# Is there an extension to the F-test?

I'm new to statistics but I come from a mathematical background. So while learning about the t-test, which compares sample means, and the f-test which compares variances is was wondering if there is an extension to the f-test (correct me if I'm wrong), similar to the ANOVA for the t-test?

ANOVA is "used to analyze the differences among means (from Wikipedia)". So, as far as I understand it, it is an extension to the t-test.

I did some research and found out about the Bartlett's test, which in my understanding would be an extension to the f-test, is that correct?

My confusion comes mainly from how the terms are used in some software. For instance g*power. Here the anova is found in the section of the f-tests, which doesnt make any sense to me...

• The usual F-test is a key player in comparing means through ANOVA. Not the sample variances are being compared, but basically variances of residuals with and without considering the group info. Dec 3, 2021 at 9:15
• Do you mean a test that tests the equality of $3+$ variances that turns into the $F$-test when you compare two variances? // @MichaelM An F-test can be used to test $H_0:\sigma_1=\sigma_2$ vs $H_a:\sigma_1\ne\sigma_2$. I think the R command is var.test.
– Dave
Dec 3, 2021 at 10:58
• Just to avoid confusion, note that some common statistical tests --- like F-test, t-test, chi-square test --- are each used for different purposes in different circumstances. Like, we use an F-test in ANOVA, but there is also an F-test that is used to compare variances of two groups. Sometimes we speak loosely, saying "t-test" to imply Student's t-test for two unpaired samples, but there are lots of other places where a t-test is used in analysis of experiments. Dec 3, 2021 at 13:58
• Building on @SalMangiafico, ANOVA has one of the most confusing names in all of science. "Analysis of variance" must be a test of the variances, right? No! ANOVA cleverly uses an F-test of variance equality to test the equality of multiple means. Despite the name, ANOVA is a test of means. However, the F-test of variance equality can be used to test variances. JBStatistics on YouTube has some videos on this. // Bartlett's test is not a generalization of the F-test, though it is a test of $\sigma_1 =\cdots =\sigma_n$. When sample sizes are unequal, var.test and bartlett.test diverge.
– Dave
Dec 3, 2021 at 14:24

You seem to be asking two questions in one. First, how come ANOVA compares 3+ means using an F-test which is designed to compare 2 variances. Second, is there a test to compare 3+ variances in the same way ANOVA compares 3+ means. I'll answer both:

1. Although the main question answered by the ANOVA does generalize that of the t-test as you have well understood (do any of these (>2) distributions have different means?), the technical way to answer the question is through an F test. The reason being that the smaller the residual variance is compared to the original variance, the further apart the means must be (and vice versa). However, the F-test only tells you whether or not there seems to be a difference between some means, but it doesn't tell you in any way which means are concerned (no matter how obvious you think it is by looking at the data. This is why ANOVA's F-test is usually completed by honest t-tests, with p-values corrected for the amount of tests:
• you call these tests post-hoc tests if you're going to compare a lot if not all possible pairs of means

• or you have the option of the more crafty contrasts, in case you know exactly where to look from the start and you don't want to ruin your statistical power by adjusting p-values for a huge number of useless tests.

1. Bartlett's test, as you mention, does exactly the job of testing the equality of variances in a number of populations. Levene's test does this job as well, and a discussion on which one to use under which conditions can be found here: Bartlett's test vs Levene's test
• This does not appear to answer the question, which concerns the comparison of $3+$ variances. Perhaps you can edit the post to clarify how what you wrote addresses that.
– Dave
Dec 3, 2021 at 11:00
• @Dave My understanding of the question is that the OP finds it confusing that ANOVA compares 3+ means using an F-test which is designed to compare 2 variances. So I addressed this point. Dec 3, 2021 at 13:22
• @Dave I can see your point, having read the question again (I was focused on the question conveyed by the last sentence of the OP). Thanks Dec 3, 2021 at 13:32
• @Dave it does answer my question. Arnaud Mortier thanks for the elaborate explanation! Dec 4, 2021 at 6:23
• @Stanissse You're welcome! Dec 6, 2021 at 14:41

The F test for ANOVA uses the ratio of two quantities, which when the population means are equal (and the assumptions hold) will both be independent estimates of $$\sigma^2$$. The one on the numerator estimates it from variation in group means, the one on the denominator estimates it from within-group variation about the group means (which will be consistent for $$\sigma^2$$ even when the population means differ).

When $$H_0$$ is false the numerator is biased upward as an estimate of $$\sigma^2$$, since it's got the effect of the variation in population group-means (under the alternative, the statistic is distributed as noncentral F).

That is, it is still effectively using a test for equality of two variance estimates, but it's using that to test equality of population means.