Multilevel model: Difference between revisions

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{{Regression bar}}

'''Multilevel models'''{{Refn|also known as '''hierarchical linear models''', '''linear mixed-effect models''', '''mixed models''', '''nested data models''', '''random coefficient''', '''random-effects models''', '''random parameter models''', or '''split-plot designs'''|group=notelower-alpha}} 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" />

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;Orthogonality of regressors to random effects

The regressors must not correlate with the random effects, <math>u_{0j}</math>. This assumption is testable but often ignored, rendering the estimator inconsistent.<ref name=":0">{{Cite journal |last1=Antonakis |first1=John |last2=Bastardoz |first2=Nicolas |last3=Rönkkö |first3=Mikko |date=2021 |title=On Ignoring the Random Effects Assumption in Multilevel Models: Review, Critique, and Recommendations |url=https://jyx.jyu.fi/bitstream/123456789/66704/2/Antonakisym.pdf |journal=Organizational Research Methods |language=en |volume=24 |issue=2 |pages=443–483 |doi=10.1177/1094428119877457 |s2cid=210355362 |issn=1094-4281|url-access= |url-status= |archive-url= |archive-date= }}</ref> If this assumption is violated, the random-effect must be modeled explicitly in the fixed part of the model, either by using dummy variables or including cluster means of all <math>X_{ij} </math> regressors.<ref name=":0" /><ref>{{Cite journal |last1=McNeish |first1=Daniel |last2=Kelley |first2=Ken |date=2019 |title=Fixed effects models versus mixed effects models for clustered data: Reviewing the approaches, disentangling the differences, and making recommendations. |url=http://doi.apa.org/getdoi.cfm?doi=10.1037/met0000182 |journal=Psychological Methods |language=en |volume=24 |issue=1 |pages=20–35 |doi=10.1037/met0000182 |pmid=29863377 |s2cid=44145669 |issn=1939-1463|url-access=subscription }}</ref><ref>{{Cite journal |last1=Bliese |first1=Paul D. |last2=Schepker |first2=Donald J. |last3=Essman |first3=Spenser M. |last4=Ployhart |first4=Robert E. |date=2020 |title=Bridging Methodological Divides Between Macro- and Microresearch: Endogeneity and Methods for Panel Data |url=http://journals.sagepub.com/doi/10.1177/0149206319868016 |journal=Journal of Management |language=en |volume=46 |issue=1 |pages=70–99 |doi=10.1177/0149206319868016 |s2cid=202288849 |issn=0149-2063|url-access=subscription }}</ref><ref>{{Cite book |last=Wooldridge |first=Jeffrey M. |url=https://books.google.com/books?id=hSs3AgAAQBAJ&dq=info:T5fz2cmyyF8J:scholar.google.com&pg=PP1 |title=Econometric Analysis of Cross Section and Panel Data, second edition |date=2010-10-01 |publisher=MIT Press |isbn=978-0-262-29679-3 |language=en}}</ref> This assumption is probably the most important assumption the estimator makes, but one that is misunderstood by most applied researchers using these types of models.<ref name=":0" />

==Statistical tests==

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== Notes ==

{{Reflist|group=notelower-alpha}}

== References ==