Mathematical Sciences: Problems in Simple and Complex Block Designs

DMS 9626115摩根 这个项目检查设计问题，比较一组 在各种封闭结构下进行处理。 最简单的 这是一个单一的阻塞变量， n个实验单元，每个单元为k个单元的区组，k小于 治疗的总次数。长期以来，人们一直认为， 这种结构良好设计的必要条件是， 分配是二进制的。该项目正在扩大范围， 非二元治疗分配的替代思想可以 产生好的设计，其中“好”可以用 标准数学形式的最优性准则 在某些情况下，设计的能力，以保持重要的 信息矩阵中的处理结构。对于以下情况， 存在一个以上的阻滞因素， 它们之间的关系是相当大的。研究者还 研究筑巢、杂交和其他 区块因子，目的是确定这些更多的类 复杂的结构，最佳的设计条件， 用现有的方法确定。建筑设计正在 也适用于各种设置。 %%% 比较一组实验条件的一般问题， 或“治疗”，当面对现有的异质性时， 在其上进行实验的材料的单位，是 在科学和工业研究的许多不同领域都很常见。 阻塞技术，早期的伟大贡献之一， 统计和实验过程，专门解决这一问题， 问题.通常使用一些可识别的滋扰值 因子，将实验单元划分为均匀 称为块的子集。发生的测量比较 然后可以以比那些更高的精度制造相同的块 在不同的区块中， 这样的比较。因此，实验得到了改进，在非常 实际上，它提供的信息更可靠。的 设计问题本身，这是本研究的主题，是为了 确定哪些治疗分配给哪些单位， 块，所述分配由最大化 总体信息质量的实验将最终 美食的备受问题自然变得更加复杂， 滋扰因素之间存在着各种关系， 介绍或识别。通过数学研究， 大量优秀的设计被生产出来， 适用于许多实验情况。应用程序运行 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. ***