Regular Fractions of Factorial Arrays
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.
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