Statistica Sinica 15(2005), 717-730

ON THE COSET PATTERN MATRICES AND

MINIMUM $\mbf M$-ABERRATION OF $\mbf{2^{n-p}}$ DESIGNS

Yu Zhu and Peng Zeng

Purdue University

Abstract: The coset pattern matrix (CPM) is formally defined as an elaborate characterization of the aliasing patterns of a fractional factorial design. The possibility of using CPM to check design isomorphism is investigated. Despite containing much information about effect aliasing, the CPM fails to determine a design uniquely. We report and discuss small nonisomorphic designs that have equivalent coset pattern matrices. These examples imply that the aliasing property and the combinatorial structure of a design depend on each other in a complex manner. Based on CPM, a new optimality criterion called the minimum $M$-aberration criterion is proposed to rank-order designs. Its connections with other existing optimality criteria are discussed.

Key words and phrases: Coset pattern matrix, Fractional factorial design, Isomorphism, Letter pattern matrix, Minimum M-aberration.