New recommended designs for screening either qualitative or quantitative factors

  • A. M. Elsawah
  • Kai-Tai FangEmail author
  • Xiao Ke
Regular Article


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.


Design isomorphism Orthogonal designs Level permutation Projection Generalized word-length pattern Hamming distance pattern Uniformity criteria 

Mathematics Subject Classification

62K05 62K15 94B05 



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.

Supplementary material

362_2019_1089_MOESM1_ESM.pdf (221 kb)
Supplementary material 1 (pdf 221 KB)


  1. Angelopoulos P, Evangelaras H, Koukouvinos C (2009) Model identification using 27 runs three level orthogonal arrays. J Appl Stat 36:33–38MathSciNetCrossRefzbMATHGoogle Scholar
  2. Chen J, Sun DX, Wu CFJ (1993) A catalogue of two-level and three-level fractional factorial designs with small runs. Int Stat Rev 61:131–135CrossRefzbMATHGoogle Scholar
  3. Chen W, Qi ZF, Zhou YD (2015) Constructing uniform designs under mixture discrepancy. Stat Probab Lett 97:76–82MathSciNetCrossRefzbMATHGoogle Scholar
  4. Clark JB, Dean AM (2001) Equivalence of fractional factorial designs. Stat Sin 11:537–547MathSciNetzbMATHGoogle Scholar
  5. Dey A, Mukerjee R (1999) Fractional factorial plans. Wiley, New YorkCrossRefzbMATHGoogle Scholar
  6. Elsawah AM (2017a) A closer look at de-aliasing effects using an efficient foldover technique. Statistics 51(3):532–557MathSciNetCrossRefzbMATHGoogle Scholar
  7. Elsawah AM (2017b) A powerful and efficient algorithm for breaking the links between aliased effects in asymmetric designs. Aust NZ J Stat 59(1):17–41MathSciNetCrossRefzbMATHGoogle Scholar
  8. Elsawah AM (2017c) Constructing optimal router bit life sequential experimental designs: new results with a case study. Commun Stat Simul Comput.
  9. Elsawah AM (2018) Choice of optimal second stage designs in two-stage experiments. Comput Stat 33(2):933–965MathSciNetCrossRefzbMATHGoogle Scholar
  10. Elsawah AM, Qin H (2015a) Mixture discrepancy on symmetric balanced designs. Stat Probab Lett 104:123–132MathSciNetCrossRefzbMATHGoogle Scholar
  11. Elsawah AM, Qin H (2015b) A new strategy for optimal foldover two-level designs. Stat Probab Lett 103:116–126MathSciNetCrossRefzbMATHGoogle Scholar
  12. Elsawah AM, Qin H (2016) Asymmetric uniform designs based on mixture discrepancy. J Appl Stat 43(12):2280–2294MathSciNetCrossRefGoogle Scholar
  13. Elsawah AM, Fang KT (2018) A catalog of optimal foldover plans for constructing U-uniform minimum aberration four-level combined designs. J Appl Stat.
  14. Evangelaras H, Koukouvinos C, Dean AM, Dingus CA (2005) Projection properties of certain three level orthogonal arrays. Metrika 62:241–257MathSciNetCrossRefzbMATHGoogle Scholar
  15. Evangelaras H, Koukouvinos C, Lappas E (2007) 18-run nonisomorphic three level orthogonal arrays. Metrika 66:31–37MathSciNetCrossRefzbMATHGoogle Scholar
  16. Fang KT, Zhang A (2004) Minimum aberration Majorization in non-isomorphic saturated asymmetric designs. J Stat Plan Inference 126:337–346CrossRefzbMATHGoogle Scholar
  17. Fang KT, Tang Y, Yin JX (2008) Lower bounds of various criteria in experimental designs. J Stat Plan Inference 138:184–195MathSciNetCrossRefzbMATHGoogle Scholar
  18. Fang KT, Ke X, Elsawah AM (2017) Construction of uniform designs via an adjusted threshold accepting algorithm. J Complex 43:28–37MathSciNetCrossRefzbMATHGoogle Scholar
