Reduced-rank models for interaction in unequally replicated two-way classifications

This paper extends reduced-rank regression models for application to interaction in unequally replicated two-way classifications having no empty cells. The a - b matrix of interaction parameters is modeled as a linear function of a rank-[varrho] matrix, where [varrho] <= min[(a - 1), (b - 1)]. For fixed [varrho], the maximum likelihood estimator of the interaction matrix is derived. The likelihood ratio test of [varrho] = m versus [varrho] = m + r and a union-intersection test of [varrho] = 0 versus [varrho] >= r are constructed and illustrated. The union-intersection test is particularly useful for simultaneous inference on product interaction contrasts.