Characterizations for Balance with Orthogonal Factorial Structure

Abstract

This chapter considers factorial designs which are balanced and have orthogonal factorial structure (OFS). Such designs have been termed balanced factorial experiments by Shah (1958, 1960a). They are also known as balanced confounded designs according to the nomenclature of Nair and Rao (1948). The main result of this section, namely Theorem 3.1.1, gives an algebraic characterization for balance with OFS in the connected case. For equireplicate and proper designs, the ‘sufficiency’ part of this result was proved by Kurkjian and Zelen (1963), while the ‘necessity’ part was proved by Kshirsagar (1966). Gupta (1983a) considered extensions to designs that are not necessarily equireplicate or proper. The following definition and lemmas will be helpful.