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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
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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==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
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
▲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|arxiv=2201.12430}}</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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