberration 2k designs. Technometrics, 22: 601–608 (1980) · Zbl 0453.62063 · doi:10.2307/1268198 [6] Wu C F J, Zhang R C. Minimum aberration designs with two-level and four-level factors. Biometrika, 80: 203–209 (1993) · Zbl 0769.62058 [7] Zhang R C, Shao Q. Minimum aberration (s 2)s n designs. Statist Sinica, 11: 213–223 (2001) · Zbl 0967.62056 [8] Mukerjee R, Wu C F J. Minimum aberration designs for mixed factorials in term of complementary sets. Statist Sinica, 11: 225–239 (2001) · Zbl 0967.62054 [9] Ai M Y, Zhang R C. Characterization of minimum aberration mixed factorials in terms of consulting designs. Statist Papers, 46(2): 157–171 (2005) · Zbl 1083.62072 · doi:10.1007/BF02762966 [10] Wu C F J, Chen Y. A graph-aided method for planning two-level experiments when certain interactions are important. Technometrics, 34: 162–175 (1992) · doi:10.2307/1269232 [11] Tang B, Ma F, Ingram D, Wang H. Bounds on the maximum number of clear two-factor interactions for 2m designs of resolution III and IV. Canad J Statist, 30: 127–136 (2002) · Zbl 0999.62059 · doi:10.2307/3315869 [12] Chen H, Hedayat A S. 2n designs with resolution III or IV containing clear two-factor interactions. J Statist Plann Inference, 75: 147–158 (1998) · Zbl 0938.62081 · doi:10.1016/S0378-3758(98)00122-0 [13] Wu H, Wu C F J. Clear two-factor interactions and minimum aberration. Ann Statist, 30: 1496–1511 (2002) · Zbl 1015.62083 · doi:10.1214/aos/1035844985 [14] Ai M Y, Zhang R C. s n designs containing clear main effects or clear two-factor interactions. Statist Probab Lett, 69: 151–160 (2004) · Zbl 1062.62143 · doi:10.1016/j.spl.2004.06.015 [15] Ke W, Tang B, Wu H. Compromise plans with clear two-factor interactions. Statist Sinica, 15: 709–715 (2005) · Zbl 1086.62083 [16] Yang G J, Liu M Q, Zhang R C. Weak minimum aberration and maximum number of clear two-factor interactions in 2 IV m designs. Sci China Ser A-Math, 48(11): 1479–1487 (2005) · Zbl 1112.62076 · doi:10.1360/04ys0165 [17] Zhao S L, Zhang R C. 2m4n designs with resolution III or IV containing clear two-factor interaction components. Statist Papers, 49: 441–454 (2008) · Zbl 1148.62063 · doi:10.1007/s00362-006-0025-4 [18] Li P F, Chen B J, Liu M Q, Zhang R C. A note on minimum aberration and clear criteria. Statist Probab Lett, 76: 1007–1011 (2006) · Zbl 1090.62080 · doi:10.1016/j.spl.2005.11.003 [19] Yang G J, Liu M Q. A note on the lower bounds on maximum number of clear two-factor interactions for 2 III m and 2 IV m designs. Comm Statist Theory Methods, 35: 849–860 (2006) · Zbl 1093.62073 · doi:10.1080/03610920500501502 [20] Chen B J, Li P F, Liu M Q, Zhang R C. Some results on blocked regular 2-level fractional factorial designs with clear effects. J Statist Plann Inference, 136(12): 4436–4449 (2006) · Zbl 1099.62082 · doi:10.1016/j.jspi.2005.07.006 [21] Yang J F, Li P F, Liu M Q, Zhang R C. $$2^{(n_1 + n_2 ) - (k_1 + k_2 )}$$ fractional factorial split-plot designs containing clear effects. J Statist Plann Inference, 136(12): 4450–4458 (2006) · Zbl 1099.62084 · doi:10.1016/j.jspi.2005.06.009 [22] Zi X M, Liu M Q, Zhang R C. Asymmetrical designs containing clear effects. Metrika, 65(1): 123–131 (2007) · Zbl 1105.62079 · doi:10.1007/s00184-006-0064-9 [23] Zi X M, Zhang R C, Liu M Q. Bounds on the maximum numbers of clear two-factor interactions for $$2^{(n_1 + n_2 ) - (k_1 + k_2 )}$$ fractional factorial split-plot designs. Sci China Ser A-Math, 49(12): 1816–1829 (2006) · Zbl 1106.62089 · doi:10.1007/s11425-006-2032-2 [24] 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 · doi:10.1111/1467-9868.00164 [25] 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 · doi:10.1214/aos/1024691471
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