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A proof of the conjecture that the Tukey-Kramer multiple comparisons procedure is conservative. (English) Zbl 0545.62047

Let \(\bar x_ i\) be distributed independently as \(N(\mu_ i,\sigma^ 2/n_ i)\) random variables (1\(\leq i\leq k)\) and let \(S^ 2\) be an independent estimate of \(\sigma^ 2\) which is distributed as a \(\sigma^ 2\chi^ 2_{\nu}/\nu\) random variable. As a natural extension of his multiple comparisons procedure for the balanced case \((n_ 1=...=n_ k)\), J. W. Tukey [The problem of multiple comparisons. Unpublished report, Princeton Univ. (1953)] proposed the following approximate 100(1-\(\alpha)\)% confidence intervals for all pairwise differences \(\mu_ i-\mu_ j: \mu_ i-\mu_ j\in [\bar x_ i-\bar x_ j\pm q^{(\alpha)}_{k,\nu}S\sqrt{(n_ i^{-1}+n_ j^{-1})/2}]\quad \forall i<j\)where \(q^{(\alpha)}_{k,\nu}\) is the upper \(\alpha\) point of the Studentized range distribution with parameters k and \(\nu\). Tukey conjectured that these intervals are actually conservative. This conjecture is analytically proved in this paper. It is shown that the minimum of the coverage probability is achieved when all the \(n_ i\) are equal in which case it is 1-\(\alpha\).
Reviewer: A.C.Tamhane

MSC:

62J15 Paired and multiple comparisons; multiple testing
62J10 Analysis of variance and covariance (ANOVA)
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