# Likelihood ratio test We can construct hypothesis tests by taking the ratio of the maximized likelihood under the alternative hypothesis and the maximized likelihood under the null hypothesis: $ \text{LRT} = -2 \log \left( \frac{\arg\max_{\theta \in \Theta_1} L(\theta)}{\arg\max_{\theta \in \Theta_0} L(\theta)} \right) $ If $H_1$ is more likely, then $\text{LRT}$ will be larger. The F-test is an example of a test that comes out of the likelihood ratio test when the likelihood has a specific form (Normal). In situations where the likelihood can be approximated by a Normal distirbution (regularity conditions), then the $\text{LRT}$ will have an approximate chi-squared distribution. --- # References [[Applied Linear Regression#6. Testing and Analysis of Variance]]