THE INTERPRETATION OF INTERACTIONS IN FACTORIAL EXPERIMENTS
THE INTERPRETATION OF INTERACTIONS IN FACTORIAL EXPERIMENTS
复制标题
因子实验中相互作用的解释
DOI：
10.1093/biomet/39.1-2.65
复制
发表时间：
1952
期刊：
影响因子：
2.7
通讯作者：
E. J. Williams
中科院分区：
文献类型：
--
作者：
E. J. Williams
当两个或两个以上因素的联合效应不具有加性时，提出一个简单的模型来表示这些效应。假设一个因素的影响在其他因素的不同水平上是成比例的，而不是相等的。第一个因子的主要效应是其他因子不同水平的简单效应的加权平均数，权重是估计的比例因子。权重以平方和与乘积的矩阵的特征向量形式给出，该矩阵对应于矩阵的最大特征根;加权主效应的平方和是该特征根的倍数，其他特征根对应于交互作用平方和的分区。该分析与一组变量的典型分析密切相关。检验的显着性（a）的剩余的相互作用和（B）的适当性的一组建议的权重进行了讨论。对于矩阵只有两个非零特征根的情形，讨论了根的联合分布求其极限形式的方法。将联合概率密度展开为一系列虚宗量的贝塞尔函数。导出了根的矩和积矩的渐近公式。当只有两个非零的潜在根时，提出了一组权重的充分性的精确测试。最后用一个数值例子说明了分析方法。
Where the joint effects of two or more factors are not additive, a simple model is proposed for representing the effects. The effects of one factor are assumed to be proportional, rather than equal, at different levels of the other factors. The main effects of the first factor are given as weighted averages of the simple effects at the different levels of the other factors, the weights being the estimated factors of proportionality. The weights are given as the latent vector of a matrix of sums of squares and products corresponding to the largest latent root of the matrix; the sum of squares for the weighted main effect is a multiple of this latent root, and the other latent roots correspond to a partition of the interaction sum of squares. The analysis is closely related to the canonical analysis of a set of variates. Tests of significance of (a) the residual interactions and (b) the adequacy of a proposed set of weights are discussed. For the case where the matrix has only two non-vanishing latent roots, the approach of the joint distribution of the roots to its limiting form is discussed. The joint probability density is expanded as a series of Bessel functions of imaginary argument. Asymptotic formulae for the moments and product-moments of the roots are derived. Exact tests for the adequacy of a proposed set of weights, when there are only two non-vanishing latent roots, are presented. The methods of analysis are illustrated with a numerical example.