The likelihood ratio test is aimed at testing a simple null hypothesis against a simple alternative hypothesis. (See Hypothesis for an explanation of “simple hypothesis”).
P(X | H0)
| r = | P(X | H1) |
where X is the observed data (sample), P(X | H) is the conditional probability of X provided the hypothesis H is true, H0 is the null hypothesis, H1 is the alternative hypothesis. See also Likelihood function.
According to the Neyman-Pearson lemma, the likelihood ratio test is the most powerful test for any significance level (probability of Type I error). See also Power of a Hypothesis Test.
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