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体系中的。例如，反应产物的数量/大小/纯度的形成通常取决于不同反应物的加成顺序(OofA)。OofA的实验可以追溯到Fisher(1937年)，当时一位女士能够(通过品尝)区分茶或牛奶是首先加入杯子中的。这可能是OofA第一次流行的实验。一般而言，有m个所需的组件，并且希望确定逐个添加这m个组件的最佳顺序。了解与生产相关的组件的最佳添加顺序至关重要。请注意，这里有m！可能的比较顺序--以品茶为例，有m！=2！=2个可能的顺序。然而，测试所有的m往往是负担不起的！处理，设计问题就出现了(例如，当m=10时，m！约为360万人)。尽管OofA效应在统计学上有些原始，但在过去的几十年里，Oofa效应在许多领域经常被提及。其中包括生物化学(Shinohara和Ogawa，1998)、食品科学(Jourdain et al.2009年)，营养科学(Karim等人。2000年)，以及药学(Rajaonarivony等人)。1993年)，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，本项目的成功成果可以直接应用于化学、生物等许多不同的领域。此外，提出的问题为其他学科的基础学术研究提供了一个千载难逢的机会。在这个项目中，PI计划建立评估OofA设计的基本理论。PI还计划在不同的设计效率和实验预算要求下，构建OofA设计类。从数学的角度来看，OofA的设计就是选择m！排列，以及由此产生的群论或组合学中的一些研究课题。例如，如果假设一个成对顺序(PWO)模型，则需要估计m(m-1)/2个参数，因此只需要m(m-1)/2+1次(而不是m！运行)。提出了更先进的模型，并将对其进行研究。计算机科学的卓有成效的应用是值得期待的，因为随着m变大，寻找有效的OofA设计的计算复杂性极大地增加。拟议的OofA实验方法虽然在许多领域都已为人所知，但在统计学上是一项新颖的新研究--它在基础工作方面与该领域以往的大多数研究截然不同，具有很高的潜在实用价值。许多案例研究和申请提供了拟议研究潜在成功的早期迹象。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.