Mathematical Sciences: Problems in Simple and Complex Block Designs

DMS 9626115 Morgan这个项目考察了在各种阻塞结构下比较一组处理的设计问题。其中最简单的是单个块变量，它将n个实验单元划分为每个k个单元的块，k小于处理的总数。长期以来，人们一直认为这种结构良好设计的必要条件是赋值为二进制。该项目正在扩大非二元处理分配的替代思想可以产生良好设计的范围，其中“好”可以用标准数学公式的最优性标准来表示，在某些情况下，设计在信息矩阵中保持重要处理结构的能力。对于存在多个阻塞因素的情况，它们之间可想象的关系范围相当大。研究者还在研究嵌套、交叉和其他阻塞因素之间的相互作用，目的是确定这些更复杂结构的类别，这些结构的最佳设计条件可以用现有的方法确定。设计结构也正在为每个设置制定。当要进行实验的材料单元存在异质性时，对一组实验条件或“处理方法”进行比较是一个普遍的问题，这在许多不同的科学和工业研究领域都很常见。阻塞技术是统计学和实验过程的早期重大贡献之一，专门解决了这个问题。通常使用一些可识别的干扰因子的值，将实验单元划分为称为块的均匀子集。然后，在同一块中进行的测量比较可以比在不同块中进行的测量比较具有更高的精度，因为这种比较消除了异质性。因此，实验得到了改进，在非常实际的意义上，它提供了更可靠的信息。设计问题本身，这也是本研究的主题，是决定哪些治疗被分配到哪些单元，这些分配是由最大限度地提高实验最终将产生的整体信息质量的愿望驱动的。随着更多的扰人因素被引入或识别，问题自然变得更加复杂。通过对这种情况的数学研究，产生了大量的好设计，然后这些设计适用于许多实验情况。应用程序运行的实验调查的范围，从制造领域到农业。＊＊＊

DMS 9626115 Morgan This project examines design problems for comparing a set of treatments under a variety of blocking structures. The simplest of these is that of a single blocking variable which partitions n experimental units into blocks of k units each, k being less than the total number of treatments. It has long been believed that a necessary condition for a good design for this structure is that the assignment be binary. This project is expanding the scope for which the alternative idea of nonbinary treatment assignment can yield good designs, where "good" may be expressed in terms of the standard mathematically formulated optimality criteria and, in some cases, ability of the design to maintain important treatment structure in the information matrix. For situations with more than one blocking factor present, the range of conceivable relationships among them is quite large. The investigator is also studying the interplay between nesting, crossing, and other blocking factors, with the goal of identifying classes of these more complex structures for which optimal design conditions can be determined with existing methods. Design constructions are being formulated for each setting as well. %%% The general problem of comparing a set of experimental conditions, or "treatments," when faced with heterogeneity in the available units of material on which the experiment is to be performed, is common to many disparate fields of scientific and industrial inquiry. The technique of blocking, one of the early great contributions of statistics and the experimental process, specifically addresses this problem. Typically using values of some identifiable nuisance factor, the experimental units are partitioned into homogeneous subsets called blocks. Comparison of measurements occurring in the same block can then be made with greater precision than those in different blocks, the heterogeneity having been eliminated from such compar isons. Thus the experiment is improved, in the very practical sense that it gives information that is more reliable. The design problem itself, which is the topic of this study, is to determine which treatments are assigned to which units in which blocks, said assignment being driven by the desire to maximize the overall quality of information the experiment will ultimately produce. The problem naturally becomes more complicated as more nuisance factors, with various relationships among one another, are introduced or identified. By studying the situation mathematically, large families of good designs are produced which are then applicable to a host of experimental situations. Applications run the gamut of experimental inquiry, from the manufacturing realm to the agricultural. ***