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Is there a way to test if the variance among groups is equal to the variance within groups, and if not, which variance is larger?

Answers on related posts all seem to point to either an F-test or a Levene's Test, but those aren't really making the comparison that I'm looking for. F tests use the ratio of variances to test for the homogeneity of means, but I'm interested in explicitly testing whether the variance within groups is larger/smaller/equal to the variance among groups. Similarly, I'm not interested in testing whether or not the within-group variance is homogeneous among groups (with Levene's Test or one of the alternatives).

My only thought is that a mixed model, with a random effect for group, gives an estimate of the variance among groups, while the combination of fixed effects and residual variance comprises the within-group variation. But it doesn't actually directly compare the two.

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This is the definition of a variance ratio test, usually accomplished with the F test (less commonly, chi square). And, your use case of testing between and within groups variances IS the ANOVA F test

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