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For coursework, I am doing a two-sided test ($H_0 beta = 0, H_a beta neq 0$). The test itself is a generalized likelihood ratio. The test statistic is the ratio LR:

$$LR=frac{L(beta=0)}{argmax_{beta in R}L(beta)}$$

Where L is the likelihood function.Then $-2ln(LR)$ follows a $chi_1^2$.

I am trying to calculate the p-value for a specific value of LR. Say I look for at a $chi_1^2$ table (or online calc) and find out that

$$ P(chi_1^2 geq -2ln(LR)) = alpha $$

Is $alpha$ my final p-value?

Or do I need to account for the fact that the test is two-sided and set the p-value $= 2*alpha$?

I think I should do the later but I am a bit unsure. Some extra intuition would help.

Edit: added the LR function.

The $p$ value is related to the rejection region, so the answer depends on how you divide $alpha$ over both sides for the test. If you reject H0 when the test statistic is less than $F(alpha/2)$ or more than $F(1-alpha/2)$, then $2alpha$ is the right answer. However, you can use any rejection region $(-infty,a] cup [b,infty)$ as long as $a$ and $b$ satisfy $F^{-1}(b) - F^{-1}(a) = 1-alpha$. For skewed unimodal distributions you can impose $f(a)=f(b)$, which leads to a different answer.

In your example of a generalized likelihood ratio test, the rejection region is an interval $[b,infty)$, even though the test is two sided. So $alpha$ is the p-value.