A Simple and Asymptotically Optimal Test for the Equality of Normal Populations: A Pragmatic Approach to One-Way Classification
提出一个结合均值相等检验和方差相等检验的简单渐近最优检验方法，用于同时检验多个正态总体的均值和方差是否相等，适用于重复测量、轮廓分析等单因素分类实验。
Snedecor and Cochran (1967, p. 324) observed that an application of different treatments to otherwise homogeneous experimental units often results in groups that are different not only in means but also in variances. The usual one-way classification procedure assumes a priori the homogeneity of variances among different groups and tests the equality of means only. Thus motivated by the problem of simultaneously testing the equality of means and the equality of variances of several normal populations, I suggest in this article a simple and an asymptotically optimal test. The suggested test is based on the combination of two independent tests, namely (a) the test for the equality of means given that all variances are equal but unspecified and (b) the test for the equality of variances when all means, not necessarily all equal, are unspecified. Tests (a) and (b) are combined by the Fisher method (1950, p. 99). Littell and Folks (1973) showed that the Fisher method of combining two or more independent tests is at least as good as any other optimal method. Using this fact and the results of Hsieh (1979) on the Bahadur optimality of the individual tests, the suggested test was found to be asymptotically optimal (details can be obtained from the author on request) in the sense of Bahadur (1960) efficiency. Further, the test can be applied easily using existing statistical tables. Examples where this test can be applied include the analysis of repeated measurements, the profile analysis of several groups, and, in general, the analysis of one-way classified experiments.