The robustness of the covariance analysis of a one-way classification
单因素分类中协方差分析的稳健性AI 解读 ↓
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单因素分类中协方差分析的稳健性
- 研究问题
- 单因素分类的协方差分析F检验对非正态性、处理与斜率交互及二次回归项有多敏感，伴随变量如何影响这种敏感性？
- 方法与思路
- 作者分析设计结构与伴随变量对非正态敏感性的作用，给出实用判断表达式，并考察斜率交互和二次回归项的影响。
- 材料陈述的贡献
- 摘要称平衡设计中的协方差分析F检验仍明显受非正态影响；大样本下斜率交互影响较小，二次回归项影响更严重。
- 阅读关注点
- 应核对非正态性涉及哪些变量、各稳健性比较的条件及样本框架，并从原PDF确认设计矩阵相关表达式。
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- 仅据OpenAlex索引摘要，稿件版本未注明；矩阵表达式的幂次疑似提取丢失，需查原PDF，不据此重构公式。摘要未列结论完整条件。
原文摘要
SUMMARY The effect of non-normality and the effect of treatment slope interaction in the analysis of covariance is studied. Unlike the analysis of variance F-test the analysis of covariance F-test is found to be appreciably affected by non-normality even in balanced classifications. The degree of sensitivity to non-normality is determined by the distribution of the concomitant variables. The violation of the assumption of no treatment-slope interaction does not seem to have serious effect on the F-test in large samples, but a much more serious effect appears to be that of a quadratic component of regression on the F-test. Atiqullah (1962) has shown that the extent to which the analysis of variance F-test is influenced by non-normality depends on the structure of the design matrix, A, in the usual linear model for the problem. In balanced set-ups the design matrix, A, possesses the property of quadratic balance which is characterized by the matrix A (A'A) 1-A' having its diagonal terms all equal. This property practically nullifies the effect of non-normality on inferences drawn from the F-test in the standard orthogonal cross-classified set-ups, such as balanced two or more-way arrangements, Latin squares, balanced incomplete block designs, etc. But in the analysis of covariance the linear model contains, besides the design matrix A, a matrix of concomitant observations and therefore the sensitivity of F-test to non-normality will be determined both by the structure of the design matrix A and the nature of the concomitant matrix X. In balanced classifications, it is thus the concomitant variables alone which will determine sensitivity to non-normality. This fact also follows from the discussion of Box & Watson (1962) who have discussed robustness to non-normality of regression tests and have shown that it is the extent of non-normality in the independent variables which determines sensitivity to non-normality. In the first part of the present paper we investigate in more detail the influence which the concomitant variables have in determining sensitivity to non-normalityin the analysis of covariance F-test and obtain expressions for judging the effect of non-normality in practical situations. In the second part of the paper we discuss how far the analysis of covariance, as usually applied on the presumption of no interaction between treatments and slope, will be sensitive to situations entailing interactions or to a model containing quadratic gradient.