In statistics, the likelihood-ratio test is a hypothesis test that involves comparing the goodness of fit of two competing statistical models, typically one found by maximization over the entire parameter space and another found after imposing some constraint, based on the ratio of their likelihoods.
If the more constrained model (i.e., the null hypothesis) is supported by the observed data, the two likelihoods should not differ by more than sampling error.
[3] In fact, the latter two can be conceptualized as approximations to the likelihood-ratio test, and are asymptotically equivalent.
[4][5][6] In the case of comparing two models each of which has no unknown parameters, use of the likelihood-ratio test can be justified by the Neyman–Pearson lemma.
The lemma demonstrates that the test has the highest power among all competitors.
The likelihood ratio test statistic for the null hypothesis
is given by:[8] where the quantity inside the brackets is called the likelihood ratio.
Often the likelihood-ratio test statistic is expressed as a difference between the log-likelihoods where is the logarithm of the maximized likelihood function
for the sampled data) and denote the respective arguments of the maxima and the allowed ranges they're embedded in.
converges asymptotically to being χ²-distributed if the null hypothesis happens to be true.
[9] The finite-sample distributions of likelihood-ratio statistics are generally unknown.
: In this case, under either hypothesis, the distribution of the data is fully specified: there are no unknown parameters to estimate.
For this case, a variant of the likelihood-ratio test is available:[11][12] Some older references may use the reciprocal of the function above as the definition.
, via the relation The Neyman–Pearson lemma states that this likelihood-ratio test is the most powerful among all level
The likelihood-ratio test rejects the null hypothesis if the value of this statistic is too small.
How small is too small depends on the significance level of the test, i.e. on what probability of Type I error is considered tolerable (Type I errors consist of the rejection of a null hypothesis that is true).
The numerator corresponds to the likelihood of an observed outcome under the null hypothesis.
Low values of the likelihood ratio mean that the observed result was much less likely to occur under the null hypothesis as compared to the alternative.
High values of the statistic mean that the observed outcome was nearly as likely to occur under the null hypothesis as the alternative, and so the null hypothesis cannot be rejected.
Suppose that we have a random sample, of size n, from a population that is normally-distributed.
Both the mean, μ, and the standard deviation, σ, of the population are unknown.
The likelihood function is With some calculation (omitted here), it can then be shown that where t is the t-statistic with n − 1 degrees of freedom.
If the distribution of the likelihood ratio corresponding to a particular null and alternative hypothesis can be explicitly determined then it can directly be used to form decision regions (to sustain or reject the null hypothesis).
In most cases, however, the exact distribution of the likelihood ratio corresponding to specific hypotheses is very difficult to determine.
[citation needed] Assuming H0 is true, there is a fundamental result by Samuel S. Wilks: As the sample size
, and if the null hypothesis lies strictly within the interior of the parameter space, the test statistic
[14] This implies that for a great variety of hypotheses, we can calculate the likelihood ratio