ANALYSIS OF RANDOMIZED EXPERIMENTS WITH ORTHOGONAL BLOCK STRUCTURE .I. BLOCK STRUCTURE AND NULL ANALYSIS OF VARIANCE
ANALYSIS OF RANDOMIZED EXPERIMENTS WITH ORTHOGONAL BLOCK STRUCTURE .I. BLOCK STRUCTURE AND NULL ANALYSIS OF VARIANCE
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DOI：
10.1098/rspa.1965.0012
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发表时间：
1965-01-01
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通讯作者：
NELDER, JA
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文献类型：
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作者：
NELDER, JA
到目前为止，几乎所有的实验设计都假设实验单元是或可以以嵌套和交叉分类的方式组合在一起。当所采用的每个嵌套分类在每个单元中嵌套的子单元数量相等时，则称实验单元具有简单块结构。每一个简单的块结构都有一个完全的随机化理论。在保留块结构的情况下，允许对单元的标签进行任何排列，这定义了一个有效的随机化过程。在零实验中，所有单位都接受同样的处理，由随机化过程产生的所有可能的产量向量的总体给出零随机化分布。这个分布的协方差矩阵，方差的零分析，以及其中各种均方的期望都可以从块结构的初始描述中推导出来。所有简单的块结构都可以用一组相互正交的幂等矩阵Ci来表示。对于非简单块结构（例如，在一个块中具有不等数量的块），也可能存在这样的矩阵集，并且这样的集合的存在定义了正交块结构。非简单正交块结构没有完整的随机化理论，从它们推断需要进一步的假设。
Nearly all the experimental designs so far proposed assume that the experimental units are or can be grouped together in blocks in ways that can be described in terms of nested and cross-classifications. When every nesting classification employed has equal numbers of subunits nested in each unit, then the experimental units are said to have asimple block structure. Every simple block structure has a complete randomization theory. Any permutation of the labelling of the units is permissible which preserves the block structure, and this defines a valid randomization procedure. In a null experiment all units receive the same treatment, and the population of all possible vectors of yields generated by the randomization procedure gives the null randomization distribution. The covariance matrix of this distribution, the null analysis of variance, and the expectations of the various mean squares in it are all derivable from the initial description of the block structure. All simple block structures can be characterized by a set of mutually orthogonal idempotent matrices Ci. Such sets of matrices may also exist for non-simple block structures (e. g. those having unequal numbers of plots in a block), and the existence of such a set defines an orthogonal block structure. Non-simple orthogonal block structures do not have a complete randomization theory and inferences from them require further assumptions.