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Fitting three-level meta-analytic models in R: A step-by-step tutorial
Open accessAbstract
Applying a multilevel approach to meta-analysis is a strong method for dealing with dependency of effect sizes.However, this method is relatively unknown among researchers and, to date, has not been widely used in meta-analytic research.Therefore, the purpose of this tutorial was to show how a three-level random effects model can be applied to meta-analytic models in R using the rma.mv function of the metafor package.This application is illustrated by taking the reader through a step-by-step guide to the multilevel analyses comprising the steps of (1) organizing a data file; (2) setting up the R environment; (3) calculating an overall effect; (4) examining heterogeneity of within-study variance and between-study variance; (5) performing categorical and continuous moderator analyses; and (6) examining a multiple moderator model.By example, the authors demonstrate how the multilevel approach can be applied to meta-analytically examining the association between mental health disorders of juveniles and juvenile offender recidivism.In our opinion, the rma.mv function of the metafor package provides an easy and flexible way of applying a multi-level structure to meta-analytic models in R. Further, the multilevel meta-analytic models can be easily extended so that the potential moderating influence of variables can be examined.
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“As a result, there no information is lost and (moderator) effects can be estimated more precisely and with maximum power in the statistical analyses (Assink and Wibbelink 2016).”
From Risk Factors for School Absenteeism and Dropout: A Meta-Analytic Review (2019) · cites this paper as Assink and Wibbelink 2016
Experimental — accuracy not yet measuredThe source backs part of the claim, not all of it — or backs it for a different population, condition or comparison.
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“As recommended by Assink and Wibbelink (2016), moderation analyses should be conducted if less than 75% of the variance can be attributed to Level 1.”
From Therapeutic games to reduce anxiety and depression in young people: A systematic review and exploratory meta‐analysis of their use and effectiveness. (2024) · cites this paper as Assink and Wibbelink (2016)
Experimental — accuracy not yet measuredThe cited source says what the claim attributes to it.
In the citing paper
“There is no need to know the correlations between outcomes extracted from one study for estimating the covariance matrix of the effect sizes since the second level in the model accounts for sampling covariation ( Assink and Wibbelink, 2016 ).”
From The effect of contextual interference on transfer in motor learning - a systematic review and meta-analysis (2024) · cites this paper as Assink and Wibbelink, 2016
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“To deal with this intra-study dependency of prevalence estimates, we opted for the three-level model with random effects for data points obtained within one article to calculate pooled prevalence estimates (29, 30).”
From The global prevalence of autism spectrum disorder: A three-level meta-analysis (2023) · cites this paper as [30]
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“we used a multilevel approach to metaanalysis (Assink & Wibbelink, 2016), which is a strong method for dealing with dependency of effect sizes, as it accounts for the hierarchical structure of the data in which effect sizes are nested within studies (Van Den Noortgate & Onghena, 2003).”
From Associations between rejection sensitivity and mental health outcomes: A meta-analytic review (2017) · cites this paper as Assink & Wibbelink, 2016
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“The Knapp and Hartung method (2003) has the advantage that it reduces Type I‐errors (Assink & Wibbelink, 2016).”
From Does Natural Mentoring Matter? A Multilevel Meta‐analysis on the Association Between Natural Mentoring and Youth Outcomes (2018) · cites this paper as Assink & Wibbelink, 2016
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“A three‐level random effects model was used to account for three levels of variance, including the sampling variance of each effect size (level 1), the variance between effect sizes extracted from the same study (level 2), and the variance between the studies (level 3).”
From Does Natural Mentoring Matter? A Multilevel Meta‐analysis on the Association Between Natural Mentoring and Youth Outcomes (2018) · cites this paper as Assink & Wibbelink, 2016
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“Likelihood‐ratio tests were used to compare the deviance scores of the full model to the deviance of models excluding level 2 or level 3 variance parameters, making it possible to determine whether significant variance is present at the two levels (Assink & Wibbelink, 2016).”
