Block Designs: Advances in Theory and Use

在许多不同的科学和工业研究领域中，一个共同的问题是，当面对要进行实验的材料单位的异质性时，比较一组实验条件或“处理”。拦截技术专门解决了这个问题。通常使用一些可识别的滋扰系数的值，实验单位被划分为称为区块的同质子集。然后，可以根据同一区块的测量结果进行更精确的比较来进行推断。设计问题是确定哪些处理被分配给哪些区块中的哪些单元，所述分配是由最大化最终将产生的信息质量的愿望驱动的。随着具有各种相互关系的更多滋扰因素的引入或识别，问题变得更加复杂。这个项目解决了各种常见实验设置的设计问题，有两个广泛的目标：(I)扩展已知的理论，以确定比目前已知或可用的更广泛的设置的最佳设计；以及(Ii)制作一个全面的设计目录，将其并入一个更大的、基于网络的资源中，其中将包括各种可下载的组合设计，对统计学家、数学家和学术界和工业界的其他科学家有用。通过将最优设计收集到一个容易获取的资源中，可以增加好设计的使用，从而提高实验科学的实践。当比较v处理时，跨科学努力范围最常见的情况是一个单一的阻塞变量将k个实验单位分成b个区块，每个区块k个单位。本项目将研究这些简单区块设计的最优化问题，以及随之而来的组合问题，在需要时通过计算扩大理论，以产生实用范围的参数组合(v；b；k)的设计，包括不可能等重的设置。具有两个阻塞因子的最频繁使用的设计是行-列设计，其中两个阻塞因子可以被可视化为矩形阵列中的行和列，以及可再解设计，在该设计中，第二阻塞因子将简单的块设计划分成块的子集，每个块子集由单个复制组成。该项目还将解决这两类设计的最优化和结构方面的问题，特别强调实践中最需要的计划：那些很少重复的计划。

A problem common to many disparate fields of scientific and industrial enquiry isthat of comparing a set of experimental conditions, or `treatments', when faced withheterogeneity in the units of material on which the experiment is to be performed. Thetechnique of blocking specifically addresses this problem. Typically using values of someidentifiable nuisance factor, the experimental units are partitioned into homogeneoussubsets called blocks. Inferences can then be based on the more precise comparisons ofmeasurements from the same block. The design problem is to determine which treat-ments are assigned to which units in which blocks, said assignment being driven bythe desire to maximize the quality of information that will ultimately be produced.The problem becomes more complicated as more nuisance factors, with various inter-relationships, are introduced or identified. This project addresses the design problemfor a variety of commonly encountered experimental settings, with two broad goals:(i) to extend the known theory for determining optimal designs for a wider range ofsettings than is now known or available; and (ii) to produce a comprehensive catalogof designs to be incorporated into a larger, web-based resource that will include a vari-ety of downloadable combinatorial designs useful for statisticians, mathematicians, andother scientists in academia and industry. By collecting optimal designs in a single,easily accessible resource, the use of good designs can be increased and the practice ofexperimental science consequently sharpened.When comparing v treatments, the most commonly encountered situation across arange of scientific endeavors is that of a single blocking variable partitioning bk experi-mental units into b blocks of k units each. This project will examine optimality problemsfor these simple block designs, and the attendant combinatorial issues, augmenting the-ory by computation where needed, to produce designs for a practical range of parametercombinations (v; b; k), including settings where equireplication is not possible. The mostfrequently used designs with two blocking factors are row-column designs, in which twoblocking factors can be visualized as row and columns in a rectangular array, and re-solvable designs, in which a second blocking factor partitions a simple block design intosubsets of blocks each consisting of a single replicate. This project will also addressproblems in optimality and construction of designs in both these classes, with specialemphasis on plans most demanded in practice: those with few replicates.