Editing Likelihood-ratio test (section)

{{Short description|Statistical test that compares goodness of fit}}
{{About|the statistical test that compares goodness of fit|a general description of the likelihood ratio|Likelihood ratio|the use of likelihood ratios in interpreting diagnostic tests|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 model]]s, typically one found by [[Mathematical optimization|maximization]] over the entire [[parameter space]] and another found after imposing some [[Constraint (mathematics)|constraint]], based on the ratio of their [[likelihood function|likelihoods]]. If the more constrained model (i.e., the [[null hypothesis]]) is supported by the [[Realization (probability)|observed data]], the two likelihoods should not differ by more than [[sampling error]].<ref>{{cite book |first=Gary |last=King |author-link=Gary King (political scientist) |title=Unifying Political Methodology : The Likelihood Theory of Statistical Inference |location=New York |publisher=Cambridge University Press |year=1989 |isbn=0-521-36697-6 |page=84 |url=https://books.google.com/books?id=cligOwrd7XoC&pg=PA84 }}</ref> Thus the likelihood-ratio test tests whether this ratio is [[Statistical significance|significantly different]] from one, or equivalently whether its [[natural logarithm]] is significantly different from zero.

The likelihood-ratio test, also known as '''Wilks test''',<ref>{{cite book |first1=Bing |last1=Li |first2=G. Jogesh |last2=Babu |title=A Graduate Course on Statistical Inference |location= |publisher=Springer |year=2019 |page=331 |isbn=978-1-4939-9759-6 }}</ref> is the oldest of the three classical approaches to hypothesis testing, together with the [[Lagrange multiplier test]] and the [[Wald test]].<ref>{{cite book |first1=G. S. |last1=Maddala |author-link=G. S. Maddala |first2=Kajal |last2=Lahiri |title=Introduction to Econometrics |location=New York |publisher=Wiley |edition=Fourth |year=2010 |page=200 }}</ref> In fact, the latter two can be conceptualized as approximations to the likelihood-ratio test, and are asymptotically equivalent.<ref>{{cite journal |first=A. |last=Buse |title=The Likelihood Ratio, Wald, and Lagrange Multiplier Tests: An Expository Note |journal=[[The American Statistician]] |volume=36 |issue=3a |year=1982 |pages=153–157 |doi=10.1080/00031305.1982.10482817 }}</ref><ref>{{cite book |first=Andrew |last=Pickles |title=An Introduction to Likelihood Analysis |location=Norwich |publisher=W. H. Hutchins & Sons |year=1985 |isbn=0-86094-190-6 |pages=[https://archive.org/details/introductiontoli0000pick/page/24 24–27] |url=https://archive.org/details/introductiontoli0000pick/page/24 }}</ref><ref>{{cite book |first=Thomas A. |last=Severini |title=Likelihood Methods in Statistics |location=New York |publisher=Oxford University Press |year=2000 |isbn=0-19-850650-3 |pages=120–121 }}</ref> In the case of comparing two models each of which has no unknown [[statistical parameters|parameters]], use of the likelihood-ratio test can be justified by the [[Neyman–Pearson lemma]]. The lemma demonstrates that the test has the highest [[statistical power|power]] among all competitors.<ref name="NeymanPearson1933">{{citation | last1 = Neyman | first1 = J. | author-link1 = Jerzy Neyman| last2 = Pearson | first2 = E. S. | author-link2 = Egon Pearson| doi = 10.1098/rsta.1933.0009 | title = On the problem of the most efficient tests of statistical hypotheses | journal = [[Philosophical Transactions of the Royal Society of London A]] | volume = 231 | issue = 694–706 | pages = 289–337 | year = 1933 | jstor = 91247 |bibcode = 1933RSPTA.231..289N | url = http://www.stats.org.uk/statistical-inference/NeymanPearson1933.pdf | doi-access = free }}</ref>