Abstract: Two-level factorial designs are considered under a conditional model with a pair of conditional and conditioning factors. Such a pair can arise in many practi- cal situations. With properly defined main effects and interactions, an appropriate effect hierarchy is introduced under the conditional model. A complementary set theory as well as an efficient computational procedure, supported by a powerful re- cursion relation, are developed to implement the resulting design strategy, leading to minimum aberration designs. This calls for careful handling of many new and subtle features of the conditional model as compared to the traditional one.

Key words and phrases: Bias, complementary set, effect hierarchy, model robust- ness, orthogonal array, regular design, universal optimality, wordlength pattern.