Symmetry and Asymmetry in Experimental Design

在比较实验设计中，实验材料的阻隔是减小方差的基本手段。研究者正在从三个不同的角度探索这一原则的有效应用，基于对称/不对称的概念，在比较的处理中，在可用的实验材料中，以及在允许的设计集的外部限制中。传统的设计最优性理论强烈依赖于治疗兴趣平等的假设，通过对实验信息的总结测量来表达，这些信息对治疗排列是不变的。第一个目标是发展一个连贯的最优性理论来评估设计时，置换是站不住脚的，也就是说，当实验者面临不对称的治疗兴趣。处理重复的平等性是另一种传统的对称性，它取决于可用块的数量及其大小，可能导致资源利用不足，从而导致信息减少。第二个目标是以全面的方式发展处理治疗复制不对称的理论。在一些实验中，要求将块划分为子集，以便每个子集中每个处理只发生一次；针对这种情况的设计被称为是可解决的。可解析性是在处理分配中增加的对称性，超出了通常要求的块设计。第三个目标是扩展理论的最佳可解决的设计，以涵盖数量的处理，重复和块大小，目前没有可用的结果。总的来说，研究者正在寻求一种比较实验设计的方法，这种方法由于具有更广泛的适用性，可以为广泛学科的实验者编写显著扩展的设计目录作为资源。随着最优性理论的发展，上述每个问题的一个组成部分是为许多块和处理组合构建和编译设计，并将这些设计编目在线，供研究人员、实验人员和统计从业者免费访问。使好的设计易于获得为更有效的实验提供了一个基本工具，通过提高实验可以产生的信息的质量，帮助科学进步和工业创新的进步。

Blocking of experimental material is a fundamental device for variance reduction in the design of comparative experiments. The investigator is exploring efficient applications of this principle from three distinct perspectives based on symmetry/asymmetry notions in the treatments to be compared, in the available experimental material, and in external restrictions on the set of permissible designs. Conventional design optimality theory rests strongly on the assumption of equality of treatment interest, expressed through summary measures of experimental information that are invariant to treatment permutation. The first goal is to develop a coherent optimality theory for evaluating designs when permutability is not tenable, that is, when experimenters are faced with asymmetry in treatment interest. Equality of treatment replications is another conventional symmetry that, depending on the number of blocks available and their sizes, can result in underutilization of resources and consequent reduction of information. The second goal is to develop theory for dealing with asymmetry of treatment replication in a comprehensive fashion. In some experiments it is demanded that blocks be partitioned into subsets so that each treatment occurs exactly once in each subset; a design for this situation is said to be resolvable. Resolvability is an added symmetry in treatment assignment, over that ordinarily required of a block design. The third goal is to extend theory for optimal resolvable designs to cover numbers of treatments, replicates and block sizes for which there are currently no available results.Collectively, the investigator is pursuing an approach to the design of comparative experiments that, by virtue of wider applicability, can compile significantly expanded design catalogs as resources for experimenters in a wide range of disciplines. A component of each of the problems described above, as the optimality theory evolves, is the building and compiling of designs for many combinations of blocks and treatments, and cataloging these online for free access by researchers, experimenters, and statistical practitioners. Making good designs easily available provides a basic tool for more efficient experimentation, aiding progress in scientific advancement and industrial innovation by increasing the quality of information that experiments can produce.