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The region surrounds the maximum-likelihood estimate, and all points (parameter sets) within that region differ at most in log-likelihood by some fixed value. The χ 2 distribution given by Wilks' theorem converts the region's log-likelihood differences into the "confidence" that the population's "true" parameter set lies inside. The art of ...
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.
In statistics, the likelihood principle is the proposition that, given a statistical model, all the evidence in a sample relevant to model parameters is contained in the likelihood function. A likelihood function arises from a probability density function considered as a function of its distributional parameterization argument.
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.
Recall that for the multinomial model, the MLE of ^ given some data is defined by ^ = Furthermore, we may represent each null hypothesis parameter ~ as ~ = Thus, by substituting the representations of ~ and ^ in the log-likelihood ratio, the equation simplifies to ((~ |) (^ |)) = = = = Relabel the variables with and with .
The relation can also be estimated by a so-called Fagan nomogram (shown at right) by making a straight line from the point of the given pre-test probability to the given likelihood ratio in their scales, which, in turn, estimates the post-test probability at the point where that straight line crosses its scale.
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:
In statistics, maximum likelihood estimation (MLE) is a method of estimating the parameters of an assumed probability distribution, given some observed data. This is achieved by maximizing a likelihood function so that, under the assumed statistical model , the observed data is most probable.