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'''Multilevel models''' (also known as '''hierarchical linear models''', '''linear mixed-effect model''', '''mixed models''', '''nested data models''', '''random coefficient''', '''random-effects models''', '''random parameter models''', or '''split-plot designs''') are [[statistical model]]s of [[parameter]]s that vary at more than one level.<ref name="Raud" /> An example could be a model of student performance that contains measures for individual students as well as measures for classrooms within which the students are grouped. These models can be seen as generalizations of [[linear model]]s (in particular, [[linear regression]]), although they can also extend to non-linear models. These models became much more popular after sufficient computing power and software became available.<ref name="Raud" />
Multilevel models are particularly appropriate for research designs where data for participants are organized at more than one level (i.e., [[nested data]]).<ref name="Fidell">{{cite book|last=Fidell|first=Barbara G. Tabachnick, Linda S.|title=Using multivariate statistics|year=2007|publisher=Pearson/A & B|___location=Boston ; Montreal|isbn=978-0-205-45938-4|edition=5th}}</ref> The units of analysis are usually individuals (at a lower level) who are nested within contextual/aggregate units (at a higher level).<ref name="Luke">{{cite book|last=Luke|first=Douglas A.|title=Multilevel modeling|year=2004|publisher=Sage|___location=Thousand Oaks, CA|isbn=978-0-7619-2879-9|edition=3. repr.}}</ref> While the lowest level of data in multilevel models is usually an individual, repeated measurements of individuals may also be examined.<ref name="Fidell" /><ref name="Gomes2022">{{cite journal |last1=Gomes |first1=Dylan G.E. |title=Should I use fixed effects or random effects when I have fewer than five levels of a grouping factor in a mixed-effects model? |journal=PeerJ |date=20 January 2022 |volume=10 |pages=e12794 |doi=10.7717/peerj.12794|pmid=35116198 |pmc=8784019 |doi-access=free }}</ref> As such, multilevel models provide an alternative type of analysis for univariate or [[multivariate analysis]] of [[repeated measures]]. Individual differences in [[growth curve (statistics)|growth curves]] may be examined.<ref name="Fidell" /> Furthermore, multilevel models can be used as an alternative to [[ANCOVA]], where scores on the dependent variable are adjusted for covariates (e.g. individual differences) before testing treatment differences.<ref name="Cohen">{{cite book|last1=Cohen|first1=Jacob|title=Applied multiple regression/correlation analysis for the behavioral sciences|publisher=Erlbaum|___location=Mahwah, NJ [u.a.]|isbn=978-0-8058-2223-6|edition=3.|date=3 October 2003}}</ref> Multilevel models are able to analyze these experiments without the assumptions of homogeneity-of-regression slopes that is required by ANCOVA.<ref name="Fidell" />
Multilevel models can be used on data with many levels, although 2-level models are the most common and the rest of this article deals only with these. The dependent variable must be examined at the lowest level of analysis.<ref name="Raud">{{cite book|last=Bryk|first=Stephen W. Raudenbush, Anthony S.|title=Hierarchical linear models : applications and data analysis methods|year=2002|publisher=Sage Publications|___location=Thousand Oaks, CA [u.a.]|isbn=978-0-7619-1904-9|edition=2. ed., [3. Dr.]}}</ref>
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* {{cite book |last1=Swamy |first1=P. A. V. B. |author-link=P. A. V. B. Swamy |last2=Tavlas |first2=George S. |chapter=Random Coefficient Models |title=A Companion to Theoretical Econometrics |editor-last=Baltagi |editor-first=Badi H. |___location=Oxford |publisher=Blackwell |year=2001 |isbn=978-0-631-21254-6 |pages=410–429 }}
* {{cite book |last1=Verbeke |first1=G. |last2=Molenberghs |first2=G. |year=2013 |title=Linear Mixed Models for Longitudinal Data |publisher=Springer }} Includes [[SAS (software)|SAS]] code
* {{cite journal |last1=Gomes |first1=Dylan G.E. |title=Should I use fixed effects or random effects when I have fewer than five levels of a grouping factor in a mixed-effects model? |journal=PeerJ |date=20 January 2022 |volume=10 |pages=e12794 |doi=10.7717/peerj.12794|pmid=35116198 |pmc=8784019 |doi-access=free }}
==External links==
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