Characterizations for Orthogonal Factorial Structure

In Chapter 3, we have considered several characterizations for factorial designs which are balanced and have OFS. The emphasis on balance, however, has a drawback that the resulting designs, although theoretically elegant and simple to interpret, may become too large and hence expensive. More specifically, the combinatorial restrictions imposed by an attempt to achieve a balance over all the interactions (cf Theorem 3.2.2) may call for a prohibitively large number of replications. Because of this reason, since the early seventies, work started on conditions for OFS alone. A brief review of these developments up to that stage was given by Chatterjee (1982). In this chapter, we propose to present a comprehensive account of several characterizations for OFS considering both connected and disconnected designs. The use of these results for construction purposes will be indicated later in the monograph. Before presenting the main results of this chapter, we introduce some preliminaries that will be useful in the subsequent development.