# Testing within-model equality of parameters — why the F-distribution? (and also a question about degrees of freedom)

Let me start out by saying I'm somewhat of a novice at statistics. I've had some formal training but it was rather patchy, and I'm aware I'm ignorant of many key concepts. I'm having to "learn on the job", as it were.

I have a linear model that looks like this:

$$y = A_0 + A_1x_1 + A_2x_2$$

Where $A_0$, $A_1$, and $A_2$ are fitted parameters, and $x_1$ and $x_2$ are the two features that each data point can have.

Here is what I need to do: test the hypothesis $H_0$ that $A_1 = A_2$ (or $A_1 - A_2 = 0$). I looked online and found a paper that explains how to do this.

From reading this, it looks as though I need to do the following:

• Calculate this test statistic:

$$\left(\frac{A_1 - A_2}{\sqrt{s_{A_1}^2 + s_{A_2}^2 - 2s_{A_1}s_{A_2}}}\right)^2$$

• And compare that quantity to this F-distribution:

$$F_{1, N-K-1}$$

Where I'm assuming K = 3 because I have three parameters.

My questions:

• Why do we use an F-distribution here? I've used F before when comparing two variances, but here the things I'm comparing aren't variances, they're just fitted parameters.

• Why are the degrees of freedom $1$ and $N-K-1$? Again, I get how the degrees of freedom work when I'm comparing variances of two samples (it's $n_1 -1$ and $n_2 -1$), and I get how it works when I'm doing a T-test on a single parameter (e.g. for testing $A_1 = 0$ I have $n-3$ degrees of freedom), but what's going on with the $1$ there? And why the extra $-1$ in the other one?

• Why is the whole quantity squared? The PDF says "since an F test is being reported, all of this is squared", but whenever I've done F-tests before I never read anything that said I should square the result ...

• Ignore the square and evaluate the test statistic with a Student t distribution. – whuber Dec 11 '17 at 21:53
• Are you saying the PDF is wrong? How comes? – dain Dec 11 '17 at 22:08
• No, it's not wrong: the recipe I gave you is equivalent to the one you quote. (It's actually a little better, because it allows two-sided tests of $H_0$ whereas the $F$ test allows only a one-sided tset.) – whuber Dec 11 '17 at 22:13
• Okay thank you. And one more thing -- the t-test degrees of freedom should still be $N-K$, with $K$ the number of parameters? – dain Dec 11 '17 at 22:18
• I doubt it. The t test usually has $N-K-1$ degrees of freedom, assuming you are performing the fitting using ordinary least squares. As a check, consider the case $K=0$: there, you are comparing the mean of the data to $0$ and--as you well know--the degrees of freedom are $N-1$. – whuber Dec 11 '17 at 22:20