A METHOD FOR JUDGING ALL CONTRASTS IN THE ANALYSIS OF VARIANCE
A METHOD FOR JUDGING ALL CONTRASTS IN THE ANALYSIS OF VARIANCE
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DOI：
10.1093/biomet/40.1-2.87
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发表时间：
1953-01-01
期刊：
影响因子：
2.7
通讯作者：
SCHEFFE, H
中科院分区：
文献类型：
--
作者：
SCHEFFE, H
对于困扰方差分析实践的以下问题，可以找到一个简单的答案：在通常的假设下，如果传统的。假设的f检验：μ1=μ2=…=μk α显著性水平拒绝，μi之间的对比有什么进一步的推论是有效的（除了对比值不全为零的推论）？假设f检验任务为−1，自由度为vdegree。对于任意1，…，ckwithwrite θ作为对比，writeand为通常的θ估计值和的方差。然后对于所有的对比，不管θ s的真实值是多少，它们都满足的概率是1 - α，其中s2是（k - 1）乘以f分布的上α点，具有k - 1和自由度。假设我们说，如果| b> s，估计的对比度与‘显著不同于0 ’，那么f检验拒绝shift，并且只有当某些显著不同于0时，如果确实如此，我们才能说哪个。更一般地说，上述不等式可以适用于所有实验中所有陈述都正确的比例的明显频率解释的所有对比。关系被认为是使用学生化范围表的较早的Tukey方法，并且在它们具有相同方差并且所有对si，j（i≠j）具有相同协方差的特殊情况下有效。对新方法的工作特性进行了分析。这篇论文是有组织的，以便希望学习方法和避免证明的读者可以跳过§§2和§5。
A simple answer is found for the following question which has plagued the practice of the analysis of variance: Under the usual assumptions, if the conventional.F-test of the hypothesisH: μ1=μ2=…=μkat the α level of significance rejectsH, what further inferences are valid about the contrasts among the μi(beyond the inference that the values of the contrasts are not all zero)? Suppose theF-test hask−1 andvdegrees of freedom. For anyc1, …,ckwithwrite θ for the contrast, and writeandfor the usual estimates of θ and the variance of. Then for the totality of contrasts, no matter what the true values of the θ's, the probability is 1−α that they all satisfywhere S2is (k−1) times the upper α point of theF-distribution withk−1 andvdegrees of freedom. Suppose we say that the estimated contrast withis ‘significantly different from zero’ if |> S. Then theF-test rejectsHif and only if someare significantly different from zero, and if it does, we can say just which. More generally, the above inequality can be employed for all the contrasts with the obvious frequency interpretation about the proportion of experiments in which all statements are correct. Relations are considered to an earlier method of Tukey using the Studentized range tables and valid in the special case where theiall have the same variance and all pairsi,j(i≠j) have the same covariance. Some results are obtained for the operating characteristic of the new method. The paper is organized so that the reader who wishes to learn the method and avoid the proofs may skip §§ 2 and 5.