On the definition of effects in fractional factorial designs

Abstract

This paper simplifies a previous exposition of Rao's 1947 proof of his inequalities for orthogonal arrays. A key issue in the proof is the way in which one defines effects in a fractional factorial design. Here we replace the definition used in the earlier exposition with a simpler one, based on a more obvious interpretation of what Rao wrote and more in line with common practice, and show that it still leads to the same mathematical results. As in Rao's original paper, all designs are assumed to be unblocked. Two applications are given illustrating alias patterns in certain nonregular fractional designs, the second affording an opportunity to compare this approach with an alternative one due to Box and Wilson.