Multilevel model: Difference between revisions

Revision as of 10:40, 8 January 2023

Revision as of 22:33, 17 August 2023

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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 |bibcode=2020PLoSO..1536860L |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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==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|arxiv=2201.12430}}</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|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==