Robustness to non-normality of regression tests
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回归检验对非正态性的稳健性
- 研究问题
- 以正态理论 F 分布为参照的回归检验，对观测值非正态性的敏感程度如何取决于回归变量的数值和实验设计？
- 方法与思路
- 作者近似推导非正态情形下回归 F 统计量的分布，以若干特殊情形说明设计变量的作用，并考察可消除近似阶影响的设计。
- 材料陈述的贡献
- 摘要称，回归变量的数值特征决定检验对非正态性的敏感度；在采用的近似阶内，可选择设计使该影响消失。
- 阅读关注点
- 应核对分布近似保留的阶数、设计变量特征的数学定义，以及等组样本均值检验与方差比较例子的适用条件。
- 阅读边界
- 依据 OpenAlex 索引摘要，稿件版本未注明；近似阶内无影响不等于精确稳健，摘要未给出近似误差或刊本变化。
原文摘要
A number of statistical procedures involve the comparison of a ‘regression’ mean square with a ‘residual’ mean square using the normal-theory F distribution for reference. The use of the procedure for the analysis of actual data implies that the distribution of the mean-square ratio is insensitive to moderate non-normality. Many investigators, in particular Pearson (1931), Geary (1947), Gayen (1950), have considered the sensitivity of this distribution to parent non-normality for important special cases and a very general investigation was carried out by David & Johnson (1951a, b). The principal object of this paper is to demonstrate the overriding influence which the numerical values of the regression variables have in deciding sensitivity to non-normality and to demonstrate the essential nature of this dependency. We first obtain a simple approximation to the distribution of the regression F statistic in the non-normal case. This shows that it is ‘the extent of non-normality’ in the regression variables (the x 's), which determines sensitivity to non-normality in the observations (the y 's). Our results are illustrated for certain familiar special cases. In particular the well-known robustness of the analysis of variance test to compare means of equal-sized groups and the notorious lack of robustness of the test to compare two estimates of variance from independent samples are discussed in this context. We finally show that it is possible to choose the regression variables so that, to the order of approximation we employ, non-normality in the y 's is without effect on the distribution of the test statistic. Our results demonstrate the effect which the choice of experimental design has in deciding robustness to non-normality.