A Method for Judging all Contrasts in the Analysis of VarianceHenry ScheffeBiometrika, Vol. 40, No. 1/2. (Jun., 1953), pp. 87-104.Stable URL:http://links.jstor.org/sici?sici=0006-3444%28195306%2940%3A1%2F2%3C87%3AAMFJAC%3E2.0.CO%3B2-DBiometrika is currently published by Biometrika Trust.Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available athttp://www.jstor.org/about/terms.html. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtainedprior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content inthe JSTOR archive only for your personal, non-commercial use.Please contact the publisher regarding any further use of this work. 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For more information regarding JSTOR, please contact [email protected]://www.jstor.orgMon Oct 22 15:42:09 2007A METHOD FOR JUDGING ALL CONTRASTS IN THE ANALYSIS OF VARIANCE* BYHENRY SCHEFP*, Columbia University 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 P-test of the hypothesis H:p, =p, =.. . =p, at the a level of significance rejects H, what further inferences are valid about the contrasts among the pi (beyond the inference that the values of the contrasts are not all zero) ? Suppose the F-test has k -1 k k and v degrees of freedom. For any c,, ...,ck with C ci = 0 write 0 for the contrast cipi, and write 1 18 and for the usual estimates of 0 and the variance of 8. Then for the totality of contrasts, no matter what the true values of the 8'9, the probability is 1 -a that they all satisfy where Sais (k- 1) times the upper a point of_the P-distribution with k- 1 and v degrees of freedom. Suppose we say that the estimated contrast f3is 'significantly different from zero' if 181 >Si?;. Then the P-test r2jects H if and only if some 8 are significantly different from zero, and if it does, we can say just which 0. Xore generally, the above inequality can be employed for all the contrasts with the obvious frequency interpretationabout the proportionof experimentsin whichall statementsare correct. Relations are considered to an earlier method of Tukey using the Studentized range tables and valid in the special case where the Piall have the same variance and all pairs &, p, (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 $5 2 and 5. The general problem is that of making inferences about the contrasts among a set of 'true means ' or 'true main effects ' p,, p,, ...,pk in the analysis of variance. For example, the pi might be the true row effects in a two-way lay-out with possibly unequal numbers of observations per cell. The pi may be unrestricted or subject to a single restriction of the form k where the hiand h are known constants with hi+ 0. A contrast is a linear function of the pi, k 1 6' = Xcipi, (2)1 determined by k known constants ci satisfying the condition The value of the linear function for a particular set ofp, will be called the value of the contrast; it will not cause any confusion in the following to use the same symbol 8both for the contrast and the value of the contrast. We make the assumptions usual in the analysis of variance, namely, that there is at hand a set of statistics ,ill,,i12, ...,,ilk, and 62, such that the ,ili have a multivariate normal dis- tribution and are statistically independent of 3,that E(,ili)=pi (i'=1, ...,k), and . . cov (,ili, jj)= aij v2 (z,~= 1,...,k), (4) where the constants aij are known, and v2 is unknown. The pi will always be 'Model I' (non-random) effects, as discussed by Eisenhart (1947) or Mood (1950). In a pure Model I * Work sponsored by the Office of Naval Research (U.S.A.).88 A method for judging all contrasts in the analysis of variance situation, (r2is the variance (r2,of a single observation ('error variance'). In a mixed model situation where there exists an exact F-test of the hypothesis H: p1 =,U2 = ... =pk, (5) (r2 will equal (r2, plus further unknown non-negative parameters. In any case, 82 is an estimate of (r3with v D.F. (degrees of freedom), that is, vS2/(r2has the ~2 distribution with vD.F. The case where (r2 is known can be treated by obvious modifications of the theory below, usually merely by putting v = coand $2 = (r2in the results. It is further assumed that if the pi are unrestricted the rank* of the covariance matrix with elements (4)is k, and that if the pi are subject to a restriction (1)then the liiare subject to the same restriction (1)and the rank of the covariance matrix is k -1. The hypothesis H in (5), equivalent to the statement that all the contrasts are zero, can be tested by the conventional F-statistic with k -1 and ~D.F.We shall refer to this test at significance level a as 'the ' F-test of H. The problem of making further inferences about the contrasts, arising when the F-test rejects H, has been considered by various writers, including R. A. Fisher (1935), D. Newman (1939), J. W. Tukey (1951), and H. K. Nandi (1951). Except for Tukey's and Nandi's, the methods involve repeated tests of significance on the same data, and are hence subject to the usual objection that little is known about the joint operating characteristic. While it is often not possible in practical applications to avoid repeated tests of significance, it is possible for the particular problem we are considering. The solution studied in this paper is based on the following probability statement about the infinite totality of contrasts:? For any contrast (2) denote its estimate by 8, the variance of 8by (r:,