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The likelihood-ratio test, also known as Wilks test, [2] is the oldest of the three classical approaches to hypothesis testing, together with the Lagrange multiplier test and the Wald test. [3] In fact, the latter two can be conceptualized as approximations to the likelihood-ratio test, and are asymptotically equivalent.
For example: 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 (()) = with = degrees of freedom, and is equal to .
The commonly used chi-squared tests for goodness of fit to a distribution and for independence in contingency tables are in fact approximations of the log-likelihood ratio on which the G-tests are based. [4] The general formula for Pearson's chi-squared test statistic is
Rather than the Wald method, the recommended method [21] to calculate the p-value for logistic regression is the likelihood-ratio test (LRT), which for these data give (see § Deviance and likelihood ratio tests below).
Numerous other tests can be viewed as likelihood-ratio tests or approximations thereof. [15] The asymptotic distribution of the log-likelihood ratio, considered as a test statistic, is given by Wilks' theorem. The likelihood ratio is also of central importance in Bayesian inference, where it is known as the Bayes factor, and is used in Bayes' rule.
Likelihood Ratio: An example "test" is that the physical exam finding of bulging flanks has a positive likelihood ratio of 2.0 for ascites. Estimated change in probability: Based on table above, a likelihood ratio of 2.0 corresponds to an approximately +15% increase in probability.
When two models are nested, models can also be compared using a chi-square difference test. The chi-square difference test is computed by subtracting the likelihood ratio chi-square statistics for the two models being compared. This value is then compared to the chi-square critical value at their difference in degrees of freedom.
In statistics, the score (or informant [1]) is the gradient of the log-likelihood function with respect to the parameter vector. Evaluated at a particular value of the parameter vector, the score indicates the steepness of the log-likelihood function and thereby the sensitivity to infinitesimal changes to the parameter