Frequency tables for the coding-invariant quality assessment of factorial designs

Quality assessment of factorial designs, particularly mixed-level factorial designs, is a nontrivial task. Existing methods for orthogonal arrays include generalized minimum aberration, a modification thereof that was proposed by Wu and Zhang for mixed two- and four-level arrays, and minimum projection aberration. For supersaturated designs, E(s2) or χ2-based criteria are widely used. Based on recent insights by Grömping and Xu regarding the interpretation of the projected aR values used in minimum projection aberration, this article proposes three new types of frequency tables for assessing the quality of level-balanced factorial designs. These are coding invariant, which is particularly important for designs with qualitative factors. The proposed tables are used in the same way as those used in minimum projection aberration and behave more favorably when used for mixed-level arrays. Furthermore, they are much more manageable than the above-mentioned approach by Wu and Zhang. The article justifies the proposed tables based on their statistical information content, makes recommendations for their use, and compares them with each other and with existing criteria. As a byproduct, it is shown that generalized minimum aberration refines the established expected χ2 criterion for level-balanced supersaturated designs.