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searchaux" style="display:none">Inexact statistical measure</div> <p>In <a href="/wiki/Statistics" title="Statistics">statistics</a>, <b>quasi-likelihood</b> methods are used to estimate parameters in a statistical model when exact likelihood methods, for example <a href="/wiki/Maximum_likelihood" class="mw-redirect" title="Maximum likelihood">maximum likelihood</a> estimation, are computationally infeasible. Due to the wrong likelihood being used, quasi-likelihood estimators lose asymptotic efficiency compared to, e.g., maximum likelihood estimators. Under broadly applicable conditions, quasi-likelihood estimators are <a href="/wiki/Consistent_estimator" title="Consistent estimator">consistent</a> and <a href="/wiki/Estimator#Asymptotic_normality" title="Estimator">asymptotically normal</a>. The asymptotic covariance matrix can be obtained using the so-called <i>sandwich estimator</i>. Examples of quasi-likelihood methods include the <a href="/wiki/Generalized_estimating_equations" class="mw-redirect" title="Generalized estimating equations">generalized estimating equations</a> and pairwise likelihood approaches. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quasi-likelihood&action=edit&section=1" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The term <b>quasi-likelihood function</b> was introduced by <a href="/wiki/Robert_Wedderburn_(statistician)" title="Robert Wedderburn (statistician)">Robert Wedderburn</a> in 1974 to describe a function that has similar properties to the log-<a href="/wiki/Likelihood_function" title="Likelihood function">likelihood function</a> but is not the log-likelihood corresponding to any actual <a href="/wiki/Probability_distribution" title="Probability distribution">probability distribution</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> He proposed to fit certain quasi-likelihood models using a straightforward extension of the algorithms used to fit <a href="/wiki/Generalized_linear_models" class="mw-redirect" title="Generalized linear models">generalized linear models</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Application_to_overdispersion_modelling">Application to overdispersion modelling</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quasi-likelihood&action=edit&section=2" title="Edit section: Application to overdispersion modelling"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Quasi-likelihood estimation is one way of allowing for <a href="/wiki/Overdispersion" title="Overdispersion">overdispersion</a>, that is, greater variability in the data than would be expected from the <a href="/wiki/Statistical_model" title="Statistical model">statistical model</a> used. It is most often used with models for <a href="/wiki/Count_data" title="Count data">count data</a> or grouped binary data, i.e. data that would otherwise be modelled using the <a href="/wiki/Poisson_distribution" title="Poisson distribution">Poisson</a> or <a href="/wiki/Binomial_distribution" title="Binomial distribution">binomial distribution</a>. </p><p>Instead of specifying a probability distribution for the data, only a relationship between the mean and the variance is specified in the form of a <a href="/wiki/Variance_function" title="Variance function">variance function</a> giving the variance as a function of the mean. Generally, this function is allowed to include a multiplicative factor known as the <b>overdispersion parameter</b> or <b>scale parameter</b> that is estimated from the data. Most commonly, the variance function is of a form such that fixing the overdispersion parameter at unity results in the variance-mean relationship of an actual probability distribution such as the binomial or Poisson. (For formulae, see the <a href="/wiki/Generalized_linear_model#Binomial_data" title="Generalized linear model">binomial data example</a> and <a href="/wiki/Generalized_linear_model#Count_data" title="Generalized linear model">count data example</a> under <a href="/wiki/Generalized_linear_models" class="mw-redirect" title="Generalized linear models">generalized linear models</a>.) </p> <div class="mw-heading mw-heading3"><h3 id="Comparison_to_alternatives">Comparison to alternatives</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quasi-likelihood&action=edit&section=3" title="Edit section: Comparison to alternatives"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Random-effects_models" class="mw-redirect" title="Random-effects models">Random-effects models</a>, and more generally <a href="/wiki/Mixed_model" title="Mixed model">mixed models</a> (<a href="/wiki/Multilevel_model" title="Multilevel model">hierarchical models</a>) provide an alternative method of fitting data exhibiting overdispersion using fully specified probability models. However, these methods often become complex and computationally intensive to fit to binary or count data. Quasi-likelihood methods have the advantage of relative computational simplicity, speed and robustness, as they can make use of the more straightforward algorithms developed to fit <a href="/wiki/Generalized_linear_models" class="mw-redirect" title="Generalized linear models">generalized linear models</a>. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quasi-likelihood&action=edit&section=4" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Quasi-maximum_likelihood_estimate" title="Quasi-maximum likelihood estimate">Quasi-maximum likelihood estimate</a></li> <li><a href="/wiki/Extremum_estimator" title="Extremum estimator">Extremum estimator</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quasi-likelihood&action=edit&section=5" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"> <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFWedderburn1974" class="citation journal cs1">Wedderburn, R. W. M. (1974). "Quasi-likelihood functions, generalized linear models, and the Gauss—Newton method". <i>Biometrika</i>. <b>61</b> (3): 439–447. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1093%2Fbiomet%2F61.3.439">10.1093/biomet/61.3.439</a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0375592">0375592</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Biometrika&rft.atitle=Quasi-likelihood+functions%2C+generalized+linear+models%2C+and+the+Gauss%E2%80%94Newton+method&rft.volume=61&rft.issue=3&rft.pages=439-447&rft.date=1974&rft_id=info%3Adoi%2F10.1093%2Fbiomet%2F61.3.439&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0375592%23id-name%3DMR&rft.aulast=Wedderburn&rft.aufirst=R.+W.+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuasi-likelihood" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quasi-likelihood&action=edit&section=6" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMcCullaghNelder,_John1989" class="citation book cs1">McCullagh, Peter; <a href="/wiki/John_Nelder" title="John Nelder">Nelder, John</a> (1989). <i>Generalized Linear Models</i> (second ed.). London: Chapman and Hall. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-412-31760-5" title="Special:BookSources/0-412-31760-5"><bdi>0-412-31760-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Generalized+Linear+Models&rft.edition=second&rft.pub=London%3A+Chapman+and+Hall&rft.date=1989&rft.isbn=0-412-31760-5&rft.aulast=McCullagh&rft.aufirst=Peter&rft.au=Nelder%2C+John&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuasi-likelihood" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHardinHilbe,_Joseph2007" class="citation book cs1">Hardin, James; <a href="/wiki/Joseph_Hilbe" title="Joseph Hilbe">Hilbe, Joseph</a> (2007). <i>Generalized Linear Models and Extensions</i> (second ed.). College Station: Stata Press.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Generalized+Linear+Models+and+Extensions&rft.edition=second&rft.pub=College+Station%3A+Stata+Press&rft.date=2007&rft.aulast=Hardin&rft.aufirst=James&rft.au=Hilbe%2C+Joseph&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuasi-likelihood" class="Z3988"></span></li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Quasi-likelihood&oldid=1175405197">https://en.wikipedia.org/w/index.php?title=Quasi-likelihood&oldid=1175405197</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Likelihood" title="Category:Likelihood">Likelihood</a></li><li><a href="/wiki/Category:Maximum_likelihood_estimation" title="Category:Maximum likelihood estimation">Maximum likelihood estimation</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 14 September 2023, at 20:21<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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