Likelihood Ratio Test

The likelihood ratio (LR) test is the MLE analogue of the F-test. To test a set of $q$ restrictions, estimate the model under the restrictions (restricted, $\mathcal L_R$) and without them (unrestricted, $\mathcal L_{UR}$), then compute $LR = 2[\log \mathcal L_{UR} - \log \mathcal L_R]$. Under the null, $LR \sim \chi^2(q)$; reject if $LR$ exceeds the critical value. Intuition: if removing the restricted variables makes the fit much worse, those variables matter.

When to use

Test any nested-model restriction estimated by MLE — joint significance of multiple variables in a logit/probit, the validity of a constraint, the addition of a polynomial term. In R, lmtest::lrtest(unrestricted, restricted) does it. Contrast with the F-test, which compares sums of squared residuals from OLS models and uses an F distribution — same logic, different distribution.