Inter-effect Orthogonality and Optimality in Hierarchical Models

The study of optimal fractional factorial plans has received considerable attention over the last two decades. The universal optimality of plans based on orthogonal arrays was shown by Cheng (1980) and Mukerjee (1982). Various extensions of this result have also been reported in the literature; see Mukerjee (1995) for a brief review. Most of these results relate to situations where all factorial effects involving the same number of factors are considered equally important and, as such, the underlying model involves the general mean and all effects involving up to a specified number of factors. The aforesaid presumption of equality in the importance of all factorial effects involving the same number of factors is, however, not tenable in many practical situations. For example, there may be a reason to believe that only one of the factors can possibly interact with the others and that interactions involving three or more factors are absent. The model then involves the general mean, all main effects and only some but not all two–factor interactions. The issue of optimality in situations of this kind for two–level factorials has been addressed recently by Hedayat and Pesotan (1992, 1997) and Chiu and John (1998); see also Wu and