Concept Architecture
Concept
Theoretically, Random Effects Model is a statistical model used in meta-analysis that assumes the true treatment effect varies across studies rather than being identical in every study. Observed study differences therefore arise from both within-study sampling error and genuine between-study heterogeneity. The model exists to estimate the average treatment effect across a distribution of underlying true effects while accounting for variability between studies.
Mathematically, the random effects model combines study-specific effect estimates using inverse-variance weighting adjusted for between-study variance. Each study is weighted according to both its sampling variance and the estimated between-study variance, denoted ��. The pooled estimate therefore reflects both within-study precision and heterogeneity across the evidence base.
In practice, the random effects model is widely applied in systematic reviews, health technology assessment and comparative effectiveness research when clinical, methodological or statistical heterogeneity is present. Estimates of between-study variance are commonly obtained using methods such as restricted maximum likelihood (REML), DerSimonian?Laird or Paule?Mandel. The resulting pooled treatment effects frequently provide inputs for health economic evaluation.
Purpose
Used to estimate the average treatment effect across studies while accounting for both within-study sampling error and genuine between-study heterogeneity.
Mathematical Formulae
Primary Formula
Pooled estimate:
??RE = (?w????) / ?w?
where:
- ??RE = pooled random-effects estimate
- ??? = study-specific effect estimate
- w? = random-effects weight
Supporting Formulae
Random-effects weight:
w? = 1 / (Var(???) + ��)
Variance of pooled estimate:
Var(??RE) = 1 / ?w?
Standard error:
SE(??RE) = �(1 / ?w?)
95% confidence interval:
??RE � 1.96 ? SE(??RE)
Related Mathematical Methods
- Between-Study Heterogeneity
- Restricted Maximum Likelihood (REML)
- DerSimonian?Laird Estimation
- Paule?Mandel Estimation
- Cochran's Q Test
- I� Statistic
- Random-Effects Meta-Analysis
- Meta-Regression
Example
A meta-analysis pools 15 studies evaluating a new anticoagulant. Moderate heterogeneity is observed (I� = 58%), and the between-study variance is estimated as �� = 0.032 using REML. Applying a random effects model produces a pooled log hazard ratio of ?0.24 (95% confidence interval ?0.38 to ?0.10), reflecting both within-study uncertainty and genuine variation between studies.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| SUMPRODUCT | =SUMPRODUCT(B2:B16,C2:C16)/SUM(B2:B16) | Calculate the pooled random-effects estimate |
| SUM | =SUM(B2:B16) | Calculate the total random-effects weight |
| SQRT | =SQRT(1/SUM(B2:B16)) | Calculate the pooled standard error |
| EXP | =EXP(A2) | Convert pooled log estimates to hazard ratios, odds ratios or risk ratios |
| CHISQ.DIST.RT | =CHISQ.DIST.RT(Q,df) | Assess heterogeneity before selecting the random-effects model |
VBA (Optional)
Automate estimation of between-study variance and random-effects pooled estimates across multiple meta-analyses and generate heterogeneity summaries.
Sources
- DerSimonian R, Laird N. Meta-Analysis in Clinical Trials. Controlled Clinical Trials. 1986.
- Higgins JPT, Thomas J, Chandler J, et al. Cochrane Handbook for Systematic Reviews of Interventions.
- Borenstein M, Hedges LV, Higgins JPT, Rothstein HR. Introduction to Meta-Analysis.
- Veroniki AA, Jackson D, Viechtbauer W, et al. Methods to Estimate the Between-Study Variance and Its Uncertainty in Meta-Analysis. Research Synthesis Methods. 2016.
- NICE. Health Technology Evaluation Manual.
- ISPOR Good Practice Reports.
Related Concepts (2)
Library
Publications
8
Network Meta-Analysis for Decision Making — Dias, Ades, Welton, Jansen & Sutton, 1st Edition ed., 2018 (John Wiley & Sons)
The definitive text on network meta-analysis (mixed treatment comparisons) for decision making, presenting a coherent Bayesian framework (implemented in WinBUGS) for synthesising evidence across multiple treatments, including inconsistency, bias adjustment, and use in cost-effectiveness models.
BookView source →Introduction to Meta-Analysis — Borenstein, Hedges, Higgins & Rothstein, 2nd Edition ed., 2021 (John Wiley & Sons)
A clear, applied introduction to meta-analysis — computing effect sizes, fixed- and random-effects models, heterogeneity, subgroup analysis, meta-regression, and publication bias — written for readers across disciplines.
BookView source →NICE DSU Technical Support Document 1: Introduction to Evidence Synthesis for Decision Making — Dias, Welton, Sutton & Ades, TSD 1 ed., 2011 (NICE Decision Support Unit (University of Sheffield))
The introductory document of the NICE DSU evidence-synthesis series, setting out the overall analytic approach — separating baseline (natural history) and relative treatment-effect models — for synthesising evidence to inform cost-effectiveness decisions.
NICE DSU Technical Support Document 2: A General Linear Modelling Framework for Pairwise and Network Meta-Analysis of Randomised Controlled Trials — Dias, Welton, Sutton & Ades, TSD 2 ed., 2011 (NICE Decision Support Unit (University of Sheffield))
The core methods document for pairwise and network meta-analysis in NICE submissions — a generalised linear modelling framework with fixed- and random-effects models for binomial, Poisson, normal and other outcome types, implemented in WinBUGS.
NICE DSU Technical Support Document 4: Inconsistency in Networks of Evidence Based on Randomised Controlled Trials — Dias, Welton, Sutton, Caldwell, Lu & Ades, TSD 4 ed., 2011 (NICE Decision Support Unit (University of Sheffield))
Guidance on assessing and handling inconsistency — conflict between direct and indirect evidence — in network meta-analysis, a key validity check for mixed treatment comparisons.
