Ordered Alternatives and the 1 1/2-Tail Test

Under proper conditions, two independent tests of the null hypothesis of homogeneity of means are provided by a set of sample averages.One test, with tail probability P 1, relates to the variation between the sample averages, while the other, with tail probability P 2, relates to the concordance of the rankings of the sample averages with the anticipated rankings under an alternative hypothesis.The quantity G = P 1 P 2 is considered as the combined test statistic and, except for the discreteness in the null distribution of P 2, would correspond to the Fisher statistic for combining probabilities.Illustration is made, for the case of four means, on how to get critical values of G or critical values of P 1 for each possible value of P 2, taking discreteness into account.Alternative measures of concordance considered are Spearman's . and Kendall's ..The concept results, in the case of two averages, in assigning two-thirds of the test size to the concordant tail, one-third to the discordant tail.