Recent researches show that indicator functions are very effective tools for studying theoretical properties of factorial designs. Initially proposed by Fontana, Pistone and Rogantin (2000) and later modified and generalized by the investigator and his collaborators, indicator functions provide a unified representation for all factorial designs, regular or non-regular, two-level, multi-level or mixed-level. The aliasing structure of a factorial design could be clearly revealed by expanding its indicator function with respect to an orthogonal polynomial basis. An important application of this approach is to study geometric structures of orthogonal arrays, which are essential for response surface methodology but have been largely overlooked in the past. The proposed research seeks not only in depth understanding of factorial designs but also discovery of many new efficient designs. The project pursues three main tasks. First, continue development of theoretical foundation of the indicator function approach and apply it to different types of designs including blocked designs and split-plot designs. Second, generalize minimum aberration criteria to all factorial designs. The criteria should be based on the intrinsic structure of a design but not on a priori specified models. Links to design efficiency and estimation capacity will be thoroughly studied, both theoretically and empirically. Third, develop and apply a sequential algorithm to construct a complete catalogue of non-isomorphic orthogonal arrays of small run sizes.
Factorial designs are vital for investigations in both life and physical sciences, as well as in agricultural and industrial studies. Non-regular designs have an advantage over regular designs at the screening and exploration stage of an investigation. Besides many traditional areas that factorial designs are applied, one particular area that will benefit is biomedical research, in which exploration and screening investigations are pervasive, especially with new technologies such as microarrays. Many new, more efficient designs are to be generated in the proposed research and be disseminated through interdisciplinary collaboration with other members of the scientific communities and interactions with industries. An integrated part of the proposed research is to adopt the indicator function approach in the curriculum of standard experimental design courses. Such change from current curriculum will promote the future applications of non-regular designs in practice. The investigator is also committed to involve students in the proposed research, especially those from underrepresented groups. In addition, the proposed research is expected to open interdisciplinary communication between algebraic geometricians and statisticians.
