On the Theory of Connected Designs: Characterization and Optimality

connected design is a compromise between locally and globally connected designs. Theorems and corollaries are given which characterize the different classes of connected designs. In our discussion on the optimality of connected designs we show that there is much to be gained by partitioning the family of connected designs in the above fashion. Our optimality criteria are S-optimality suggested by Shah, which selects the design with minimum trace of the information matrix squared and (M, S)-optimality which selects the S optimal design from the class of designs with maximum trace of the information matrix. Using these optimality criteria, we have been able to derive some new results which we hope to be of interest to the users and reseachers in the field of optimum design theory. To be specific, let BD {v, b, (rj),(k.)} denote a block design on a set of v treatments with b blocks of size k., u = 1, 2, .. , b and treatment i is replicated ri times. Then we have shown that for the family of connected block designs BD {v, b, (ri), k} with (i) less than k - 1 treatments having replication equal to one and binary (0, 1) the S-optimum design is pseudo-globally connected; (ii) the S-optimum design is globally connected if ri > 1 and the designs are binary; and (iii) at least one treatment with replication greater than b, then the (M, S)-optimum design is pseudo-globally connected. In the final part of this paper we mention some unsolved problems in this area.