几种复杂试验设计的理论和构造

Design of experiments is one of the most important branches of statistics. It not only has theoretical meaning but also is of great practical value. With the rapid development of modern science and technology, all walks of life are making more and more high requests on experimental design, especially on effective designs and analysis methods suitable for various experiments and emerging fields. Order-of-addition experiments aim at studying the optimal orders of addition among components. Order-of-addition experiments are widely applied in many areas. Currently, the most appealing designs used to study order-of-addition experiments are pair-wise ordering designs. As a class of screening designs, composite designs are also a type of sequential designs. This project will mainly focus on some issues on order-of-addition experiments and composite designs, including (1) design construction and theory analysis for order-of-addition orthogonal arrays; (2) construction of optimal minimal-point pair-wise ordering designs under D-efficiency criterion, and analysis of isomorphism for pair-wise ordering designs; (3) construction and theory on composite designs based on uniform designs. This project is based on the active topics and hot issues in the current international studies of experimental design. Through this project, we expect to have some breakthroughs of theory and methods, and construct a series of optimal designs for practical use.

试验设计是统计学的重要分支之一，它不仅在理论上有重要意义，在实际领域也具有重大应用价值。随着现代科学和新技术的飞速发展，各行业对试验设计的要求也越来越高，迫切需要适用于各类复杂试验以及新兴领域的有效试验设计和分析方法。添加顺序试验旨在研究成分间的最优添加顺序，应用于多个领域。目前，研究添加顺序试验使用较多的是配对排序设计。作为因子筛选设计，复合设计不仅可以筛选重要因子，也是一类序贯设计。本项目将对添加顺序试验以及复合设计中的若干问题展开研究，具体包括：（1）添加顺序正交表的构造和理论分析；（2）D-效率准则下的最优的最少点配对排序设计的构造方法及同构性分析；（3）基于均匀设计的复合设计的构造及理论研究。这些课题都是基于当前国际试验设计研究前沿和热点提出的。通过本项目的研究争取对这几种复杂设计的构造有一定的理论和方法突破，并构造出一系列优良的设计表，供实际使用。

本项目系统地研究了添加顺序试验和因析试验中的一些热点问题，取得了丰富的研究成果，具体地（1）给出了全天添加顺序试验设计的D-效率，这为选取部分添加顺序试验设计提供了基准；（2）构造了一类最少点的添加顺序试验设计，这类设计可以最大程度地节省试验花费；（3）给出了两类D-效率最优的部分添加顺序试验设计的构造方法，研究了这两类设计的平衡性质；（4）构造了一系列最优的分区组的两水平因析设计，在先验信息下，这一系列最优设计能够保证试验者尽可能多地估计出感兴趣的效应。在本项目资助下，已发表SCI论文7篇; 参加国际会议交流1人次，邀请国内外学者来访10余人次；项目负责人共培养硕士生14名，其中4人顺利毕业，10名在读。圆满完成了项目申请时的计划。