The Maximum Familywise Error Rate of Fisher's Least Significant Difference Test
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Fisher 最小显著差异检验的最大全族错误率
- 研究问题
- 单因素布局中，Fisher 最小显著差异检验的最大全族错误率会超过名义水平多少，能否通过简单修改加以控制？
- 方法与思路
- 作者对平衡设计和三总体非平衡设计推导精确表达式，对更多总体的非平衡设计给出上界，并据此修改检验程序。
- 材料陈述的贡献
- 摘要称原检验的最大全族错误率可能远超名义水平；修改后在所述第一类模型中等于名义水平，第二类模型中不超过它。
- 阅读关注点
- 应核对两类模型的定义、最大全族错误率对哪些参数取最大，以及修改程序与独立正态变量极差定理的衔接。
- 阅读边界
- 所给索引摘要未标稿件版本。摘要未列完整模型假设和修改公式，错误率控制结论不能自动推广到任意分布、依赖结构或检验流程。
原文摘要
In this article an investigation is made of the maximum familywise error rate (MFWER) of Fisher's least significant difference (LSD) test for testing the equality of k population means in a one-way layout. An exact expression for the MFWER is derived (see Theorem 1) for all balanced models and for an unbalanced model with k = 3 populations (Type I models). A close upper bound for the MFWER is derived for all unbalanced models with four or more populations (Type II models). These expressions are used to illustrate that the MFWER may greatly exceed the nominal size α of the LSD test. In addition, a simple modification of the LSD test is proposed to control the MFWER. This modified procedure has MFWER equal to the nominal level α for Type I models and no greater than α for Type II models (Theorem 2) and is, therefore, recommended as an improvement over the LSD test. The key to the analysis is two theorems concerning the ranges of independent normal random variables, which are contained in the Appendix. The first of these theorems (Theorem A.1) is concerned with the conservative nature of unbalanced models as compared with balanced models and was published in Hayter (1984), although in a different context. The second theorem (Theorem A.2) presents a new chain of inequalities concerning the ranges of independent normal random variables partitioned into groups of differing sizes. It is believed that this new latter theorem may be of independent interest.