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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.
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.
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
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:
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).
The log diagnostic odds ratio is sometimes used in meta-analyses of diagnostic test accuracy studies due to its simplicity (being approximately normally distributed). [ 4 ] Traditional meta-analytic techniques such as inverse-variance weighting can be used to combine log diagnostic odds ratios computed from a number of data sources to produce ...
The way it is done there is that we have two approximately Normal distributions (e.g., p1 and p2, for RR), and we wish to calculate their ratio. [b] However, the ratio of the expectations (means) of the two samples might also be of interest, while requiring more work to develop. The ratio of their means is: