Mathematical Sciences: Problems in Simple and Complex Block Designs

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DMS 9626115 Morgan This project examines design problems for comparing a set of treatments under a variety of blocking structures. The simplest of these is that of a single blocking variable which partitions n experimental units into blocks of k units each, k being less than the total number of treatments. It has long been believed that a necessary condition for a good design for this structure is that the assignment be binary. This project is expanding the scope for which the alternative idea of nonbinary treatment assignment can yield good designs, where "good" may be expressed in terms of the standard mathematically formulated optimality criteria and, in some cases, ability of the design to maintain important treatment structure in the information matrix. For situations with more than one blocking factor present, the range of conceivable relationships among them is quite large. The investigator is also studying the interplay between nesting, crossing, and other blocking factors, with the goal of identifying classes of these more complex structures for which optimal design conditions can be determined with existing methods. Design constructions are being formulated for each setting as well. %%% The general problem of comparing a set of experimental conditions, or "treatments," when faced with heterogeneity in the available units of material on which the experiment is to be performed, is common to many disparate fields of scientific and industrial inquiry. The technique of blocking, one of the early great contributions of statistics and the experimental process, specifically addresses this problem. Typically using values of some identifiable nuisance factor, the experimental units are partitioned into homogeneous subsets called blocks. Comparison of measurements occurring in the same block can then be made with greater precision than those in different blocks, the heterogeneity having been eliminated from such compar isons. Thus the experiment is improved, in the very practical sense that it gives information that is more reliable. The design problem itself, which is the topic of this study, is to determine which treatments are assigned to which units in which blocks, said assignment being driven by the desire to maximize the overall quality of information the experiment will ultimately produce. The problem naturally becomes more complicated as more nuisance factors, with various relationships among one another, are introduced or identified. By studying the situation mathematically, large families of good designs are produced which are then applicable to a host of experimental situations. Applications run the gamut of experimental inquiry, from the manufacturing realm to the agricultural. ***