Simultaneous Test for Means: An Unblind Way to the F-test in One-way Analysis of Variance

Abstract

After rejecting the null hypothesis in the analysis of variance, the next step is to make the pairwise comparisons to find out differences in means. The purpose of this paper is threefold. The foremost aim is to suggest expression for calculating decision limit that enables us to collect the test and pairwise comparisons in one step. This expression is proposed as the ratio of between square for each treatment and within sum of squares for all treatments. The second aim is to obtain the sampling distribution of the proposed ratio under the null hypothesis. This sampling distribution is derived exactly as the beta distribution of the second type. The third aim is to use beta distribution of second type and adjusted p-values to create adjusted points and decision limit. Therefore, reject the null hypothesis of equal means if any adjusted point falls outside the decision limit. Simulation study is conducted to compute type I error. The results show that the proposed method controls the type I error near the nominal values using Benjamini-Hochbergadjusted p-values and it is very comparative to classical one-way analysis of variance plus pairwise comparisons. Two applications are given to show the benefits of the proposed method