Optimal step-type designs for comparing test treatments with a control

The problem of obtaining A-optimal designs for comparing v test treatments with a control in b blocks of size k each is considered. A condition on the parameters (u, b, k) is identified for which optimal step-type designs can be obtained. Families of such designs are given. Methods of searching for highly efficient designs are proposed for situations in which it is difficult to determine an A-optimal design. Under the usual additive homoscedastic model, an A-optimal design minimizes the average variance of the least squares estimators of the control-test treatment comparisons. Majumdar and Notz (1983) gave a method for finding A-optimal designs. Their optimal designs can basically be of two types, using the terminology of Hedayat and Majumdar (1984): rectangular (R), in which every block has the same number of replications of the control, and step (5), in which some blocks contain the control t times and the others t + 1 times. Optimal R-type designs were studied by Hedayat and Majumdar (1985). Families of such designs, particularly when each block has one replication of the control, were given in that paper. In this article, we intend to study optimal S-type designs. Step-type designs are more complicated than rectangular-type designs; the latter are balanced incomplete block designs in the test treatments augmented by an equal number of controls in each block, but the former do not have such a simple characterization. Consequently, both the optimality and the construction of such designs are more involved.