On maximizing a design estimation efficiency using hidden projection properties

Hadamard matrices have traditionally been used for screening main effects only, because of their complex aliasing structures. The hidden projection property, as introduced by Wang and Wu (Statistica Sinica 5 (1995) 235-250), suggests that complex aliasing allows some interactions to be entertained and estimated without making additional runs in order to form full or fractional factorial designs as the geometric approach inclines. However, in most cases, because the original data are sufficiently noisy to mask the significance of any two-factor interaction we need to add (fewer) runs that give the maximum amount of information for this purpose. In this paper, we list what runs give the maximum amount of information needed in order to entertain and estimate some two-factor interactions for all the inequivalent projections of Hadamard matrices of order n=16,20 and 24 when they are projected into four and five dimensions.