Abstract
By the affine resolvable design theory, there are 68 non-isomorphic classes of symmetric orthogonal designs involving 13 factors with 3 levels and 27 runs. This paper gives a comprehensive study of all these 68 non-isomorphic classes from the viewpoint of the uniformity criteria, generalized word-length pattern and Hamming distance pattern, which provides some interesting projection and level permutation behaviors of these classes. Selecting best projected level permuted subdesigns with \(3\le k\le 13\) factors from all these 68 non-isomorphic classes is discussed via these three criteria with catalogues of best values. New recommended uniform minimum aberration and minimum Hamming distance designs are given for investigating either qualitative or quantitative \(4\le k\le 13\) factors, which perform better than the existing recommended designs in literature and the existing uniform designs. A new efficient technique for detecting non-isomorphic designs is given via these three criteria. By using this new approach, in all projections into \(1\le k\le 13\) factors we classify each class from these 68 classes to non-isomorphic subclasses and give the number of isomorphic designs in each subclass. Close relationships among these three criteria and lower bounds of the average uniformity criteria are given as benchmarks for selecting best designs.
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References
Angelopoulos P, Evangelaras H, Koukouvinos C (2009) Model identification using 27 runs three level orthogonal arrays. J Appl Stat 36:33–38
Chen J, Sun DX, Wu CFJ (1993) A catalogue of two-level and three-level fractional factorial designs with small runs. Int Stat Rev 61:131–135
Chen W, Qi ZF, Zhou YD (2015) Constructing uniform designs under mixture discrepancy. Stat Probab Lett 97:76–82
Clark JB, Dean AM (2001) Equivalence of fractional factorial designs. Stat Sin 11:537–547
Dey A, Mukerjee R (1999) Fractional factorial plans. Wiley, New York
Elsawah AM (2017a) A closer look at de-aliasing effects using an efficient foldover technique. Statistics 51(3):532–557
Elsawah AM (2017b) A powerful and efficient algorithm for breaking the links between aliased effects in asymmetric designs. Aust NZ J Stat 59(1):17–41
Elsawah AM (2017c) Constructing optimal router bit life sequential experimental designs: new results with a case study. Commun Stat Simul Comput. https://doi.org/10.1080/03610918.2017.1397164
Elsawah AM (2018) Choice of optimal second stage designs in two-stage experiments. Comput Stat 33(2):933–965
Elsawah AM, Qin H (2015a) Mixture discrepancy on symmetric balanced designs. Stat Probab Lett 104:123–132
Elsawah AM, Qin H (2015b) A new strategy for optimal foldover two-level designs. Stat Probab Lett 103:116–126
Elsawah AM, Qin H (2016) Asymmetric uniform designs based on mixture discrepancy. J Appl Stat 43(12):2280–2294
Elsawah AM, Fang KT (2018) A catalog of optimal foldover plans for constructing U-uniform minimum aberration four-level combined designs. J Appl Stat. https://doi.org/10.1080/02664763.2018.1545013
Evangelaras H, Koukouvinos C, Dean AM, Dingus CA (2005) Projection properties of certain three level orthogonal arrays. Metrika 62:241–257
Evangelaras H, Koukouvinos C, Lappas E (2007) 18-run nonisomorphic three level orthogonal arrays. Metrika 66:31–37
Fang KT, Zhang A (2004) Minimum aberration Majorization in non-isomorphic saturated asymmetric designs. J Stat Plan Inference 126:337–346
Fang KT, Tang Y, Yin JX (2008) Lower bounds of various criteria in experimental designs. J Stat Plan Inference 138:184–195
Fang KT, Ke X, Elsawah AM (2017) Construction of uniform designs via an adjusted threshold accepting algorithm. J Complex 43:28–37
Fries A, Hunter WG (1980) Minimum aberration \(2^{k-p}\) designs. Technometrics 8:601–608
Hall M Jr (1961) Hadamard matrix of order 16. Jet Propuls Lab Res Summ 1:21–26
Hedayat AS, Sloane NJ, Stufken J (1999) Orthogonal arrays: theory and application. Springer, Berlin
Hickernell FJ (1998a) A generalized discrepancy and quadrature error bound. Math Comput 67:299–322
Hickernell FJ (1998b) Lattice rules: how well do they measure up? In: Hellekalek P, Larcher G (eds) Random and quasi-random point sets. Lecture notes in statistics, vol 138. Springer, New York, pp 109-166
Lam C, Tonchev VD (1996) Classification of affine resolvable \(2-(27, 9, 4)\) designs. J Stat Plan Inference 56:187–202
Ma CX, Fang KT (2001) A note on generalized aberration in fractional designs. Metrika 53:85–93
Ma CX, Fang KT, Lin DKJ (2001) On isomorphism of factorial designs. J Complex 17:86–97
Sartono B, Goos P, Schoen ED (2012) Classification of three-level strength-3 arrays. J Stat Plan Inference 142(4):794–809
Sun DX, Wu CFJ (1993) Statistical properties of Hadamard matrices of order 16. In: Kuo W (ed) Quality through engineering design. Elsevier, Amsterdam, pp 169–179
Tang B, Deng LY (2001) Minimum \(G_2\)-aberration for nonregular fractional factorial designs. Ann Stat 27:1914–1926
Tang Y, Xu H (2013) An effective construction method for multi-level uniform designs. J Stat Plan Inference 143:1583–1589
Tang Y, Xu H (2014) Permuting regular fractional factorial designs for screening quantitative factors. Biometrika 101(2):333–350
Tang Y, Xu H, Lin DKJ (2012) Uniform fractional factorial designs. Ann Stat 40:891–907
Wang Y, Fang KT (1981) A note on uniformdistribution and experimental design. Chin Sci Bull 26:485–489
Xu H (2005) A datalogue of three-level regular fractiional factorial designs. Metrika 62:259–281
Xu H, Wu CFJ (2001) Generalized minimum aberration for asymmetrical fractional factorial designs. Ann Stat 29:549–560
Xu H, Zhang J, Tang Y (2014) Level permutation method for constructing uniform designs under the wrap-around \(L_2\)-discrepancy. J Complex 30:46–53
Yang X, Yang G-J, Su Y-J (2018) Lower bound of a verage centered \(L_2\)-discrepancy for U-type designs. Commun Stat Theory Methods. https://doi.org/10.1080/03610926.2017.1422761
Zhou YD, Xu H (2014) Space-filling fractional factorial designs. J Am Stat Assoc 109(507):1134–1144
Zhou YD, Fang KT, Ning JH (2013) Mixture discrepancy for quasi-random point sets. J Complex 29:283–301
Acknowledgements
The authors greatly appreciate valuable comments and suggestions of the referees and the Associate Editor that significantly improved the paper. The authors greatly appreciate the kind support of Prof. Ping He during this work. This work was partially supported by the UIC Grants (Nos. R201409, R201712, R201810 and R201912) and the Zhuhai Premier Discipline Grant.
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Elsawah, A.M., Fang, KT. & Ke, X. New recommended designs for screening either qualitative or quantitative factors. Stat Papers 62, 267–307 (2021). https://doi.org/10.1007/s00362-019-01089-9
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DOI: https://doi.org/10.1007/s00362-019-01089-9
Keywords
- Design isomorphism
- Orthogonal designs
- Level permutation
- Projection
- Generalized word-length pattern
- Hamming distance pattern
- Uniformity criteria