NICE DSU Technical Support Document 7: Evidence Synthesis of Treatment Efficacy in Decision Making — A Reviewer’s Checklist — Ades, Caldwell, Reken, Welton, Sutton & Dias, TSD 7 ed., 2011 (NICE Decision Support Unit (University of Sheffield))
A reviewer’s checklist for appraising evidence syntheses of treatment efficacy used in decision making, covering the assumptions and reporting expected of pairwise and network meta-analyses submitted to NICE.
NICE DSU Technical Support Document 20: Multivariate Meta-Analysis of Summary Data for Combining Treatment Effects on Correlated Outcomes and Evaluating Surrogate Endpoints — Bujkiewicz, Achana, Papanikos, Riley & Abrams, TSD 20 ed., 2019 (NICE Decision Support Unit (University of Sheffield))
Guidance on multivariate and network meta-analysis of correlated outcomes and on the evaluation of surrogate endpoints, extending standard synthesis methods to jointly model multiple related treatment effects.
NICE DSU Technical Support Document 25: Evidence Synthesis of Diagnostic Test Accuracy for Decision Making — Dias, Ren, Bujkiewicz, et al., TSD 25 ed., 2024 (NICE Decision Support Unit (University of Sheffield))
Guidance on synthesising diagnostic test accuracy evidence (sensitivity and specificity) for use in decision models, including bivariate and hierarchical meta-analysis methods.
Media
1
Introduction to Network Meta-Analysis (ISPOR Statistical Methods SIG) — Emma Hawe & Sofia Dias, 2-part webinar ed., 2023 (ISPOR)
A two-part ISPOR Special Interest Group webinar: an introduction to network meta-analysis by Emma Hawe, followed by special topics in NMA by Sofia Dias — core methods for indirect and mixed treatment comparison.
Webinar RecordingView source →
Frequently Asked Questions (7)
What is a random effects model?
A meta-analysis model allowing genuine variation in the true treatment effect across studies, adding a between-study variance parameter to the estimate.
Source: DerSimonian R, Laird N. Meta-analysis in clinical trials. Controlled Clinical Trials. 1986;7(3):177-188. doi:10.1016/0197-2456(86)90046-2.
What is a random-effects model?
A random-effects model is a meta-analysis model that allows the true treatment effect to vary genuinely across studies, rather than assuming a single common effect, by adding a between-study variance parameter to the analysis. Under this model, each study estimates its own true effect drawn from a distribution, and the meta-analysis estimates the mean of that distribution along with the between-study variance. The random-effects model gives relatively more weight to smaller studies and wider confidence intervals when heterogeneity is present. So a random-effects model synthesises studies while allowing for real variation in their true effects.
Source: DerSimonian & Laird 1986
Why does a random-effects model treat each study's true effect as different?
A random-effects model treats the studies in a meta-analysis as each estimating its own true effect, drawn from a distribution of effects rather than a single common value. It does this because studies differ in patients, settings, and methods in ways that plausibly change how a treatment works, so assuming one identical effect would be unrealistic. By estimating both the average effect and its spread, it produces a wider, more cautious interval and gives smaller studies relatively more weight. Allowing the true effect to vary is its premise. Borenstein and colleagues (2009) describe this model.
Source: Borenstein et al. 2009
What assumption underlies the random-effects model?
The random-effects model assumes that the true treatment effects vary across studies, following a distribution, so that each study estimates a different underlying effect, and the observed variation reflects both this genuine between-study variation and sampling error. This contrasts with the fixed-effect model, which assumes a single common effect. The random-effects assumption is appropriate when the studies differ clinically or methodologically in ways that could genuinely change the effect. So the random-effects model rests on the assumption that the true effect is not the same in every study but varies, which is captured by the between-study variance parameter.
Source: DerSimonian & Laird 1986
How does the random-effects model weight studies?
The random-effects model weights studies using both their within-study precision and the between-study variance, so that, compared with the fixed-effect model, it gives relatively more weight to smaller studies and less dominance to the largest ones. As heterogeneity increases, the weights become more similar across studies. This weighting reflects that each study estimates a different true effect. So the random-effects model combines studies with weights that incorporate the between-study variation, producing a pooled estimate of the mean effect and a wider confidence interval than the fixed-effect model when heterogeneity is present, reflecting the additional uncertainty from the variation in true effects.
Source: Higgins et al. 2003
When is a random-effects model appropriate?
A random-effects model is appropriate when the studies differ in ways that could genuinely cause the true effect to vary, such as differences in populations, interventions, or settings, so that assuming a single common effect is unrealistic, and when heterogeneity is present. It accounts for this variation and gives a pooled estimate of the mean effect with appropriately wider uncertainty. Where the studies are homogeneous and a common effect is plausible, a fixed-effect model may suffice. So a random-effects model is appropriate under genuine heterogeneity, and the diversity of the studies and the assessed heterogeneity guide its use over the fixed-effect model.
Source: DerSimonian & Laird 1986
How does the random-effects model differ from the fixed-effect model?
The random-effects model assumes the true effects vary across studies and includes a between-study variance, giving relatively more weight to smaller studies and wider confidence intervals under heterogeneity, while the fixed-effect model assumes a single common effect, attributes variation to chance, and weights purely by precision. The random-effects model suits heterogeneous studies and estimates a mean effect, whereas the fixed-effect model suits homogeneous studies and estimates the common effect. So the two differ in whether they allow the true effect to vary, which affects the weighting, the width of the interval, and the interpretation of the pooled estimate.
Source: DerSimonian & Laird 1986
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 3 Dec 2025
Content version: 1.0.0
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- HE-ES-ESM-049
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