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Mixed two- and four-level fractional factorial split-plot designs with clear effects. (English) Zbl 1238.62089
Summary: Mixed-level designs have become widely used in practical experiments. When the levels of some factors are difficult to be changed or controlled, fractional factorial split-plot (FFSP) designs are often used. It is important to know when a mixed-level FFSP design with resolution III or IV has clear effects. This paper investigates the conditions of a resolution III or IV FFSP design with both two-level and four-level factors to have various clear factorial effects, including two types of main effects and three types of two-factor interaction components. The structures of such designs are shown and illustrated with examples.

##### MSC:
 62K15 Factorial statistical designs
##### Keywords:
resolution; whole-plot factor; sub-plot factor
Full Text:
##### References:
 [1] Addelman, S., Orthogonal main-effect plans for asymmetrical factorial experiments, Technometrics, 4, 21-46, (1962) · Zbl 0116.36704 [2] Ai, M.Y.; Zhang, R.C., $$s^{n - m}$$ designs containing clear main effects or clear two-factor interactions, Statistics & probability letters, 69, 151-160, (2004) · Zbl 1062.62143 [3] Bingham, D.; Sitter, R.R., Some theoretical results for fractional split-plot designs, Annals of statistics, 27, 1240-1255, (1999) · Zbl 0957.62065 [4] Bisgaard, S.; Steinberg, D.M., The design and analysis of $$2^{k - p} \times s$$ prototype experiments, Technometrics, 39, 52-62, (1997) · Zbl 0869.62069 [5] 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, Journal of statistical planning and inference, 136, 4436-4449, (2006) · Zbl 1099.62082 [6] Chen, H.; Hedayat, A.S., $$2^{n - m}$$ designs with resolution III and IV containing clear two-factor interactions, Journal of statistical planning and inference, 75, 147-158, (1998) · Zbl 0938.62081 [7] Hedayat, A.; Pu, K.; Stufken, J., On the construction of asymmetrical orthogonal arrays, Annals of statistics, 20, 2142-2152, (1992) · Zbl 0784.62076 [8] Huang, P.; Chen, D.; Voelkel, J., Minimum aberration two-level split-plot designs, Technometrics, 40, 314-326, (1998) · Zbl 1064.62552 [9] Mukerjee, R.; Fang, K.T., Fractional factorial split-plot designs with minimum aberration and maximum estimation capacity, Statistica sinica, 12, 381-397, (2002) [10] 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, Canadian journal of statistics, 30, 127-136, (2002) · Zbl 0999.62059 [11] Wu, C.F.J., Construction of $$2^m 4^n$$ designs via a grouping scheme, Annals of statistics, 17, 1880-1885, (1989) · Zbl 0695.62198 [12] 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) [13] Wu, C.F.J.; Zhang, R.C.; Wang, R.G., Construction of asymmetrical orthogonal arrays of the type $$\mathit{OA}(s^k$$, $$(s^{r_1})^{n_1} \ldots(s^{r_l})^{n_l})$$, Statistica sinica, 2, 203-219, (1992) [14] Yang, G.J.; Liu, M.Q., A note on the lower bounds on maximum number of clear two-factor interactions for $$2_{\operatorname{III}}^{m - p}$$ and $$2_{\operatorname{IV}}^{m - p}$$ designs, Communications in statistics—theory and methods, 35, 849-860, (2006) · Zbl 1093.62073 [15] Yang, G.J.; Liu, M.Q.; Zhang, R.C., Weak minimum aberration and maximum number of clear two-factor interactions in $$2_{\operatorname{IV}}^{m - p}$$ designs, Science in China (series A), 48, 1479-1487, (2005) · Zbl 1112.62076 [16] 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, Journal of statistical planning and inference, 136, 4450-4458, (2006) · Zbl 1099.62084 [17] Zhang, R.C.; Shao, Q., Minimum aberration $$(S^2) S^{n - k}$$ designs, Statistica sinica, 11, 213-223, (2001) · Zbl 0967.62056 [18] Zhao, S.L.; Zhang, R.C., Bound on the maximum number of clear two-factor interactions for $$2^{n -(n - k)}$$ designs, Acta Mathematica scientia, 28, 949-954, (2008) · Zbl 1198.62074 [19] Zhao, S.L.; Zhang, R.C., $$2^m 4^n$$ designs with resolution III or IV containing clear two-factor interaction components, Statistical papers, 49, 441-454, (2008) · Zbl 1148.62063 [20] Zhao, S.L.; Zhang, R.C.; Liu, M.Q., Some results on $$4^m 2^n$$ designs with clear two-factor interaction components, Science in China (series A), 51, 1297-1314, (2008) · Zbl 1143.62043 [21] 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, Science in China (series A), 49, 1816-1829, (2006) · Zbl 1106.62089
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