Journal of Statistical Theory and Practice

, Volume 1, Issue 1, pp 101–115 | Cite as

On the Geometric Equivalence and Non-equivalence of Symmetric Factorial Designs

  • T. I. KatsaounisEmail author
  • C. A. Dingus
  • A. M. Dean


Two factorial designs with quantitative factors are called geometrically equivalent if the design matrix of one can be transformed into the design matrix of the other by row and column permutations, and reversal of symbol order in one or more columns. In this paper, we compare two known methods for the determination of geometric equivalence and propose a modified method based on the “split weights” of the rows of a design matrix. We also propose and evaluate new screening methods for geometric non-equivalence. Most of the paper concentrates on symmetric designs with factors at three levels, but extensions to designs with factors at four or more levels and to asymmetric designs are indicated.

MSC 2000 subject classification



Deseq algorithm design equivalence design isomorphism factorial experiment geometric equivalence indicator function 


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  1. Ai, M.Y., Zhang, R.C., 2004. Projection justification of generalized minimum aberration for asymmetrical fractional factorial designs. Metrika 60, 279–285.MathSciNetCrossRefGoogle Scholar
  2. Chen, J., Sun, D.X., Wu, C.F.J., 1993. A catalogue of two-level and three-level fractional factorial designs with small runs. International Statistical Review 61, 131–145.CrossRefGoogle Scholar
  3. Cheng, S. W., Ye, K.Q., 2004. Geometric isomorphism and minimum aberration for factorial designs with quantitative factors. Ann. Statist. 32, 2168–2185.MathSciNetCrossRefGoogle Scholar
  4. Clark, J. B., Dean, A.M., 2001. Equivalence of fractional factorial designs. Statistica Sinica 11, 537–547.MathSciNetzbMATHGoogle Scholar
  5. Deng, L.Y., Tang, B., 2002. Design selection and classification for Hadamard matrices using generalized minimum aberration criteria. Technometrics 44, 173–184.MathSciNetCrossRefGoogle Scholar
  6. Draper, N.R., Mitchell, T.J., 1967. The construction of saturated 2Rkp designs Ann. Math. Statist. 38, 1110–1126.MathSciNetCrossRefGoogle Scholar
  7. Draper, N.R. Mitchell, T.J., 1968. Construction of the set of 256-run designs of resolution ≥5 and the set of even 512-run designs of resolution ≥6 with special reference to the unique saturated designs. Ann. Math. Statist. 39, 246–255.MathSciNetCrossRefGoogle Scholar
  8. Draper, N.R., Mitchell, T.J., 1970. Construction of a set of 512-run designs of resolution ≥5 and a set of even 1024-run designs of resolution ≥6. Ann. Math. Statist. 41, 876–887.MathSciNetCrossRefGoogle Scholar
  9. Evangelaras, H., Koukouvinos, C., Dean, A.M., Dingus, C.A., 2005. Projection properties of certain three level orthogonal arrays. Metrika 62, 241–257.MathSciNetCrossRefGoogle Scholar
  10. Fontana, R., Pistone, G., Rogantin, M.P., 2000. Classification of two-level factorial fractions. J. Statist. Plann. Inf. 87, 149–172.MathSciNetCrossRefGoogle Scholar
  11. Hickernell, F.J., 1998. A generalized discrepancy and quadrature error bound. Math. Computation 67, 299–322.MathSciNetCrossRefGoogle Scholar
  12. Johnson, R.A., Wichern, D.W., 2002. Applied Multivariate Statistical Analysis, 5th edition. Prentice Hall, New Jersey.zbMATHGoogle Scholar
  13. Katsaounis, T.I., Dean, A.M., 2007. A survey and evaluation of methods for determination of combinatorial equivalence of factorial designs. J. Statist. Plann. Inf. In press.Google Scholar
  14. Ma, C.X., Fang, K.T., Lin, D.K.J., 2001. On the isomorphism of fractional factorial designs. Journal of Complexity 17, 86–97.MathSciNetCrossRefGoogle Scholar
  15. MacWilliams, F.J., Sloane, N.J.A., 1977. The Theory of Error Correcting Codes. North Holland, Amsterdam.zbMATHGoogle Scholar
  16. Tang, B., 2001. Theory of J-characteristics for fractional factorial designs and projection justification of minimum G2-aberration. Biometrika 88, 401–407.MathSciNetCrossRefGoogle Scholar
  17. Xu, H., 2003. Minimum moment aberration for nonregular designs and supersaturated designs. Statistica Sinica 13, 691–708.MathSciNetzbMATHGoogle Scholar
  18. Xu, H., 2005. A catalogue of three-level regular fractional factorial designs. Metrika 62, 259–281.MathSciNetCrossRefGoogle Scholar
  19. Xu, H., Deng, L.Y., 2005. Moment aberration projection for nonregular fractional factorial designs. Technometrics 47, 121–131.MathSciNetCrossRefGoogle Scholar
  20. Ye, K.Q., 2003. Indicator function and its application in two-level factorial designs. Ann. Statist. 31, 984–994.MathSciNetCrossRefGoogle Scholar

Copyright information

© Grace Scientific Publishing 2007

Authors and Affiliations

  1. 1.Department of MathematicsThe Ohio State UniversityMansfieldUSA
  2. 2.Battelle Memorial InstituteColumbusUSA
  3. 3.Department of StatisticsThe Ohio State UniversityColumbusUSA

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