Characterization of three-order confounding via consulting sets

The aliased effect-number pattern, proposed by Zhang et al. (Stat Sin 18:1689–1705, 2008), is often used to express the overall confounding between factorial effects in two-level regular designs. The confounding relationships of main effects and two-factor interactions have been well revealed in literature, but little is known about three and higher-order interactions. To fill the gaps, this paper aims to study the confounding of three-order interactions in any two-level design. When the factor number of the design is larger, we derive the confounding among lower-order and three-order interactions via the complementary method. Further, confounding formulas can be obtained owing to the consulting sets of the structured designs. These two approaches can cover all designs in the sense of isomorphism. As an application of the previous results, the confounding numbers of three-order interactions are tabulated for 16, 32, and 64-run optimal designs under the general minimum lower-order confounding criterion.