# p-value of a test statistic on a two-sided test

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.

• @LinAlg, tks. I realized that in general tests (of a transformation, standardization of X) we need to look at both sides. But in this case, we have a likelihood ratio of a constrained likelihood (in the numerator) divided by an unconstrained likelihood (in the denominator). Thus the smaller the value the greatest the evidence against H0. – LucasMation Nov 17 '18 at 20:39
• By the way, I forgot to mention that -2ln(LR) is what has Chi^2 distribution, not LR itself. Will edit that now. – LucasMation Nov 17 '18 at 20:40
• @LinAlg, please incorporate my point avobe into your answer so I can mark it as a solution to the question. – LucasMation Nov 17 '18 at 20:41
• could you explicitly add what likelihood was in the denominator? – LinAlg Nov 17 '18 at 21:05
• @LinAlg see above – LucasMation Nov 19 '18 at 2:48

## 1 Answer

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 $$2\alpha$$ 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.