A Test of an Experimental Hypothesis of Negligible Difference between Means

Haitovsky's recent note (1) in The American Statistician shows that R2 (the corrected multiple correlation coefficient) will increase when a single variable is removed from a regression equation if and only if the corresponding t-statistic is less than unity. This is a special case of the mnore general property that R2 will increase when a group of variables are removed if and only if the corresponding F-statistic is less than unity. The generalization of t-tests to F-tests in the analysis of variance framework is well known, and it is nice to be able to tie in the R2 statistic for pedagogical purposes. We shall give the proof in terms of the estimated equation error variance. Of course, R2 bears an exact mirror relation to the estimated variance, that being R = 1 (v/c) where v is the sum of squared errors divided by degrees of freedom and c is the varianice of the dependent variable. Starting with k original variables and removinig r of thlem, we may study the resulting changes in -the error variance estimates from the analysis of variance form: