Aberration in qualitative multilevel designs

F. Rapallo, R. Fontana, M. P. Rogantin

Generalized Word Length Pattern (GWLP) is an important and widely-used tool for comparing fractional factorial designs. We consider qualitative factors, and we code their levels using the roots of the unity. We write the GWLP of a fraction using the polynomial indicator function, whose coefficients encode many properties of the fraction. We show that the coefficient of a simple or interaction term can be written using the counts of its levels. This apparently simple remark leads to major consequence, including a convolution formula for the counts. We also show that the mean aberration of a term over the permutation of its levels provides a connection with the variance of the level counts. In the symmetric prime case, this theory leads to an alternative expression of the GWLP. As case studies, we consider non-isomorphic orthogonal arrays that have the same GWLP. The different distributions of the mean aberrations suggest that they could be used to discriminate between fractions.

Palabras clave: Algebraic statistics, Complex coding, Fractional factorial designs, Generalized word-length pattern, Indicator function

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