Concept Architecture
Concept
Theoretically, Fixed Effect Model is a statistical model used in meta-analysis that assumes all included studies estimate a single common true treatment effect. Any observed differences between study results are attributed solely to within-study sampling error rather than genuine variation in treatment effects. The model exists to provide the most precise pooled estimate when the assumption of a common underlying effect is considered appropriate.
Mathematically, the fixed effect model estimates the pooled treatment effect as an inverse-variance weighted average of individual study estimates. Studies with greater precision receive larger weights because they contribute more information to the estimation of the common effect. The variance of the pooled estimate is derived directly from the sum of the inverse-variance weights.
In practice, the fixed effect model is applied when clinical, methodological and statistical heterogeneity are considered negligible. It is commonly used in systematic reviews, health technology assessment and evidence synthesis when studies are sufficiently similar in design and patient populations. Where substantial between-study heterogeneity exists, a random-effects model is generally preferred.
Purpose
Used to estimate a common treatment effect across multiple studies under the assumption that all studies measure the same underlying effect and observed differences arise solely from sampling variability.
Mathematical Formulae
Primary Formula
??FE = (?w????) / ?w?
where:
- ??FE = pooled fixed effect estimate
- ??? = effect estimate from study i
- w? = inverse-variance weight
Supporting Formulae
Inverse-variance weight:
w? = 1 / Var(???)
Variance of pooled estimate:
Var(??FE) = 1 / ?w?
Standard error:
SE(??FE) = �(1 / ?w?)
95% confidence interval:
??FE � 1.96 ? SE(??FE)
Related Mathematical Methods
- Inverse-Variance Weighting
- Fixed Effect Meta-Analysis
- Cochran's Q Test
- Confidence Interval Estimation
- Mantel-Haenszel Method
- Generic Inverse Variance Method
Example
Four clinical trials evaluate a new asthma treatment. Individual study log risk ratios are pooled using inverse-variance weighting under a fixed effect model. The weighted analysis produces a pooled log risk ratio of ?0.18 with a standard error of 0.06, corresponding to a risk ratio of 0.84 (95% confidence interval 0.75 to 0.94). Because statistical heterogeneity is negligible (I� = 4%), the fixed effect model is considered appropriate.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| SUMPRODUCT | =SUMPRODUCT(B2:B6,C2:C6)/SUM(B2:B6) | Calculate the pooled fixed effect estimate |
| SUM | =SUM(B2:B6) | Calculate the total inverse-variance weight |
| SQRT | =SQRT(1/SUM(B2:B6)) | Calculate the standard error of the pooled estimate |
| EXP | =EXP(A2) | Convert pooled log estimates to relative risks, odds ratios or hazard ratios |
VBA (Optional)
Automate inverse-variance weighted fixed effect meta-analyses across multiple evidence syntheses and generate pooled estimates with confidence intervals.
Sources
- Borenstein M, Hedges LV, Higgins JPT, Rothstein HR. Introduction to Meta-Analysis.
- Higgins JPT, Thomas J, Chandler J, et al. Cochrane Handbook for Systematic Reviews of Interventions.
- Sutton AJ, Abrams KR, Jones DR, Sheldon TA, Song F. Methods for Meta-Analysis in Medical Research.
- NICE. Health Technology Evaluation Manual.
- ISPOR Good Practice Reports.
Related Concepts (2)
Library
Publications
2
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 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.
Frequently Asked Questions (7)
What is a fixed effect model?
A meta-analysis model assuming all included studies estimate the same true effect, with observed variation attributed entirely to sampling error.
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 single true value does a fixed-effect model assume the studies share?
A fixed-effect model assumes that all the studies in a meta-analysis are estimating one and the same true effect, so any differences among their results arise only from sampling error within each study. On that premise it combines them by weighting each purely according to its precision, letting the largest studies carry the most influence. This is reasonable when the studies are alike in their patients and methods, but wrong when the effect genuinely varies between them. One shared true effect is its central assumption. Borenstein and colleagues (2009) describe this model.
Source: Borenstein et al. 2009
What is a fixed-effect model?
A fixed-effect model is a meta-analysis model that assumes all the included studies estimate the same single true effect, so that the observed variation among their results is attributed entirely to sampling error, or chance. Under this assumption, the studies differ only because of random variation, and the pooled estimate is a weighted average of the study effects, weighted by their precision. The fixed-effect model gives more weight to larger studies. So a fixed-effect model combines studies on the assumption of a common underlying effect, treating differences among them as chance, which contrasts with the random-effects model that allows the true effect to vary.
Source: DerSimonian & Laird 1986
What assumption underlies the fixed-effect model?
The fixed-effect model rests on the assumption that all the studies estimate the same single true effect, so any differences in their observed results are due only to sampling error rather than to genuine differences in the effect. This assumption is appropriate when the studies are sufficiently similar that a common effect is plausible and heterogeneity is minimal. If the true effects genuinely vary across studies, the assumption is violated, and a random-effects model is more suitable. So the fixed-effect model assumes homogeneity of the true effect, which determines when it is appropriate and distinguishes it from the random-effects model.
Source: DerSimonian & Laird 1986
How does the fixed-effect model weight studies?
The fixed-effect model weights studies by their precision, typically the inverse of their variance, so that more precise studies, usually larger ones with more events, receive more weight in the pooled estimate, while smaller, less precise studies receive less. Because it assumes a common effect, all the weight reflects statistical precision alone. This inverse-variance weighting produces a pooled estimate dominated by the larger studies. So the fixed-effect model combines studies by inverse-variance weighting, giving greater influence to more precise studies, which is appropriate under its assumption that all estimate the same true effect.
Source: DerSimonian & Laird 1986
When is a fixed-effect model appropriate?
A fixed-effect model is appropriate when the studies are sufficiently similar in their populations, interventions, and methods that they can plausibly be assumed to estimate the same true effect, and when heterogeneity among them is minimal. In such cases, treating the variation as chance is reasonable. Where the true effects are likely to vary across studies, owing to clinical or methodological diversity, a random-effects model is more suitable. So a fixed-effect model is appropriate under homogeneity of the true effect, and the assessment of heterogeneity and the similarity of the studies guides whether it or the random-effects model should be used.
Source: Higgins et al. 2003
How does the fixed-effect model differ from the random-effects model?
The fixed-effect model assumes all studies estimate a single common true effect, attributing variation to chance and weighting by precision, while the random-effects model assumes the true effects vary across studies and incorporates the between-study variance, giving relatively more weight to smaller studies and producing wider confidence intervals when heterogeneity is present. The fixed-effect model suits homogeneous studies, and the random-effects model suits heterogeneous ones. So the two differ in whether they assume a common effect or allow it 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: 2 Dec 2025
Content version: 1.0.0
Canonical Identity
- Persistent URI
- https://healtheconomics.wiki/concept/fixed-effect-model
- Term code
- HE-ES-ESM-019
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