On the equivalence of definitions for regular fractions of mixed-level factorial designs

The notion of regularity for fractional factorial designs was originally defined only for two-level factorial designs. Recently, rather different definitions for regular fractions of mixed-level factorial designs have been proposed by Collombier [1996. Plans d’Expérience Factoriels. Springer, Berlin], Wu and Hamada [2000. Experiments. Wiley, New York] and Pistone and Rogantin [2008. Indicator function and complex coding for mixed fractional factorial designs. J. Statist. Plann. Inference 138, 787–802]. In this paper we prove that, surprisingly, these definitions are equivalent. The proof of equivalence relies heavily on the character theory of finite Abelian groups. The group-theoretic framework provides a unified approach to deal with mixed-level factorial designs and treat symmetric factorial designs as a special case. We show how within this framework each regular fraction is uniquely characterized by a defining relation as for two-level factorial designs. The framework also allows us to extend the result that every regular fraction is an orthogonal array of a strength that is related to its resolution, as stated in Dey and Mukerjee [1999. Fractional Factorial Plans. Wiley, New York] to mixed-level factorial designs.