Likelihood Ratio

A quantity used to test nested hypotheses. Let  be a nested hypothesis with  degrees of freedom within  (which has  degrees of freedom), then calculate the maximum likelihood of a given outcome, first given , then given . Then

Comparison of  to the critical value of the chi-squared distribution with  degrees of freedom then gives the significance of the increase in likelihood.

The term likelihood ratio is also used (especially in medicine) to test nonnested complementary hypotheses as follows,

REFERENCES:

Feller, W. An Introduction to Probability Theory and Its Applications, Vol. 2, 3rd ed. New York: Wiley, pp. 211-212, 1971.

Hoel, P. G. "Likelihood Ratio Tests." §9.1.4 in Introduction to Mathematical Statistics, 3rd ed. New York: Wiley, pp. 220-228, 1962.