A METHOD FOR JUDGING ALL CONTRASTS IN THE ANALYSIS OF VARIANCE*

Abstract

A simple answer is found for the following question which has plagued the practice of the analysis of variance: Under the usual assumptions, if the conventional. F -test of the hypothesis H : μ 1 =μ 2 =…=μ k at the α level of significance rejects H , what further inferences are valid about the contrasts among the μ i (beyond the inference that the values of the contrasts are not all zero)? Suppose the F -test has k −1 and v degrees of freedom. For any c 1 , …, c k with ∑ 1 k c i = 0 write θ for the contrast ∑ 1 k c i μ i , ⁠ , and write ^ θ and σ ^ θ ^ 2 for the usual estimates of θ and the variance of ^ θ ⁠ . Then for the totality of contrasts, no matter what the true values of the θ's, the probability is 1−α that they all satisfy θ ^ - S σ ^ θ ^ ≤ θ ≤ θ ^ + S σ ^ θ ^ , where S 2 is ( k −1) times the upper α point of the F -distribution with k −1 and v degrees of freedom. Suppose we say that the estimated contrast with ^ θ is ‘significantly different from zero’ if | ^ θ > S ^ σ ^ θ . Then the F -test rejects H if and only if some ^ θ are significantly different from zero, and if it does, we can say just which ^ θ ⁠ . More generally, the above inequality can be employed for all the contrasts with the obvious frequency interpretation about the proportion of experiments in which all statements are correct. Relations are considered to an earlier method of Tukey using the Studentized range tables and valid in the special case where the ^ μ i all have the same variance and all pairs ^ μ i , ^ μ j (i≠j) have the same covariance. Some results are obtained for the operating characteristic of the new method. The paper is organized so that the reader who wishes to learn the method and avoid the proofs may skip §§ 2 and 5.