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Regular Fractions of Factorial Arrays

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mODa 11 - Advances in Model-Oriented Design and Analysis

Part of the book series: Contributions to Statistics ((CONTRIB.STAT.))

Abstract

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.

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Acknowledgements

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.

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Authors and Affiliations

  1. Beuth University of Applied Sciences Berlin, Dep. II, Luxemburger Str. 10, 13353, Berlin, Germany

    Ulrike Grömping

  2. School of Mathematics & Statistics, University of St Andrews, St Andrews, KY16 9SS, UK

    R. A. Bailey

Authors
  1. Ulrike Grömping
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  2. R. A. Bailey
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Corresponding author

Correspondence to Ulrike Grömping .

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Editors and Affiliations

  1. Department of Statistics, Technische Universität Dortmund, Dortmund, Germany

    Joachim Kunert

  2. Department of Statistics, Technische Universität Dortmund, Dortmund, Germany

    Christine H. Müller

  3. Department of Statistics, London School of Economics, London, United Kingdom

    Anthony C. Atkinson

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Grömping, U., Bailey, R.A. (2016). Regular Fractions of Factorial Arrays. In: Kunert, J., Müller, C., Atkinson, A. (eds) mODa 11 - Advances in Model-Oriented Design and Analysis. Contributions to Statistics. Springer, Cham. https://doi.org/10.1007/978-3-319-31266-8_17

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