Concept Architecture
Concept
Theoretically, Copas Model is a statistical selection model used to assess and adjust for publication bias in meta-analysis. It models the probability that a study is observed or published as a function of its precision and underlying effect estimate, thereby accounting for the possibility that smaller or less statistically significant studies are underrepresented. The model exists to evaluate the robustness of pooled treatment effects to potential selection bias.
Mathematically, the Copas model combines a random-effects meta-analysis with a selection mechanism that specifies the probability of study inclusion. The framework jointly estimates the overall treatment effect, between-study heterogeneity and publication process using likelihood-based methods. Sensitivity analyses are performed by varying assumptions about the selection mechanism to evaluate the stability of pooled estimates.
In practice, the Copas model is applied after a conventional meta-analysis when publication bias is suspected. It is primarily used as a sensitivity analysis alongside funnel plots, Begg tests and Egger tests. In health economics and health technology assessment, the Copas model provides an alternative estimate of pooled treatment effects that accounts for potential selective publication of evidence.
Purpose
Used to assess and adjust for publication bias in meta-analysis by modelling the probability of study selection and evaluating the robustness of pooled treatment-effect estimates under alternative assumptions about selective publication.
Mathematical Formulae
Primary Formula
Selection model:
P(S? = 1) = �(� + ? / SE?)
where:
- P(S? = 1) = probability that study i is observed
- � = cumulative standard normal distribution
- � = selection intercept
- ? = selection parameter
- SE? = standard error of study i
Supporting Formulae
Random-effects model:
?? ~ N(?, ��)
Observed effect:
Y? ~ N(??, SE?�)
Related Mathematical Methods
- Random-Effects Meta-Analysis
- Selection Models
- Maximum Likelihood Estimation
- Sensitivity Analysis
- Funnel Plot Analysis
- Egger Test
- Begg Test
Example
A meta-analysis of 24 oncology trials shows asymmetry in the funnel plot, suggesting possible publication bias. A Copas model is fitted to examine the impact of selective publication. After accounting for the assumed selection process, the pooled log hazard ratio changes from ?0.31 to ?0.24, indicating that the original treatment effect may have been modestly overestimated.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| NORM.S.DIST | =NORM.S.DIST(A2,TRUE) | Evaluate the cumulative normal selection function |
| LN | =LN(B2) | Calculate log-likelihood components |
| EXP | =EXP(C2) | Transform model parameters where required |
| Solver | Objective: maximise log-likelihood | Estimate Copas model parameters by maximum likelihood |
VBA (Optional)
Automate sensitivity analyses by repeatedly fitting Copas selection models under alternative assumptions regarding study selection and publication bias.
Sources
- Copas JB, Shi JQ. Meta-Analysis, Funnel Plots and Sensitivity Analysis. Biostatistics. 2000.
- Copas JB, Jackson D. A Bound for Publication Bias Based on the Fraction of Unpublished Studies. Biometrics. 2004.
- 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.
- NICE. Health Technology Evaluation Manual.
- ISPOR Good Practice Reports.
Related Concepts (2)
Library
Publications
1
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 →
Frequently Asked Questions (6)
What is the Copas model?
A statistical model adjusting a meta-analysis result for potential publication bias by modelling how publication likelihood depends on results.
Source: Copas 1999
How does the Copas model adjust a result for publication bias?
The Copas model adjusts a meta-analysis for publication bias by explicitly modelling the chance that a study gets published as a function of its results, on the premise that studies with weaker or null findings are less likely to appear. By estimating how many such studies are probably missing and what they would have shown, it produces a corrected effect that allows for the unseen results. This gives a sense of how far the visible literature might overstate the truth. Modelling what selection hid is its approach. Copas and Shi (2001) describe this model.
Source: Copas & Shi 2001
How does the Copas model work?
The Copas model works by specifying a selection mechanism that relates the probability of a study being published to its results, typically its effect estimate and standard error, alongside the model for the true effects. By modelling the selection process, it estimates how the observed studies may be a biased subset and adjusts the pooled effect toward what it would be without selective publication. The degree of adjustment depends on assumptions about the strength of selection, which are varied in sensitivity analyses. So the Copas model estimates a bias-adjusted effect by explicitly modelling how publication depends on study results.
Source: Copas 1999
Why is the Copas model used?
The Copas model is used to go beyond detecting publication bias to estimating its impact and adjusting the pooled effect accordingly, providing an indication of how much the meta-analysis result might change if unpublished studies were included. This is valuable because tests such as Egger's indicate possible bias but do not quantify its effect on the estimate. By modelling the selection process, the Copas model offers a bias-adjusted estimate and a sensitivity analysis of how the result depends on the extent of selective publication. So it is used to assess the robustness of meta-analysis conclusions to publication bias.
Source: Sterne, Gavaghan & Egger 2000
What are the assumptions of the Copas model?
The Copas model assumes a particular form for the selection mechanism relating publication probability to study results, and its adjustment depends on assumptions about the strength and pattern of this selection, which cannot be fully known, since the unpublished studies are unobserved. It also assumes a model for the distribution of true effects. Because the selection is not directly observable, the results are sensitive to these assumptions. So the Copas model rests on assumptions about how publication depends on results, and its estimates are explored across a range of selection assumptions rather than treated as a single definitive correction.
Source: Copas 1999
What are the limitations of the Copas model?
The limitations of the Copas model arise because the selection mechanism it models cannot be observed directly, so the adjustment depends on unverifiable assumptions about how publication relates to results, and the results are sensitive to these; it is more complex to apply than simple tests for publication bias; and it may be unstable with few studies. So the Copas model provides an informative sensitivity analysis of the potential impact of publication bias rather than a definitive correction, and its output is interpreted as showing how conclusions might change under different assumptions about selective publication, rather than as a precise bias-adjusted estimate.
Source: Copas 1999
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 2 Dec 2025
Content version: 1.0.0
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- Term code
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