Bayesian-inspired minimum contamination designs under a double-pair conditional effect model

In two-level fractional factorial designs,conditional main effects can provide insights by which to analyze factorial effects and facilitate the de-aliasing of fully aliased two-factor interactions.Con-ditional main effects are of particular interest in situations where some factors are nested within others.Most of the relevant literature has focused on the development of data analysis tools that use conditional main effects,while the issue of optimal factorial design for a given linear model involving conditional main effects has been largely overlooked.Mukerjee,Wu and Chang[Statist.Sinica 27(2017)997-1016]established a framework by which to optimize designs under a con-ditional effect model.Although theoretically sound,their results were limited to a single pair of conditional and conditioning factors.In this paper,we extend the applicability of their frame-work to double pairs of conditional and conditioning factors by providing the corresponding parameterization and effect hierarchy.We propose a minimum contamination-based criterion by which to evaluate designs and develop a complementary set theory to facilitate the search of minimum contamination designs.The catalogues of 16-and 32-run minimum contamination designs are provided.For five to twelve factors,we show that all 16-run minimum contamination designs under the conditional effect model are also minimum aberration according to Fries and Hunter[Technometrics 22(1980)601-608].