Aberration in qualitative multilevel designs

TL;DR

This paper introduces new methods for analyzing the Generalized Word Length Pattern (GWLP) in qualitative multilevel designs, providing computationally simple formulas and interpretative tools that enhance discrimination between similar fractional factorial designs.

Contribution

It develops a convolution formula for level counts, links mean aberrations to variance, and derives a new GWLP computation method for symmetric prime-level designs.

Findings

New convolution formula for counts

Connection between mean aberration and variance

Efficient GWLP computation for symmetric designs

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 ${\mathcal F}$ 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 ${\mathcal F}$. 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.