A theory on constructing 2n-m designs with general minimum lower order confounding

Abstract

When designing an experiment, it is important to choose a design that is optimal under model uncertainty. The general minimum lower-order confounding (GMC) criterion can be used to control aliasing among lower-order factorial effects. A characterization of GMC via complementary sets was considered in Zhang and Mukerjee (2009a); however, the problem of constructing GMC designs is only par- tially solved. We provide a solution for two-level factorial designs with n factors and N = 2 n−m runs subject to a restriction on (n, N ): 5N/16 + 1 ≤ n ≤ N − 1. The construction is quite simple: every GMC design, up to isomorphism, consists of the last n columns of the saturated 2 (N −1)−(N −1−n+m) design with Yates order. In addition, we prove that GMC designs differ from minimum aberration designs when (n, N ) satisfies either of the following conditions: (i) 5N/16 + 1 ≤ n ≤ N/2 − 4, or (ii) n ≥ N/2, 4 ≤ n + 2 r − N ≤ 2 r−1 − 4 with r ≥ 4.