An integrative framework for geometric and hidden projections in three-level fractional factorial designs

Abstract

Geometric and hidden projection are two important properties to consider for three-level fractional factorials. The effective development of methods that can address tasks involving both these properties requires a unified design framework that integrates them. We highlight one integrative framework that is based on indicator functions for three-level designs under the linear-quadratic parameterization system. We demonstrate the broad scope and utility of this framework by using it to develop a new procedure for addressing a task comprised of three serial problems in geometric and hidden projection. The first in the series is to construct the sets of smallest follow-up runs to a projection of a three-level design that yield specified aliasing relations. Second is to immediately characterize the partial aliasing relations among factorial effects in the resulting augmented designs. The final problem is to identify the augmented designs that maximize popular optimality criteria with respect to the second-order model among all of the candidate augmentations. The application of our procedure is illustrated with sample projections of definitive screening designs onto four, five, and six factors. The integrative framework that we present can ultimately facilitate the systematic development of methods that advance comprehensive understanding of multiple properties for general three-level designs.