ANOVA F Test of Non-Null Hypothesis

Abstract

ANOVA, a test of a null hypothesis, is limited in assessing the statistical significance of differences. This paper considers an ANOVA F test of the non-null hypothesis for comparing k group means. A margin is chosen for the difference of means between each group and the kth group. A non-null hypothesis is defined to be the difference equal to the margin instead of zero. Data are thus prepared under the non-null hypothesis. Then follows the derivation of the one-way ANOVA non-null F test and its power. It reduces to the classical F test on setting the margin equal to zero. The observed size of it is identical to that of the F test and is near the nominal level of significance. The observed power is close to the power in balanced designs. With the non-null F test, it enables inferences to extend to the equivalence of group means or the clinical significance of differences. An example is taken to analyze both non-inferiority trials and k-sample equivalence trials.