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.

  • $\begingroup$ @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. $\endgroup$ – LucasMation Nov 17 '18 at 20:39
  • $\begingroup$ By the way, I forgot to mention that -2ln(LR) is what has Chi^2 distribution, not LR itself. Will edit that now. $\endgroup$ – LucasMation Nov 17 '18 at 20:40
  • $\begingroup$ @LinAlg, please incorporate my point avobe into your answer so I can mark it as a solution to the question. $\endgroup$ – LucasMation Nov 17 '18 at 20:41
  • $\begingroup$ could you explicitly add what likelihood was in the denominator? $\endgroup$ – LinAlg Nov 17 '18 at 21:05
  • $\begingroup$ @LinAlg see above $\endgroup$ – LucasMation Nov 19 '18 at 2:48

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.


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