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Multiple comparison procedures for detecting differences in simply ordered means. (English) Zbl 1429.62315

Summary: In the one-way analysis of variance, we study multiple-comparison procedures that detect differences in the treatment means when they are known to be simply ordered (i.e., \(\mu_{1}\leqslant \mu_{2}\leqslant \cdots \leqslant \mu_k\)). Several existing procedures as well as four procedures proposed here are compared by Monte Carlo techniques. In 1990, Hayter developed a one-sided Studentized range procedure (H), and the max-min technique exploited in 2002 by Liu, Lee, and Peng provides an improved Hayter procedure (M). In her 2001 doctoral dissertation, Liu developed a procedure (L) that is based on the differences in the averages of two blocks of means. A procedure (R), which utilizes the restricted maximum-likelihood estimates of the means corrected for their biases and standard deviations, and an analogue to Fisher’s LSD (T) are proposed. For T, the usual \(F\) test is replaced by an order-restricted test and one-sided \(t\)-tests are used. Declaring \(\mu_i<\mu_j\) and \(\mu_l=\mu_{l^{\prime}}\) with \(l\leqslant i<j \leqslant ^{\prime}\) is a logical contradiction. Procedures H, R, and T do make such logical errors, but M and L do not. We study R\(^{\prime}\) and T\(^{\prime}\), modifications of R and T, that are free of these logical errors. Based on the Monte-Carlo study, we make the following recommendation: to examine all possible differences in the treatment means when they are simply ordered, M should be used if it is desirable to keep a tight control on the familywise error rate; but T\(^{\prime}\) should be used if the power of detecting existing differences in the means is of primary concern. If the simple order may be violated, H should be employed.

MSC:

62J10 Analysis of variance and covariance (ANOVA)
62F25 Parametric tolerance and confidence regions
62J15 Paired and multiple comparisons; multiple testing
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