An extension of the maximum F ratio with unequal degrees of freedom

Abstract

A set of tables for the upper 5 and 1 percent critical points of the maximum F ratio test for homogeneity of variance is presented. These tables cover the situation where the set, k, of mean square estimates the variance can be divided into two groups of k₁ and k₂, such that k= k₁ + k₂ . That is, there are k independent mean square estimates of variance with v₁ degrees of freedom and k₂ independent mean square estimates of variance with v₂ degrees of freedom. These tables are for k between 2 and 12, and the degrees of freedom are between 2 and 16 in increments of 2. A more exact approximation is also developed to include the cases where the degrees of freedom do not satisfy the assumptions used to develop the tables. This approximation uses the lower and upper degrees as well as k and the present tables; whereas, the previous approximation used to k, v (line over v) and tables of the critical points of the maximum F ratio test where all the degrees of freedom were equal. The approximation developed herein is shown to be 'better' that the previously used approximation when compared with the exact F (subscript max.) values for small k. An extension of the approximation is also presented. This discussion shows how the approximation can be generalized to develop approximations for other tests of homogeneity of variance. More specifically, the approximation is applied to Bartlett's M statistic. These approximations can be used to decrease computational labor involved in some of the tests for homogeneity of variance.