F-test in multiple linear regression

Asked Feb 08 '19 at 15:29

Active Feb 08 '19 at 17:20

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I'm currently reading Introduction to statistical learning. When trying to prove the collective significance of a regression linear model, we use the F-test with the following formula.

$$F=\frac{(TSS-RSS)/p}{RSS/(n-p-1)}$$

Whre $TSS=\sum(y_i-/y)²$, and $RSS=\sum(y_i-\hat y_i)²$

What I understood from this formula is that we would want that $F$ be larger than $1$ in order to reject the null hypothesis, because that means the amount of variability in $y$ explained by our model ($TSS-RSS$) is larger than the amount that the model couldn't remove ($RSS$).

Anyway, I am stuck on proving the result in the following section:

If the linear model assumptions are correct, one can show that: $E{RSS/(n − p − 1)} = σ²$ and that, provided $H0$ is true, $E{(TSS − RSS)/p} = σ²$.

with: $H_0$: $b_1=b_2=...=b_p=0$

$b_i$ is the coefficient for the ith regressor.

So, what I want is a proof or a hint for the quote above. And additional explanations would be appreciated.

edited Feb 08 '19 at 17:20

asked Feb 08 '19 at 15:29

Youssef Esseddiq

σ² is the variance of the error term. – Youssef Esseddiq Feb 08 '19 at 15:32
Do you understand how the F-test works in an ANOVA? – gung - Reinstate Monica Feb 08 '19 at 15:36
No not very much, I know F-test is used to compare models fitted to a certain dataset..it would be very helpful if you could explain it. – Youssef Esseddiq Feb 08 '19 at 15:42
You may glean some insight from my answer here: [How does the standard error work?](https://stats.stackexchange.com/a/33627/7290) – gung - Reinstate Monica Feb 08 '19 at 15:45
It looks like you mis-stated the null hypothesis. – Isabella Ghement Feb 08 '19 at 16:58
gung could you please provide some insights about what are the populations we trying to compare using anova in this situation? – Youssef Esseddiq Feb 09 '19 at 13:04

F-test in multiple linear regression

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