Aberration in qualitative multilevel designs

作者：Fontana Roberto; Rapallo Fabio^*; Rogantin Maria Piera

来源：Journal of Statistical Planning and Inference, 2016, 174: 1-10.

DOI：10.1016/j.jspi.2016.02.003

• Algebraic statistics
• Complex coding
• Counts of factor levels
• Fractional factorial designs
• Generalized word-length pattern
• Indicator function

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 s(m) 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.

• 出版日期2016-7