Multilevel model: Difference between revisions

Revision as of 08:58, 13 May 2022

Revision as of 11:54, 18 May 2022

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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 }}</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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When there are multiple level 1 independent variables, the model can be expanded by substituting vectors and matrices in the equation.

When the relationship between the response <math> Y_{ij} </math> and predictor <math> X_{ij} </math> can not be described by the linear relationship, then one can find some non linear functional relationship between the response and predictor, and extend the model to [[nonlinear mixed-effects model]]. For example, when the response <math>Y_{ij} </math> is the cumulative infection trajectory of the <math>i</math>-th country, and <math> X_{ij} </math> represents the <math>j</math>-th time points, then the ordered pair <math>(X_{ij},Y_{ij})</math> for each country may show a shape similar to [[logistic function]].<ref>{{Cite journal |last1=Lee|first1=Se Yoon |first2=Bowen |last2=Lei|first3=Bani|last3=Mallick| title = Estimation of COVID-19 spread curves integrating global data and borrowing information|journal=PLOS ONE|year=2020|volume=15 |issue=7 |pages=e0236860 |doi=10.1371/journal.pone.0236860 |arxiv=2005.00662|pmid=32726361 |pmc=7390340 |doi-access=free}}</ref><ref name="ReferenceA">{{Cite journal |last1=Lee|first1=Se Yoon |first2=Bani|last2=Mallick| title = Bayesian Hierarchical Modeling: Application Towards Production Results in the Eagle Ford Shale of South Texas|journal=Sankhya B|year=2021|volume=84 |pages=1–43 |doi=10.1007/s13571-020-00245-8|doi-access=free}}</ref>

==Level 2 regression equation==

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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 |lastlast1=Antonakis |firstfirst1=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=http://journals.sagepub.com/doi/10.1177/1094428119877457 |journal=Organizational Research Methods |language=en |volume=24 |issue=2 |pages=443–483 |doi=10.1177/1094428119877457 |s2cid=210355362 |issn=1094-4281}}</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 |lastlast1=McNeish |firstfirst1=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}}</ref><ref>{{Cite journal |lastlast1=Bliese |firstfirst1=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}}</ref><ref>{{Cite book |last=Wooldridge |first=Jeffrey M. |url=https://books.google.chcom/books?hl=en&lr=&id=hSs3AgAAQBAJ&oi=fnd&pg=PP1&dq=info:T5fz2cmyyF8J:scholar.google.com&otspg=VYQRpv__Pw&sig=xPSL2Xuq7Amu7VdxKba872ZO88A&redir_esc=y#v=onepage&q&f=falsePP1 |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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==Bayesian nonlinear mixed-effects model==

[[File:Bayesian research cycle.png|500px|thumb|right|Bayesian research cycle using Bayesian nonlinear mixed effects model: (a) standard research cycle and (b) Bayesian-specific workflow <ref>{{Cite journal |last1=Lee|first1=Se Yoon| title = Bayesian Nonlinear Models for Repeated Measurement Data: An Overview, Implementation, and Applications |journal=Mathematics|year=2022|volume=10 |issue=6 |page=898 |doi=10.3390/math10060898|doi-access=free}}</ref>.]]

Multilevel modeling is frequently used in diverse applications and it can be formulated by the Bayesian framework. Particularly, Bayesian nonlinear mixed-effects models have recently received significant attention. A basic version of the Bayesian nonlinear mixed-effects models is represented as the following three-stage:

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The panel on the right displays Bayesian research cycle using Bayesian nonlinear mixed-effects model.<ref>{{Cite journal |last1=Lee|first1=Se Yoon| title = Bayesian Nonlinear Models for Repeated Measurement Data: An Overview, Implementation, and Applications |journal=Mathematics|year=2022|volume=10 |issue=6 |page=898 |doi=10.3390/math10060898|doi-access=free}}</ref> A research cycle using the Bayesian nonlinear mixed-effects model comprises two steps: (a) standard research cycle and (b) Bayesian-specific workflow. Standard research cycle involves literature review, defining a problem and specifying the research question and hypothesis. Bayesian-specific workflow comprises three sub-steps: (b)–(i) formalizing prior distributions based on background knowledge and prior elicitation; (b)–(ii) determining the likelihood function based on a nonlinear function <math> f </math>; and (b)–(iii) making a posterior inference. The resulting posterior inference can be used to start a new research cycle.

==See also==

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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 }}

==External links==