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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.
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
For the test of independence, also known as the test of homogeneity, a chi-squared probability of less than or equal to 0.05 (or the chi-squared statistic being at or larger than the 0.05 critical point) is commonly interpreted by applied workers as justification for rejecting the null hypothesis that the row variable is independent of the ...
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. If the chi-square difference is smaller than the chi-square critical value, the new model fits the data ...
Pinheiro and Bates (2000) showed that the true distribution of this likelihood ratio chi-square statistic could be substantially different from the naïve – often dramatically so. [4] The naïve assumptions could give significance probabilities ( p -values) that are, on average, far too large in some cases and far too small in others.
The likelihood ratio of the two models is computed by G 2 = 2ln[L v / L c] Alternatively, the ratio can be expressed by G 2 = -2ln[L c / L v] where L v and L c are inverted and then multiplied by -2ln. G 2 approximately follows a chi square distribution, especially with larger samples.
An additional reason that the chi-squared distribution is widely used is that it turns up as the large sample distribution of generalized likelihood ratio tests (LRT). [8] LRTs have several desirable properties; in particular, simple LRTs commonly provide the highest power to reject the null hypothesis ( Neyman–Pearson lemma ) and this leads ...
In practice, the likelihood ratio is often used directly to construct tests — see likelihood-ratio test.However it can also be used to suggest particular test-statistics that might be of interest or to suggest simplified tests — for this, one considers algebraic manipulation of the ratio to see if there are key statistics in it related to the size of the ratio (i.e. whether a large ...