Orthogonalized moment aberration for mixed-level multi-stratum factorial designs with partially-relaxed orthogonal block structures

Abstract Multi-stratum factorial designs, such as block designs and row–column designs, are widely used for screening treatment factors in experiments involving complex structures of experimental units due to multiple sources of error. In this study, we propose a unified model-free approach, termed orthogonalized moment aberration, to compare the similarities between level combinations of treatment factors assigned to heterogeneous experimental units. The proposed approach, which uses kernel functions to evaluate the rows of design matrices rather than the columns, can assess a wide variety of mixed-level regular/nonregular factorial designs with an extensive class of heterogeneous experimental unit structures called partially-relaxed orthogonal block structures. This approach is flexible in that it can be adapted to various scenarios by choosing different kernel functions, with certain choices yielding well-known minimum aberration criteria proposed in the literature. Although model-free, the proposed method is justified by using linear mixed-effect models and Gaussian process models. Theoretical results and numerical examples presented in this article collectively demonstrate that the proposed approach can generate multi-stratum factorial designs with high D-efficiencies within a Bayesian framework.