几类因子试验的最优设计与数据分析

Experimental design has wide application in various experiments. We will make research on the following topics: (1) The essence of almost designs constructed in the literature on blocked designs belongs to designs with single block variable. We will investigate the construction methods of the optimal blocked designs with multi block variables under the minimum aberration (MA), clear effects, and general minimum lower order confounding (GMC) criteria. (2) There are two phases in the randomization of a split-plot experiment and the error variances in the two phases are usually not equal to each other. The point has not been considered in the literature in the constructing of the optimal split-plot designs. Under the assumption of inequality of the error variances in the two phases, we investigate the construction methods of optimal split-plot designs under the MA and GMC criteria. (3) We generalize the definition of second-order saturated design and the technique of projection of two-level design to those of multi-level design, and then use the new definition and technique to investigate the construction methods of the optimal multi-level designs under the MA and GMC criteria. (4) The error variances are assumed to equal to each other in all of the experimental points in the literature. In fact, this does not hold in many experiments. We suppose that the experimental points have different error variances in the replicated experiment and investigate the method of data analysis of the catalogue of experiments.

试验设计在各类试验中有广泛应用。本项目拟研究以下内容：(1)几乎所有分区组设计的文献构造的设计的本质都是单区组变量分区组设计，本项目研究多区组变量分区组设计在最小低阶混杂（MA）、纯净效应、一般最小低阶混杂（GMC）准则下的最优设计的构造方法。(2)裂区试验的随机化有两个阶段，一般情况下这两个阶段中的随机误差方差不相等。现有文献在最优裂区设计的构造中一般都没有考虑这一点。本项目在两阶段随机误差方差不等的假设下，研究在MA和GMC准则下的最优裂区设计的构造方法。(3)把二水平设计的二阶饱和设计和投影设计等概念和技术推广到高水平设计，应用这些新的概念和技术研究在MA和GMC准则下的最优高水平设计的构造方法。(4)在试验设计的文献中通常假设所有试验点有共同的误差方差，在很多试验中这一条件并不满足。本项目假设重复试验的不同试验点具有不同的随机误差方差，研究这类试验的数据分析方法。

试验设计在各类试验中有广泛应用。本项目主要研究了以下内容：(1)多区组变量分区组设计的最优设计的构造方法，给出了最小低阶混杂准则下最优多区组变量分区组设计的理论构造方法，列出了具有32个水平组合的包含最多纯净两因子交互效应成分的多区组变量分区组设计；(2)最优裂区设计的构造方法，提出了整区因子和子区因子不同等重要情形下的最优设计准则并构造了大量最优部分因子裂区设计表，得到了混水平裂区设计包含纯净两因子交互效应成分的充要条件，为下一步构造具有最多纯净两因子交互效应成分的设计奠定了理论基础；(3)最优高水平设计的构造方法，构造了满足4≤n≤20的所有一般最小低阶混杂准则下的最优3^(n-m)设计并以列表的形式给出，方便使用；(4)给出了不同试验点上的试验误差方差不相等时的试验数据分析方法。