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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 likelihood ratio is central to likelihoodist statistics: the law of likelihood states that the degree to which data (considered as evidence) supports one parameter value versus another is measured by the likelihood ratio. In frequentist inference, the likelihood ratio is the basis for a test statistic, the so-called likelihood-ratio test.
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
Diagram relating pre- and post-test probabilities, with the green curve (upper left half) representing a positive test, and the red curve (lower right half) representing a negative test, for the case of 90% sensitivity and 90% specificity, corresponding to a likelihood ratio positive of 9, and a likelihood ratio negative of 0.111.
Thus the likelihood-ratio test tests whether this ratio is significantly different from one, or equivalently whether its natural logarithm is significantly different from zero. 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 ...
Likelihoodist statistics or likelihoodism is an approach to statistics that exclusively or primarily uses the likelihood function.Likelihoodist statistics is a more minor school than the main approaches of Bayesian statistics and frequentist statistics, but has some adherents and applications.
If it turns out that the sensitivity is high then any person who has the disease is likely to be classified as positive by the test. On the other hand, if the specificity is high, any person who does not have the disease is likely to be classified as negative by the test. An NIH web site has a discussion of how these ratios are calculated. [3]
Here, denotes the likelihood ratio, and the distribution has degrees of freedom equal to the difference in dimensionality of and , where is the full parameter space and is the subset of the parameter space associated with .