Pairwise Comparisons of Generally Correlated Means

原文摘要

A commonly occurring statistical inference problem in practice is that of making simultaneous comparisons among three or more treatment means μi (1 ≤ i ≤ k) for certain experimental designs. One way to do this, the T method proposed by Tukey (1953), constructs simultaneous confidence intervals for all pairwise differences of the treatment means μi – μj (1 ≤ i, j ≤ k, i ≈ j). The joint confidence level of these confidence intervals, however, depends in a very complicated fashion on the covariance structure of the treatment mean estimates μi , and, therefore, for designs where the covariance structure is not “simple,” the joint confidence level is not readily apparent. In general, this is the case for any unbalanced design or for designs in which the treatment mean estimates μi have unequal correlations. Tukey conjectured in 1953 that whatever the correlation structure of the treatment mean estimates μi , the T method would always provide a conservative set of confidence intervals, that is, that the actual joint confidence level of the confidence intervals would always be at least as great as the nominal joint confidence level 1 – α. In this article a discussion is undertaken of the evidence that the T method in general provides a conservative set of simultaneous confidence intervals. It is shown that the coverage probability of the simultaneous confidence intervals depends on the covariance structure only through the k(k − 1)/2 variances of the pairwise differences of the treatment mean estimates. A set of sufficient, but not necessary, conditions on these variances is given, which ensures that the T-method confidence intervals are conservative. In addition, the application of the T method to various common experimental designs that produce correlated treatment mean estimates is discussed. An integral expression is derived for calculating the exact joint confidence level of the T-method confidence intervals or for calculating confidence intervals of joint confidence level exactly equal to a given value. This expression is used to evaluate the coverage probabilities for a wide variety of covariance structures with k = 4. In each case considered, the T-method confidence intervals are conservative, and, furthermore, the amount of conservativeness is very small unless the population mean estimates have radically different variances and covariances.

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