General Theory of Minimum Aberration and its Applications

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Proposal ID: DMS-0204594PI: Boxin TangTitle: General theory of minimum aberration and its applicationsAbstractThe investigator and his colleagues consider the problem of design selection for fractional factorials. In situations where there is little or no knowledge about effects that are potentially important, it is appropriate to select designs using the minimum aberration criterion. Very often, the experimenter may have reasons to believe that certain two-factor interactions are important. Appropriate designs under such circumstances are those allowing joint estimation of all main effects and these potentially important interactions.If the effects not in the postulated model, which consists of the main effects and potentially important interactions, cannot be completely ignored, they bias the estimates of the effects in the model. In the proposed research, this problem of design selection is solved by developing a general theory of minimum aberration. The general criterion of aberration to be introduced generalizes the usual criterion of minimum aberration, and it has the robust property that the best designs given by this criterion sequentially minimize the contamination of nonnegligible effects on the estimation of the effects in the postulated model. Developing a general theory of minimum aberration is also motivated by the desire of unifying various versions of minimum aberration in the literature. Is it possible to derive these versions of minimum aberration from a general theory that is based on a sound statistical principle? The general theory of aberration provides a positive answer to this question. What is more important is that such a general theory enables researchers to derive other versions of minimum aberration that may be more appropriate for given design situations. In the proposal, many research problems are discussed and in some cases, solutions are outlined. Design of experiments is an area of study in statistics that concerns efficient data collection for industrial experiments and many other areas of scientific investigation. Fractional factorial designs are a class of experimental plans and their importance, both theoretical and practical, cannot be overstated. As exploratory designs, they are directly useful in identifying important factors at the early stage of an investigation. They also form a basis on which other more sophisticated designs can be built. In the proposed research, the investigator and his colleagues study the problem of selecting the best fractional factorial designs when certain prior knowledge regarding the effects of factors is available. The problem is solved by developing a general theory of design selection criteria. In addition to solving the above practical problem, the general theory unifies various existing design selection criteria, and allows researchers to derive other criteria that are more appropriate for their situations. Involving students in research activities is an important aspect of the proposed research. The proposed research sheds new light on the study of fractional factorial designs, leads to new economical designs for the experiments in the physical, chemical, and engineering sciences, and most importantly, promotes the application of factorial designs in the areas such as biotechnology and medical research that have huge potential to benefit further from the design methodology.