  19. Fries A, Hunter WG (1980) Minimum aberration \(2^{k-p}\) designs. Technometrics 8:601–608zbMATHGoogle Scholar
  20. Hall M Jr (1961) Hadamard matrix of order 16. Jet Propuls Lab Res Summ 1:21–26Google Scholar
  21. Hedayat AS, Sloane NJ, Stufken J (1999) Orthogonal arrays: theory and application. Springer, BerlinCrossRefzbMATHGoogle Scholar
  22. Hickernell FJ (1998a) A generalized discrepancy and quadrature error bound. Math Comput 67:299–322MathSciNetCrossRefzbMATHGoogle Scholar
  23. 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 109-166Google Scholar
  24. Lam C, Tonchev VD (1996) Classification of affine resolvable \(2-(27, 9, 4)\) designs. J Stat Plan Inference 56:187–202MathSciNetCrossRefzbMATHGoogle Scholar
  25. Ma CX, Fang KT (2001) A note on generalized aberration in fractional designs. Metrika 53:85–93MathSciNetCrossRefzbMATHGoogle Scholar
  26. Ma CX, Fang KT, Lin DKJ (2001) On isomorphism of factorial designs. J Complex 17:86–97MathSciNetCrossRefzbMATHGoogle Scholar
  27. Sartono B, Goos P, Schoen ED (2012) Classification of three-level strength-3 arrays. J Stat Plan Inference 142(4):794–809MathSciNetCrossRefzbMATHGoogle Scholar
  28. Sun DX, Wu CFJ (1993) Statistical properties of Hadamard matrices of order 16. In: Kuo W (ed) Quality through engineering design. Elsevier, Amsterdam, pp 169–179Google Scholar
  29. Tang B, Deng LY (2001) Minimum \(G_2\)-aberration for nonregular fractional factorial designs. Ann Stat 27:1914–1926zbMATHGoogle Scholar
  30. Tang Y, Xu H (2013) An effective construction method for multi-level uniform designs. J Stat Plan Inference 143:1583–1589MathSciNetCrossRefzbMATHGoogle Scholar
  31. Tang Y, Xu H (2014) Permuting regular fractional factorial designs for screening quantitative factors. Biometrika 101(2):333–350MathSciNetCrossRefzbMATHGoogle Scholar
  32. Tang Y, Xu H, Lin DKJ (2012) Uniform fractional factorial designs. Ann Stat 40:891–907MathSciNetCrossRefzbMATHGoogle Scholar
  33. Wang Y, Fang KT (1981) A note on uniformdistribution and experimental design. Chin Sci Bull 26:485–489Google Scholar
  34. Xu H (2005) A datalogue of three-level regular fractiional factorial designs. Metrika 62:259–281MathSciNetCrossRefGoogle Scholar
  35. Xu H, Wu CFJ (2001) Generalized minimum aberration for asymmetrical fractional factorial designs. Ann Stat 29:549–560MathSciNetCrossRefzbMATHGoogle Scholar
  36. Xu H, Zhang J, Tang Y (2014) Level permutation method for constructing uniform designs under the wrap-around \(L_2\)-discrepancy. J Complex 30:46–53CrossRefzbMATHGoogle Scholar
  37. 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.
  38. Zhou YD, Xu H (2014) Space-filling fractional factorial designs. J Am Stat Assoc 109(507):1134–1144MathSciNetCrossRefzbMATHGoogle Scholar
  39. Zhou YD, Fang KT, Ning JH (2013) Mixture discrepancy for quasi-random point sets. J Complex 29:283–301MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of ScienceZagazig UniversityZagazigEgypt
  2. 2.Division of Science and TechnologyBNU-HKBU United International CollegeZhuhaiChina
  3. 3.The Key Lab of Random Complex Structures and Data AnalysisThe Chinese Academy of SciencesBeijingChina
  4. 4.Department of MathematicsSouthern University of Science and TechnologyShenzhenChina
  5. 5.Department of MathematicsHong Kong Baptist UniversityKowloon TongHong Kong

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