Study on Order-of-Addition Experiments

在许多实验中，许多反应物是按顺序而不是同时加入系统的。形成，例如，反应产物的数量/大小/纯度通常取决于不同反应物的添加顺序（OofA）。OofA实验可以追溯到Fisher(1937)，一位女士能够（通过品尝）分辨出是茶还是牛奶首先加入了杯子。这可能是第一个流行的OofA实验。一般来说，有m个必需的组件，人们希望确定将这m个组件依次添加的最优顺序。了解生产中相关部件的最佳添加顺序是至关重要的。注意这里有m！要比较的可能顺序—对于品茶示例，有m!=2！=2种可能的顺序。然而，要测试所有的软件通常是负担不起的。处理，然后出现设计问题(例如，当m = 10时，m！大约是360万)。虽然在统计学中有点原始，但在过去几十年中，OofA效应在许多领域经常被提及。这包括生物化学（Shinohara and Ogawa 1998）、食品科学（Jourdain et al. 2009）、营养科学（Karim et al. 2000）和制药科学（Rajaonarivony et al. 1993），仅举几例。由于加法顺序效应存在于各种真实实验和计算机实验中，因此本项目的成功成果可以直接应用于许多不同的领域，如化学和生物学。此外，提出的问题为其他学科的基础学术研究提供了一个绝佳的机会。在这个项目中，PI计划建立评估OofA设计的基本理论。PI还计划根据不同的设计效率和实验预算要求构建OofA设计类。从数学的角度来看，OofA的设计是选择对称群的一个子集由所有的m组成！由此可能产生置换，以及群论或组合学中的一些研究课题。例如，如果假设一个双序（PWO）模型，则需要估计m(m-1)/2个参数，因此只需运行m(m-1)/2+1次（而不是m!）运行)。提出了更先进的模型，并将进行研究。由于随着m的增大，寻找有效的OofA设计的计算复杂度将大大增加，因此预计计算机科学的一个卓有成效的应用。提出的OofA实验方法虽然在很多领域都是已知的，但在统计学上是一项新颖的研究，它在基础工作上与该领域大多数先前的研究有很大的不同，具有很高的潜在实用价值。许多案例研究和应用提供了拟议研究潜在成功的早期迹象。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.