In statistics, a likelihood ratio test is a statistical test used for comparing the goodness of fit of two models, one of which (the null model) is a special case of the other (the alternative model). The test is based on the likelihood ratio, which expresses how many times more likely the data are under one model than the other. This likelihood ratio, or equivalently its logarithm, can then be used to compute a p-value, or compared to a critical value to decide whether to reject the null model in favour of the alternative model. When the logarithm of the likelihood ratio is used, the statistic is known as a log-likelihood ratio statistic, and the probability distribution of this test statistic, assuming that the null model is true, can be approximated using Wilks’ theorem.
Contents
- Simple hypotheses
- Composite hypotheses
- Interpretation
- Distribution Wilks theorem
- Extensions
- Use
- Coin tossing
- References
In the case of distinguishing between two models, each of which has no unknown parameters, use of the likelihood ratio test can be justified by the Neyman–Pearson lemma, which demonstrates that such a test has the highest power among all competitors.
Simple hypotheses
A statistical model is often a parametrized family of probability density functions or probability mass functions
Note that under either hypothesis, the distribution of the data is fully specified; there are no unknown parameters to estimate. The likelihood ratio test is based on the likelihood ratio, which is often denoted by
or
where
The values
The Neyman-Pearson lemma states that this likelihood ratio test is the most powerful among all level
Composite hypotheses
A null hypothesis is often stated by saying the parameter
The likelihood function is
Here, the
A likelihood ratio test is any test with critical region (or rejection region) of the form
Interpretation
Being a function of the data
The numerator corresponds to the likelihood of an observed outcome under the null hypothesis. The denominator corresponds to the maximum likelihood of an observed outcome varying parameters over the whole parameter space. The numerator of this ratio is less than the denominator. The likelihood ratio hence is between 0 and 1. Low values of the likelihood ratio mean that the observed result was 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 the null hypothesis cannot be rejected.
The likelihood-ratio test requires nested models – models in which the more complex one can be transformed into the simpler model by imposing a set of constraints on the parameters. If the models are not nested, then a generalization of the likelihood-ratio test can usually be used instead: the relative likelihood.
Distribution: Wilks’ theorem
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 accept/reject the null hypothesis). In most cases, however, the exact distribution of the likelihood ratio corresponding to specific hypotheses is very difficult to determine. A convenient result by Samuel S. Wilks, says that as the sample size
Extensions
Wilks’ theorem assumes that the true but unknown values of the estimated parameters are in the interior of the parameter space. This is commonly violated in random or mixed effects models, for example, when one of the variance components is negligible relative to the others. In some such cases, one variance component is essentially zero relative to the others or the models are not properly nested. Pinheiro and Bates showed in 2000 that the true distribution of this likelihood ratio chi-square statistic could be substantially different from the naïve
In general, to test random effects, they recommend using Restricted maximum likelihood (REML). For fixed effects testing, they say, “a likelihood ratio test for REML fits is not feasible, because” changing the fixed effects specification changes the meaning of the mixed effects, and the restricted model is therefore not nested within the larger model.
As a demonstration, they set either one or two random effects variances to zero in simulated tests. In those particular examples, the simulated p-values with k restrictions most closely matched a 50-50 mixture of
Pinheiro and Bates also simulated tests of different fixed effects. In one test of a factor with 4 levels (degrees of freedom = 3), they found that a 50-50 mixture of
To be clear: These limitations on Wilks’ theorem do not negate any power properties of a particular likelihood ratio test. The only issue is that a
Use
Each of the two competing models, the null model and the alternative model, is separately fitted to the data and the log-likelihood recorded. The test statistic (often denoted by D) is twice the log of the likelihoods ratio, i.e., it is twice the difference in the log-likelihoods:
The model with more parameters (here alternative) will always fit at least as well—i.e., have the same or greater log-likelihood—than the model with fewer parameters (here null). Whether the fit is significantly better and should thus be preferred is determined by deriving the probability or p-value of the difference D. Where the null hypothesis represents a special case of the alternative hypothesis, the probability distribution of the test statistic is approximately a chi-squared distribution with degrees of freedom equal to
Here is an example of use. If the null model has 1 parameter and a log-likelihood of −8024 and the alternative model has 3 parameters and a log-likelihood of −8012, then the probability of this difference is that of chi-squared value of
Coin tossing
An example, in the case of Pearson’s test, we might try to compare two coins to determine whether they have the same probability of coming up heads. Our observation can be put into a contingency table with rows corresponding to the coin and columns corresponding to heads or tails. The elements of the contingency table will be the number of times the coin for that row came up heads or tails. The contents of this table are our observation X.
Here Θ consists of the possible combinations of values of the parameters
Similarly, the maximum likelihood estimates of
which does not depend on the coin i.
The hypothesis and null hypothesis can be rewritten slightly so that they satisfy the constraints for the logarithm of the likelihood ratio to have the desired nice distribution. Since the constraint causes the two-dimensional H to be reduced to the one-dimensional
For the general contingency table, we can write the log-likelihood ratio statistic as