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Likelihood-ratio test

This article is about the statistical test that compares goodness of fit. For a general description of the likelihood ratio, see Likelihood ratio. For the use of likelihood ratios in interpreting diagnostic tests, see Likelihood ratios in diagnostic testing.

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. If the more constrained model (i.e., the null hypothesis) is supported by the observed data, the two likelihoods should not differ by more than sampling error.[1] 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 test and the Wald test.[3] In fact, the latter two can be conceptualized as approximations to the likelihood-ratio test, and are asymptotically equivalent.[4][5][6] In the case of comparing two models each of which has no unknown parameters, use of the likelihood-ratio test can be justified by the Neyman–Pearson lemma. The lemma demonstrates that the test has the highest power among all competitors.[7]

  1. ^ King, Gary (1989). Unifying Political Methodology : The Likelihood Theory of Statistical Inference. New York: Cambridge University Press. p. 84. ISBN 0-521-36697-6.
  2. ^ Li, Bing; Babu, G. Jogesh (2019). A Graduate Course on Statistical Inference. Springer. p. 331. ISBN 978-1-4939-9759-6.
  3. ^ Maddala, G. S.; Lahiri, Kajal (2010). Introduction to Econometrics (Fourth ed.). New York: Wiley. p. 200.
  4. ^ Buse, A. (1982). "The Likelihood Ratio, Wald, and Lagrange Multiplier Tests: An Expository Note". The American Statistician. 36 (3a): 153–157. doi:10.1080/00031305.1982.10482817.
  5. ^ Pickles, Andrew (1985). An Introduction to Likelihood Analysis. Norwich: W. H. Hutchins & Sons. pp. 24–27. ISBN 0-86094-190-6.
  6. ^ Severini, Thomas A. (2000). Likelihood Methods in Statistics. New York: Oxford University Press. pp. 120–121. ISBN 0-19-850650-3.
  7. ^ Neyman, J.; Pearson, E. S. (1933), "On the problem of the most efficient tests of statistical hypotheses" (PDF), Philosophical Transactions of the Royal Society of London A, 231 (694–706): 289–337, Bibcode:1933RSPTA.231..289N, doi:10.1098/rsta.1933.0009, JSTOR 91247
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