On the Theory of Connected Designs: Characterization and Optimality

Abstract

Connectedness is an important property which every block design must possess if it is to provide an unbiased estimator for all elementary treatment contrasts under the usual linear additive model. We have classified the family of connected designs into three subclasses: locally connected, globally connected and pseudo-globally connected designs. Basically, a locally connected design is one in which not all the observations participate in the estimation. A globally connected design is one in which all observations participate in the estimation. Finally, a pseudo-globally 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 researchers in the field of optimum design theory. To be specific, let BD $\{v, b, (r_i), (k_u)\}$ denote a block design on a set of $v$ treatments with $b$ blocks of size $k_u, u = 1,2, \cdots, b$ and treatment $i$ is replicated $r_i$ times. Then we have shown that for the family of connected block designs BD $\{v, b, (r_i), k\}$ 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 $r_i > 1$ 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.