Likelihood Ratio Test:
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").
The likelihood ratio test is based on the likelihood ratio r as the test statistic:

where X is the observed data (sample), P(X  H) is the conditional probability of X provided the hypothesis H is true, H_{0} is the null hypothesis, H_{1} is the alternative hypothesis. See also Likelihood function.
According to the NeymanPearson 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.
If the symbols do not display properly, try
the graphic version of this page