Regular Fractions of Factorial Arrays

  • Ulrike GrömpingEmail author
  • R. A. Bailey
Conference paper
Part of the Contributions to Statistics book series (CONTRIB.STAT.)


For symmetric arrays of two-level factors, a regular fraction is a well-defined concept, which has been generalized in various ways to arrays of s-level factors with s a prime or prime power, and also to mixed-level arrays with arbitrary numbers of factor levels. This paper introduces three further related definitions of a regular fraction for a general array, based on squared canonical correlations or the commuting of projectors. All classical regularity definitions imply regularity under the new definitions, which also permit further arrays to be considered regular. As a particularly natural example, non-cyclic Latin squares, which are not regular under several classical regularity definitions, are regular fractions under the proposed definitions. This and further examples illustrate the different regularity concepts.



Ulrike Grömping’s initial work was supported by Deutsche Forschungsgemeinschaft (Grant GR 3843/1-1). Ulrike Grömping wishes to thank Hongquan Xu for fruitful discussions on an earlier version of this work. The collaboration with Rosemary Bailey was initiated at a workshop funded by Collaborative Research Center 823 at TU Dortmund University.


  1. 1.
    Bailey, R.A.: Orthogonal partitions in designed experiments. Des. Codes Cryptogr. 8, 54–77 (1996)zbMATHGoogle Scholar
  2. 2.
    Bernstein, D.S.: Matrix Mathematics. Princeton University Press, Princeton (2005)Google Scholar
  3. 3.
    Bose, R.C.: Mathematical theory of the symmetrical factorial design. Sankhya 8, 107–166 (1947)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Brien, C.: dae: functions useful in the design and ANOVA of experiments. R package version 2.7-2. (2015)
  5. 5.
    Dey, A., Mukerjee, R.: Fractional Factorial Plans. Wiley, New York (1999)CrossRefzbMATHGoogle Scholar
  6. 6.
    Eendebak, P., Schoen, E.: Complete series of non-isomorphic orthogonal arrays. (2010)
  7. 7.
    Grömping, U.: Frequency tables for the coding invariant ranking of orthogonal arrays. (2013)
  8. 8.
    Grömping, U.: DoE.base: Full factorials, orthogonal arrays and base utilities for DoE packages. R package version 0.27-1, (2015)
  9. 9.
    Grömping, U., Xu, H.: Generalized resolution for orthogonal arrays. Ann. Stat. 42, 918–939 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    James, A.T., Wilkinson, G.N.: Factorization of the residual operator and canonical decomposition of nonorthogonal factors in the analysis of variance. Biometrika 58, 279–294 (1971)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Kobilinsky, A., Bouvier, A., Monod, H.: planor: Generation of regular factorial designs. R package version 0.2-3. (2015)
  12. 12.
    Kobilinsky, A., Monod, H., Bailey, R.A.: Automatic generation of generalised regular factorial designs. Preprint Series 7, Isaac Newton Institute, Cambridge (2015)Google Scholar
  13. 13.
    Kuhfeld, W.: Orthogonal arrays. (2010)
  14. 14.
    Puntanen, S., Styan, G.P.H., Isotalo, J.: Matrix Tricks for Linear Statistical Models. Springer, Heidelberg (2011)CrossRefzbMATHGoogle Scholar
  15. 15.
    Tjur, T.: Analysis of variance models in orthogonal designs. Int. Stat. Rev. 52, 33–81 (1984)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Xu, H., Wu, C.F.J.: Generalized minimum aberration for asymmetrical fractional factorial designs. Ann. Stat. 29, 1066–1077 (2001)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Beuth University of Applied Sciences BerlinBerlinGermany
  2. 2.School of Mathematics & StatisticsUniversity of St AndrewsSt AndrewsUK

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