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The likelihood ratio is a function of the data ; therefore, it is a statistic, although unusual in that the statistic's value depends on a parameter, . The likelihood-ratio test rejects the null hypothesis if the value of this statistic is too small.
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 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
G-tests are likelihood-ratio tests of statistical significance that are increasingly being used in situations where Pearson's chi-square tests were previously recommended. [7] The general formula for G is = (),
A chi-squared test (also chi-square or χ 2 test) is a statistical hypothesis test used in the analysis of contingency tables when the sample sizes are large. In simpler terms, this test is primarily used to examine whether two categorical variables ( two dimensions of the contingency table ) are independent in influencing the test statistic ...
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.
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 ...
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 ...