From Does Natural Mentoring Matter? A Multilevel Meta‐analysis on the Association Between Natural Mentoring and Youth Outcomes (2018) · cites this paper as Assink & Wibbelink, 2016
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“This analysis considers three levels of variance components distributed across the model, including variance between effect sizes from the same study and variance between studies; this therefore allows for an examination of how effect sizes vary between participants (Level 1), outcomes (Level 2) and studies (Level 3; Assink & Wibbelink, 2016).”
From Therapeutic games to reduce anxiety and depression in young people: A systematic review and exploratory meta‐analysis of their use and effectiveness. (2024) · cites this paper as Assink & Wibbelink, 2016
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In the citing paper
“The analyses suggested that there was significant variability ( p = .001) between effect sizes (Level 2) and between studies (Level 3), indicating that moderator analyses should be conducted (Assink & Wibbelink, 2016).”
From Therapeutic games to reduce anxiety and depression in young people: A systematic review and exploratory meta‐analysis of their use and effectiveness. (2024) · cites this paper as Assink & Wibbelink, 2016
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“Dependency of effect sizes normally means that effect sizes within studies are correlated (as these are expected to show a certain similarity); this creates an overlap of information and inflates information produced by the analysis, which can result in an overconfidence in its results (Assink & Wibbelink, 2016).”
From Therapeutic games to reduce anxiety and depression in young people: A systematic review and exploratory meta‐analysis of their use and effectiveness. (2024) · cites this paper as Assink & Wibbelink, 2016
In the citing paper
“A hierarchical multi-level multivariate meta-analytical approach for data plotting was used to account for statistical dependence in effect sizes from multiple within-study comparison arms (cannabis use frequencies) to a common control group (non-use)”
From Risk-thresholds for the association between frequency of cannabis use and the development of psychosis: a systematic review and meta-analysis (2023) · cites this paper as Assink & Wibbelink, 2016
In the citing paper
“As studies reported more than one effect size (for example, depression and suicidal ideation), we opted for a multilevel meta-analysis with three levels (for example, ref. 47), a random-effect model, which enables modelling the dependent effect sizes from the same study.”
From Prevalence and severity of mental health problems in early-career researchers: a systematic review and meta-analysis (2026) · cites this paper as [47]
In the citing paper
“For parameter estimation, restricted maximum likelihood estimation was applied, and for standard errors, P values and 95% CIs, Knapp and Hartung’s adjustment47,227 was used.”
From Prevalence and severity of mental health problems in early-career researchers: a systematic review and meta-analysis (2026) · cites this paper as [47]
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Metadata
- Authors
- Assink, M., & Wibbelink, C. J. M.
- Affiliations
- University of Amsterdam (NL)
- Corresponding author
- Assink, Mark
- Venue
- The Quantitative Methods for Psychology
- Published
- 2016-10-01
- Volume
- 12
- Issue
- 3
- Pages
- 154-174
- ISSN
- 2292-1354 (ISSN-L)
- OpenAlex
- W2532414435
- Semantic Scholar
- 95f645af7a355ecf2b9a7958c12734ad1768e44f
- Cited by
- 1,330
- References
- 47
- Citations by year
- 2026: 345 · 2025: 244 · 2024: 208 · 2023: 146 · 2022: 134 · 2021: 93 · 2020: 81 · 2019: 44 · 2018: 22 · 2017: 9 · 2016: 2
- FWCI
- 21.40
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Locations
- The Quantitative Methods for Psychology (publishedVersion)
- UvA-DARE (University of Amsterdam) (publishedVersion, cc-by)
- DOAJ (DOAJ: Directory of Open Access Journals) (submittedVersion)
- Data Archiving and Networked Services (DANS) (submittedVersion, other-oa)
- Funders
- Universiteit Maastricht
- Topics
- Social Sciences › Decision Sciences › Statistics, Probability and Uncertainty › Meta-analysis and systematic reviews
- Physical Sciences › Mathematics › Statistics and Probability › Statistical Methods and Bayesian Inference
- Social Sciences › Psychology › Experimental and Cognitive Psychology › Psychology Research and Bibliometrics
- Keywords
- multilevel meta-analysis, R programming language, effect size heterogeneity, between-study variance, moderator analysis
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Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
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