Study on Order-of-Addition Experiments

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In many experiments, a number of reactants are added into the system sequentially rather than simultaneously. The formation, e.g., the amount/size/purity of the reaction products often depend on the order-of-addition (OofA) of different reactants. The OofA experiment can be traced back to Fisher (1937), where a lady was able to distinguish (by tasting) from whether the tea or the milk was first added to the cup. This is probably the first popular OofA experiment. In general, there are m required components and one hopes to determine the optimal sequence for adding these m components one after another. Knowing the optimal addition order of components related in production is crucial. Note that there are m! possible orders to be compared-for the tea-tasting example, there are m!=2!=2 possible orders. However, it is often unaffordable to test all the m! treatments, and the design problem arises (for example, when m = 10, m! is about 3.6 millions). Although it is somewhat primitive in Statistics, the OofA effect is frequently mentioned in many areas during the past decades. This includes bio-chemistry (Shinohara and Ogawa 1998), food science (Jourdain et al. 2009), nutritional science (Karim et al. 2000), and pharmaceutical science (Rajaonarivony et al. 1993), just to name a few. As the order-of-addition effect occurs in all kinds of real experiments as well as computer experiments, successful outcomes of this project can be directly used in many different areas, such as chemistry and biology. Furthermore, the proposed problems provide a golden opportunity for fundamental academic studies in other subjects.In this project, the PI plans to establish the fundamental theory of evaluating OofA designs. The PI also plans to construct classes of OofA designs, under different requirements of design efficiency and experimental budget. From the mathematical point of view, the OofA design is to select a subset of the symmetric group formed by all the m! permutations, and some research topics in group theories or combinatorics may arise thereby. For example, if a pairwise-order (PWO) model is assumed, there are m(m-1)/2 parameters to be estimated, and thus only m(m-1)/2+1 runs are needed (instead of m! runs). More advanced models are proposed and will be investigated as well. A fruitful application of computer science is anticipated, because the computational complexity for finding efficient OofA designs tremendously increases as m becomes large. The proposed approach on OofA experiment, although known in many fields, is a novel and new study in Statistics - it is very different in fundamental work for most prior research in this area, with high potential practical values. Many case studies and applications provide an early indication of the potential success of the proposed study.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.