Construction and optimality of affine-resolvable designs
Construction and optimality of affine-resolvable designs
复制标题
仿射解析设计的构造和优化
DOI：
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发表时间：
1995
期刊：
影响因子：
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通讯作者：
H. Monod
中科院分区：
文献类型：
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作者：
R. A. Bailey;H. Monod
摘要仿射可分解设计是从正交表中构造的，并且在可分解设计中相对于通常的准则是最优的。表格和文本显示了只要处理次数正确地整除区组大小的平方，如何构建最多七次重复的此类设计。使用仿射可分解设计的实验数据可以通过使用伪因子来简单地分析。在许多使用不完全区组设计的实验中，需要可解析设计，即区组可以被划分为包含每个处理一次的组的设计。有时，这些块自然地被分组为更大的块，其中包含每种处理恰好一次。微生物学的一个例子是一个实验，在这个实验中，将经过处理的细菌放在培养箱中，这样细菌就可以繁殖;块是培养箱中的架子，大块是培养箱。使用可分解设计确保大块与处理正交。在其他情况下，特别是在农业试验中，为了管理的目的，相邻的区块被分成几个大区块。那么，可分解设计不仅保持管理效果与处理正交，而且还对整个大块的损失提供某种保护，因为所有处理都同样受到这种损失的不利影响。大的区组通常被称为重复，尽管有点含糊，因为它们的存在与处理分配无关。假设在大小为k的区组中需要v个处理的可分解设计，其中对于某个大于1的整数s，v = sk，并且需要r个重复。当r = 2时，威廉姆斯，帕特森和约翰（1976）证明了最优可分解设计等价于最优不完全区组设计。因此，关于不可分解设计的最优性结果可用于寻找最有效的可分解设计。
SUMMARY Affine-resolvable designs are constructed from orthogonal arrays and shown to be optimal among resolvable designs with respect to the usual criteria. Tables and text show how to construct such designs in up to seven replicates whenever the number of treatments properly divides the square of the block size. Data from experiments which use affine- resolvable designs can be simply analysed by using pseudofactors. In many experiments where an incomplete-block design is used, a resolvable design, that is, a design in which the blocks can be partitioned into sets containing each treatment once, is desirable. Sometimes the blocks are naturally grouped into larger blocks contain- ing each treatment exactly once. An example from microbiology is an experiment in which jars of treated bacteria are placed in incubators so that the bacteria may multiply; blocks are the shelves within the incubators and large blocks are the incubators. Use of a resolv- able design ensures that the large blocks are orthogonal to treatments. In other cases, particularly in agricultural trials, neighbouring blocks are grouped into large blocks for management purposes. Then a resolvable design not only keeps management effects ortho- gonal to treatments, it also gives some protection against the loss of a whole large block, for all treatments are disadvantaged equally by such a loss. The large blocks are often called replicates, although somewhat ambiguously, because their existence is independent of the treatment allocation. Suppose that a resolvable design is needed for v treatments in blocks of size k, where v = sk for some integer s greater than 1, and that r replicates are required. For r = 2, Williams, Patterson & John (1976) showed that an optimal resolvable design is equivalent to an optimal incomplete-block design for s treatments in s blocks of size k. Thus optimality results about nonresolvable designs may be used to find the most efficient resolvable