Aberration in qualitative multilevel designs

Abstract

Generalized Word Length Pattern (GWLP) is an important and widelyused 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.

On the aberrations of mixed level orthogonal arrays with removed runs

“…Finally, the connections between our approach based on the aberrations and the D-optimality criterion for some classical statistical models are worth exploring. Some efficient algorithms for finding such Orthogonal Arrays would be helpful, for example by exploiting the notion of mean aberration for mixed level Orthogonal Arrays introduced in [7].…”

“…Given a design point f in a factorial design D = D 1 × · · · × D m , the j-th element of the GWLP of f is(7) A j (f ) = A={a1,...,aj} A⊆{1,...,m} (s a1 − 1) · · · (s aj − 1) , where s 1 , . .…”

Algebraic characterization of regular fractions under level permutations

“…In this section we present some relevant results of the algebraic theory of fractional designs. The interested reader can find further information, including the proofs of the propositions, in [8], [20], [9].…”

“…A necessary condition for isomorphism between two fractions is that they have the same Generalized Word Length Pattern (GWLP) and this condition does not depend on the level coding, see [23]. [9] derive a formula, computationally easy and of clear interpretation, for computing the GWLP of a fraction, based on the mean aberration.…”

Unions of Orthogonal Arrays and Their Aberrations via Hilbert Bases

“…In this work we use results from Combinatorics and Algebraic Geometry to ease the computation of the GWLP. The connection between the GWLP and the geometric structure of the design points is studied in [9], but we adopt here a different point of view. In particular, we show that the set of all OAs with given strength form are the points with integer entries of a cone defined through linear constraints.…”