On Splitting the Tails Unequally: a New Perspective on One-Versus Two-Tailed Tests

Abstract

The controversy that has raged since the early fifties regarding the admissibility of one-tailed tests of hypotheses was examined. From the review of that literature, it was concluded that the main advantage of the one-tailed test was the gain in power for the prediction while its main disadvantage was its inability to test for significance if the results were opposite to prediction. It is argued here that splitting α unequally between the two tails, placing most of the rejection region on the side of the prediction but a smaller fraction on the opposite side provides both power and the ability to detect opposite-to-prediction outcomes. This compromise procedure requires a finer choice in the splitting of α than the dichotomous choice of putting either all or exactly half of α in the favored tail, i.e., the choice between a one- or a two-tailed test. Rules for the most effective split, based on Bayesian considerations, are prescribed. The fraction of α in the predicted tail should be equal to the investigator's a priori probability that the predicted order, as opposed to the reversed order, of sample means will be obtained. A table of t-values is presented which gives critical regions for significance, both "expected" and "unexpected," at specified levels of a priori probability.