Allocating factors to the columns of an orthogonal array when certain interactions are important

Fractional factorial plans represented by orthogonal arrays of strength two are known to be universally optimal under a model that includes the mean and all main effects, when all other factorial effects are assumed to be absent. However, if the number of factors in the experiment is smaller than the number of columns of a saturated or, tight orthogonal array and the experimenter is interested in estimating certain 2-factor interactions as well, one can possibly entertain these 2-factor interactions in the model apart from the mean and all main effects. The problem then is to allocate factors to the columns of the orthogonal array, so that the user-specified 2-factor interactions, in addition to the mean and the main effects, are optimally estimable. This problem is investigated in this paper with reference to the orthogonal array OA(2n,2n-1,2,2)OA(2n,2n-1,2,2), which exists for every integer n⩾2n⩾2. A method for the allocation of factors to factor representations is proposed that ensures the optimal estimation of the mean, all main effects and specified 2-factor interactions. The method is illustrated by considering in detail the cases n=3,4n=3,4.