Symmetry and Asymmetry in Experimental Design

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Blocking of experimental material is a fundamental device for variance reduction in the design of comparative experiments. The investigator is exploring efficient applications of this principle from three distinct perspectives based on symmetry/asymmetry notions in the treatments to be compared, in the available experimental material, and in external restrictions on the set of permissible designs. Conventional design optimality theory rests strongly on the assumption of equality of treatment interest, expressed through summary measures of experimental information that are invariant to treatment permutation. The first goal is to develop a coherent optimality theory for evaluating designs when permutability is not tenable, that is, when experimenters are faced with asymmetry in treatment interest. Equality of treatment replications is another conventional symmetry that, depending on the number of blocks available and their sizes, can result in underutilization of resources and consequent reduction of information. The second goal is to develop theory for dealing with asymmetry of treatment replication in a comprehensive fashion. In some experiments it is demanded that blocks be partitioned into subsets so that each treatment occurs exactly once in each subset; a design for this situation is said to be resolvable. Resolvability is an added symmetry in treatment assignment, over that ordinarily required of a block design. The third goal is to extend theory for optimal resolvable designs to cover numbers of treatments, replicates and block sizes for which there are currently no available results.Collectively, the investigator is pursuing an approach to the design of comparative experiments that, by virtue of wider applicability, can compile significantly expanded design catalogs as resources for experimenters in a wide range of disciplines. A component of each of the problems described above, as the optimality theory evolves, is the building and compiling of designs for many combinations of blocks and treatments, and cataloging these online for free access by researchers, experimenters, and statistical practitioners. Making good designs easily available provides a basic tool for more efficient experimentation, aiding progress in scientific advancement and industrial innovation by increasing the quality of information that experiments can produce.