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 F 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. Moreover, using mean aberrations for symmetric sm designs with s prime, we derive a new formula for computing the GWLP of F . It is computationally easy, does not use complex numbers and also provides a clear way to interpret 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 as a further tool to discriminate between fractions.

Aberration in qualitative multilevel designs / Fontana, Roberto; Rapallo, Fabio; Rogantin, Maria Piera. - In: JOURNAL OF STATISTICAL PLANNING AND INFERENCE. - ISSN 0378-3758. - STAMPA. - 174:(2016), pp. 1-10. [10.1016/j.jspi.2016.02.003]

Aberration in qualitative multilevel designs

FONTANA, ROBERTO;
2016

Abstract

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 F 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. Moreover, using mean aberrations for symmetric sm designs with s prime, we derive a new formula for computing the GWLP of F . It is computationally easy, does not use complex numbers and also provides a clear way to interpret 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 as a further tool to discriminate between fractions.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11583/2642342
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