Optimal Blocking for Three Treatments and BIBD Robustness - Two Problems in Design Optimality

Abstract

Design optimality plays a central role in the area of statistical experimental design. In gen- eral, problems in design optimality are composed of two vital, but separable, components. One of these is determining conditions under which a design is optimal (such as criterion bounds, values of design parameters, or special structure in the information matrix). The other is construction of designs satisfying those conditions. Most papers deal with either optimality conditions, or design construction in accordance with desired combinatorial prop- erties, but not both. This dissertation determines optimal designs for three treatments in the one-way and multi-way heterogeneity settings, first proving optimality through a series of bounding arguments, then applying combinatorial techniques for their construction. Among the results established are optimality with respect to the well known E and A criteria. A- and E-optimal block designs and row-column designs with three treatments are found, for any parameter set. E-optimal hyperrectangles with three treatments are also found, for any parameter set. Systems of distinct representatives theory is used for the construction of optimal designs. Efficiencies relative to optimal criterion values are used to determine robustness of block designs against loss of a small number of blocks. Nonisomorphic bal- anced incomplete block designs are ranked based on their robustness. A complete list of most robust BIBDs for vâ ¤10, râ ¤15 is compiled.