Invariance of generalized wordlength patterns

TL;DR

This paper proves that the generalized wordlength pattern (GWLP) of a fractional factorial design remains invariant regardless of the underlying abelian group structure used for indexing, unlike the J-characteristics.

Contribution

It demonstrates the invariance of GWLP under different group structures, clarifying the foundational assumptions of design analysis.

Findings

GWLP is independent of the choice of abelian group structure

J-characteristics depend on the specific group structure used

Implications for design theory and analysis methods

Abstract

The generalized wordlength pattern (GWLP) introduced by Xu and Wu (2001) for an arbitrary fractional factorial design allows one to extend the use of the minimum aberration criterion to such designs. Ai and Zhang (2004) defined the $J$-characteristics of a design and showed that they uniquely determine the design. While both the GWLP and the $J$-characteristics require indexing the levels of each factor by a cyclic group, we see that the definitions carry over with appropriate changes if instead one uses an arbitrary abelian group. This means that the original definitions rest on an arbitrary choice of group structure. We show that the GWLP of a design is independent of this choice, but that the $J$-characteristics are not. We briefly discuss some implications of these results.