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When I do an F test(think of an F test you'd learn about in a first year stats paper) I get a ratio of variance between/variance within.

I understand what the indivudual components are i.e the variance between and the variance within. However, I dont understand why they are used as a ratio, why are they compared to one another?

So if I got thefollowing from my F test , 120/20, it seems the F statistic is saying for every 6 units of variance between there is 1 unit of variation within...and it is this that I dont understand. What does it actually mean, or how can it be thought of intuitively.

thanks

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If the null hypothesis is true, the two variance estimates should be estimating the same thing ($\sigma^2$), but because of random variation, even when the null hypothesis is true the ratio can be some distance from 1.

However, an unusually large value indicates that the "between" estimate of variance is larger (relative to the "within" one) than would be consistent with the null being true.

e.g. in the case of one-way ANOVA, if the means differ, this inflates the "between" estimate of variance to have expected value larger than $\sigma^2$ (while the within estimate isn't affected, it still has expectation $\sigma^2$). This drives the distribution of the ratio upward (it's no longer from the usual central F-distribution that we have under the null, but - if the assumptions were otherwise true - it would have a noncentral F distribution).

Depending on sample sizes, 120/20 might just be the ratio of two very noisy estimates of the same $\sigma^2$, or the numerator might be estimating something much bigger than $\sigma^2$ (if the df residual is large enough*, the first explanation becomes untenable for F that big.)

* at 5% significance level, roughly speaking denominator df of 6 would be "large enough" (and at 1%, 8 would be "large enough") to find an F-ratio of "6" inconsistent with the null.

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  • $\begingroup$ Thanks, this has highlighted what I dont understand. Namely, why do we expect the two variance estimates to be estimating ther same thing ? I am unsure why teh between and within varaince woudl be equal in teh case where the null hypothesis is true...maybe I am overthinking it. $\endgroup$ – B.Miller Sep 25 '15 at 1:30

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