Block Designs: Advances in Theory and Use

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A problem common to many disparate fields of scientific and industrial enquiry isthat of comparing a set of experimental conditions, or `treatments', when faced withheterogeneity in the units of material on which the experiment is to be performed. Thetechnique of blocking specifically addresses this problem. Typically using values of someidentifiable nuisance factor, the experimental units are partitioned into homogeneoussubsets called blocks. Inferences can then be based on the more precise comparisons ofmeasurements from the same block. The design problem is to determine which treat-ments are assigned to which units in which blocks, said assignment being driven bythe desire to maximize the quality of information that will ultimately be produced.The problem becomes more complicated as more nuisance factors, with various inter-relationships, are introduced or identified. This project addresses the design problemfor a variety of commonly encountered experimental settings, with two broad goals:(i) to extend the known theory for determining optimal designs for a wider range ofsettings than is now known or available; and (ii) to produce a comprehensive catalogof designs to be incorporated into a larger, web-based resource that will include a vari-ety of downloadable combinatorial designs useful for statisticians, mathematicians, andother scientists in academia and industry. By collecting optimal designs in a single,easily accessible resource, the use of good designs can be increased and the practice ofexperimental science consequently sharpened.When comparing v treatments, the most commonly encountered situation across arange of scientific endeavors is that of a single blocking variable partitioning bk experi-mental units into b blocks of k units each. This project will examine optimality problemsfor these simple block designs, and the attendant combinatorial issues, augmenting the-ory by computation where needed, to produce designs for a practical range of parametercombinations (v; b; k), including settings where equireplication is not possible. The mostfrequently used designs with two blocking factors are row-column designs, in which twoblocking factors can be visualized as row and columns in a rectangular array, and re-solvable designs, in which a second blocking factor partitions a simple block design intosubsets of blocks each consisting of a single replicate. This project will also addressproblems in optimality and construction of designs in both these classes, with specialemphasis on plans most demanded in practice: those with few replicates.