Symmetry and Asymmetry in Experimental Design

在对比实验设计中，实验材料的分组是减少方差的基本手段。研究人员正在从三个不同的角度探索这一原则的有效应用，这三个角度基于对称/不对称概念，在被比较的处理中，在现有的实验材料中，以及在对一组允许的设计的外部限制中。传统的设计最优理论强烈地依赖于处理利益均等的假设，该假设通过对处理排列不变的实验信息的汇总度量来表达。第一个目标是开发一个连贯的最优化理论，用于当置换不成立时，即当实验者面临治疗兴趣的不对称时，评估设计。平等对待复制是另一个传统的对称，取决于可用区块的数量和大小，可能导致资源利用不足，从而减少信息。第二个目标是发展全面处理治疗复制的不对称性的理论。在一些实验中，要求将块划分成子集，以便每个处理在每个子集中恰好发生一次；针对这种情况的设计被认为是可解决的。可分解性是在处理任务中增加的对称性，而不是通常要求的区块设计。第三个目标是扩展最优可分解设计的理论，以涵盖目前还没有可用结果的处理、重复和区组大小的数量。总体而言，研究人员正在寻求一种比较试验设计的方法，这种方法凭借更广泛的适用性，可以汇编显著扩展的设计目录，作为广泛学科中的实验人员的资源。随着最优化理论的发展，上述每个问题的一个组成部分是建立和汇编许多区块和治疗组合的设计，并在网上将这些设计编目，供研究人员、实验者和统计从业者免费获取。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质量，帮助科学进步和产业创新。

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.