Comparing two nested models by the ratio of their maximum likelihoods; its distribution follows Wilks' theorem.The likelihood ratio test is a statistical method for comparing two nested models — that is, models where one is a special case of the other. It works by taking the ratio of the maximum likelihoods of the two models, which quantifies how much better the more complex model fits the data compared to the simpler one. This test is fundamental in physics and other sciences for deciding whether adding extra parameters (like a new particle or a new interaction) is justified by the data, or whether the simpler model is sufficient.
The likelihood ratio test was introduced by Jerzy Neyman and Egon Pearson in 1928, building on earlier work by Ronald Fisher on maximum likelihood estimation.
For each model, you compute the maximum likelihood — the probability of the observed data given the best-fit parameters. The likelihood ratio is the maximum likelihood of the simpler model divided by that of the more complex model. A small ratio means the complex model fits much better. To decide if the improvement is significant, you compare the test statistic (often twice the negative log of the ratio) to a reference distribution. Under certain conditions, this statistic follows a chi-squared distribution, which lets you assign a p-value and decide whether the extra complexity is warranted.
It is widely used in particle physics to claim discoveries, such as comparing a background-only model to one with a new particle signal, and in cosmology for model selection.
The test statistic's distribution was famously proven by Samuel Wilks in 1938, but the result only holds asymptotically — for large sample sizes — and can misbehave when parameters are on boundaries.
