All sets of means in analysis of variance

TLDR: In this article, the authors proposed a new procedure that includes the analysis of variance F-test as well as decisions on pairwise contrasts by Scheff's method, which is consistent with the decisions of the F-Test as well and with those of Scheffe's method of judging all contrasts, and might be regarded as a practical way of applying the latter.

Abstract: contained in the set of all means involved in an analysis of variance. Significance decisions are based on sums of squares between means within sets, using the same critical value as for the overall F-test. The decisions are shown to be transitive in the sense that any set containing a significant subset is itself significant. However, decisions may be incomplete in that a set may be significant and yet none of its subsets be significant, so that the form of the heterogeneity of the set cannot be inferred with the specified degree of confidence. The new procedure includes the analysis of variance F-test as well as decisions on pairwise contrasts by Scheff 's method. It is consistent with the decisions of the F-test as well as with those of Scheffe's method of judging all contrasts, and might be regarded as a practical way of applying the latter. The probability of making at least one type I error among all the decisions does not exceed the significance level of the overall F-test. Together with Scheff6's method the new procedure may be regarded as providing detailed decisions implicit in significant F-tests. Probabilities of type I errors for sets of any given number of means are defined, and their importance in evaluating multiple comparisons methods pointed out. Tukey's method is seen to imply a range procedure which has advantages when the sample sizes are equal. Steps wise methods such as those of Duncan and of Newman and Keuls are compared with the above procedures in terms of their error probabilities and other properties.