Use this Skill to pool effect sizes across studies: fixed/random-effects models, heterogeneity tests (I², Cochran Q), forest plot, funnel plot, Egger test, and subgroup analysis using pymare or manual computation.
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Use this Skill to pool effect sizes across studies: fixed/random-effects models, heterogeneity tests (I², Cochran Q), forest plot, funnel plot, Egger test, and subgroup analysis using pymare or manual computation.
Meta-Analysis: Pooling Effect Sizes Across Studies
TL;DR — Pool effect sizes from multiple studies using fixed-effects or
random-effects models, quantify heterogeneity (I², Cochran Q, τ²), produce
publication-quality forest plots and funnel plots, test for publication bias
(Egger, trim-and-fill), and run subgroup / moderator analyses.
When to Use This Skill
Use this Skill when you have:
A completed systematic review with ≥ 2 quantitative studies on the same outcome
A set of effect sizes (Cohen's d, Hedges' g, OR, RR, correlation r) and their
standard errors or sample sizes
A need to communicate pooled estimates with forest or funnel plots
Questions about heterogeneity between studies or subgroup differences
Task
Use case
Fixed-effects pooling
Studies estimate the same true effect; low heterogeneity
Random-effects pooling
True effects vary across studies; I² > 25%
Heterogeneity decomposition
Understand sources of between-study variance
Forest plot
Visualize study-level and pooled estimates
Funnel plot + Egger test
Detect small-study effects / publication bias
Subgroup analysis
Test whether effect differs by moderator variable
Background & Key Concepts
Effect Size Types
Measure
Formula
Use case
Cohen's d
(M₁ − M₂) / SD_pooled
Two-group continuous outcome
Hedges' g
d × correction factor J(df)
Small samples (n < 20 per group)
Odds Ratio (OR)
(a/b) / (c/d) in 2×2 table
Binary outcome, case-control
Risk Ratio (RR)
(a/(a+b)) / (c/(c+d))
Binary outcome, cohort/RCT
Correlation r
Pearson r
Association between two continuous vars
All are converted to a common scale (log-OR or Fisher's z) for pooling, then
back-transformed for presentation.
Fixed vs Random Effects
Fixed-effects (inverse-variance weighting): assumes one true underlying effect
shared by all studies. Precision-weighted average. Appropriate when studies are
highly homogeneous.
Random-effects (DerSimonian-Laird or REML): assumes true effects vary across
studies drawn from a distribution N(μ, τ²). Accounts for between-study variance τ².
Gives wider, more honest confidence intervals when heterogeneity exists.
Heterogeneity Statistics
Cochran Q: Sum of squared deviations from pooled estimate, weighted by study
precision. Chi-squared distributed with k−1 df. H₀: all studies share one true effect.
I²: Percentage of total variability due to between-study heterogeneity.
I² = (Q − df)/Q × 100. Thresholds: 0–25% low, 25–50% moderate, > 50% high.
τ²: Between-study variance. Estimated by DerSimonian-Laird method-of-moments
or REML.
The trim-and-fill method imputes missing studies on the asymmetric side of the
funnel and re-estimates the pooled effect. Use pymare for this:
from pymare import Dataset
from pymare.estimators import DerSimonianLaird, TrimAndFill
deftrim_and_fill_pymare(es: list, se: list, study_ids: list) -> dict:
"""
Apply trim-and-fill using pymare.
Args:
es: List of effect sizes.
se: List of standard errors.
study_ids: List of study identifier strings.
Returns:
Dictionary with original and adjusted pooled estimates.
"""
dataset = Dataset(
y=es,
v=[s**2for s in se],
n=study_ids, # used as labels only
)
# Original DL estimate
dl = DerSimonianLaird()
dl.fit(dataset)
original = dl.summary_
# Trim-and-fill adjusted
taf = TrimAndFill()
taf.fit(dataset)
adjusted = taf.summary_
return {"original": original, "adjusted": adjusted}
Troubleshooting
Problem
Cause
Fix
tau2 is negative
Q < df (less variation than expected)
Truncate τ² at 0 (as implemented above)
Forest plot labels overlap
Too many studies
Reduce figsize height per study or use abbreviations
I² = 100%
One study has extreme effect or tiny SE
Check data entry; consider influence diagnostics
Egger p-value inflated
Small k (fewer than 10 studies)
Use Begg's rank correlation test instead
pymare import error
Package not installed correctly
pip install pymare --upgrade
Wide pooled CI
High τ² (genuine heterogeneity)
Report τ² and prediction interval; investigate moderators
External Resources
Borenstein et al. (2009) Introduction to Meta-Analysis (Wiley)
