Build forest plots (HR, OR, RR, beta-coefficient summaries with CIs) and funnel plots (meta-analysis publication-bias diagnostics) using forestplot, metafor, ggforest, and MendelianRandomization with proper axis-scaling, summary-diamond placement, subgroup nesting, and Egger / trim-and-fill asymmetry tests. Use when summarizing effects across subgroups, trials, or instruments — meta-analysis, Mendelian randomization, subgroup HRs.
Build forest plots (HR, OR, RR, beta-coefficient summaries with CIs) and funnel plots (meta-analysis publication-bias diagnostics) using forestplot, metafor, ggforest, and MendelianRandomization with proper axis-scaling, summary-diamond placement, subgroup nesting, and Egger / trim-and-fill asymmetry tests. Use when summarizing effects across subgroups, trials, or instruments — meta-analysis, Mendelian randomization, subgroup HRs.
Before using code patterns, verify installed versions match. If versions differ:
R: packageVersion('<pkg>') then ?function_name
If code throws ImportError, AttributeError, or TypeError, introspect the installed package and adapt the example to match the actual API rather than retrying.
Forest and Funnel Plots
"Summarize effects across studies / subgroups" -> Render each effect estimate (HR, OR, RR, β) as a square (size = inverse variance / weight), horizontal bar (95% CI), and label, with an optional summary diamond at the bottom from a meta-analysis pool (fixed-effect or random-effects). The funnel plot diagnoses publication bias by plotting effect size vs precision; asymmetry indicates missing small-study-with-null-result publications (Egger 1997).
The Single Most Important Modern Insight -- Heterogeneity Is the First Question
A pooled effect estimate is meaningless if the underlying studies are heterogeneous. The Higgins I² statistic (Higgins-Thompson 2002 Stat Med 21:1539) quantifies between-study variance; the conventional 25/50/75% interpretation tiers come from the Cochrane Handbook §10.10.2 (Higgins et al editors), NOT the original Higgins-Thompson paper which cautioned against rigid cutoffs. A meta-analysis with I² > 75% and a pooled effect must explain the heterogeneity (subgroup analysis, meta-regression) — pooling without explanation is statistically defensible but biologically unhelpful.
A forest plot's bottom must report: pooled estimate + 95% CI + I² + τ² (between-study variance) + Q-test p-value. Without these, the plot is a list of effects, not a meta-analysis.
Decision Tree by Analysis Type
Analysis
Tool
Pooling model
Forest method
Single-trial subgroup HRs
survminer::ggforest
None (subgroup display)
Coxph object
Meta-analysis of binary outcomes
metafor::rma -> forest()
DerSimonian-Laird or REML random-effects
Standard forest
Meta-analysis of continuous outcomes
metafor::rma(yi, vi)
REML random-effects
Standard forest
Mendelian randomization
MendelianRandomization::mr_forest
Multiple MR methods
MR-specific forest
Subgroup forest with interaction p
metafor::rma + addpoly + interaction model
Subgroup REML
Nested forest
Network meta-analysis
netmeta::forest.netmeta
Bayesian or frequentist
Network forest
Cumulative meta-analysis (over time)
metafor::cumul + forest
–
Cumulative forest
Fixed-Effect vs Random-Effects Meta-Analysis
Model
Assumption
When appropriate
Pooled estimate weight
Fixed-effect (Mantel-Haenszel, IVS)
All studies estimate the same true effect
Single mechanism, homogeneous design
1 / within-study variance
Random-effects (DerSimonian-Laird, REML)
Studies estimate distinct true effects from a common distribution
Heterogeneous designs / populations
1 / (within + between variance)
Use random-effects by default. Fixed-effect assumes all studies estimate the same parameter, which is almost never true across multi-center trials with different populations. REML is the modern default (Viechtbauer 2005); DerSimonian-Laird is the older default still commonly seen.
metafor::rma + forest -- The Reference Implementation
Goal: Pool study-level effect estimates with random-effects meta-analysis; render a forest plot with study weights, individual effect+CI, and pooled summary diamond.
Approach: Compute per-study yi (effect) and vi (sampling variance); fit REML random-effects model; pass to forest() with prediction interval if heterogeneity is non-trivial.
library(metafor)# Input: per-study effect (yi) and variance (vi)# For OR: yi = log(OR), vi = SE(log(OR))^2# For HR: yi = log(HR), vi = SE(log(HR))^2
res <- rma(yi = log_or, vi = log_or_se^2,
data = studies, slab = paste(author, year),
method ='REML')# I^2 and tau^2 in the summary
summary(res)# I^2 (residual heterogeneity)# tau^2 (estimated amount of (residual) heterogeneity)# Q-test for heterogeneity
forest(res,
atransf =exp,# display OR on natural scale
at =log(c(0.25,0.5,1,2,4)),# ticks at meaningful OR values
refline =0,# log(1) for OR/HR/RR
xlab ='Odds Ratio (95% CI)',
header =c('Study','OR [95% CI]'),
mlab = bquote(paste('RE Model (Q = ', .(round(res$QE,2)),', df = ', .(res$k -1),', p = ', .(format.pval(res$QEp, digits =2)),'; ', I^2,' = ', .(round(res$I2,1)),'%)')),
addpred =TRUE)# prediction interval per Higgins 2009
addpred = TRUE adds a 95% prediction interval — where a new study's effect is expected to fall (Higgins-Thompson-Spiegelhalter 2009 JRSS-A). This is the most honest summary when I² > 30%.
ggforest for Cox Subgroup Forests
library(survminer)
fit <- coxph(Surv(time, status)~ treatment + age + sex + stage, data = df)
ggforest(fit,
data = df,
main ='Subgroup HRs',
cpositions =c(0.02,0.22,0.4),
fontsize =0.7,
refLabel ='Reference',
noDigits =2)
ggforest produces a publication-ready subgroup forest from a coxph object. For pre-specified subgroup analyses (treatment × subgroup interaction), test interaction explicitly and annotate the p-value.
Small-k Regime -- When Meta-Analysis Asymptotics Break
For k < 5 studies, the REML-based 95% CI from metafor::rma() is severely anti-conservative — it relies on chi-square asymptotics that fail with few studies. Use Hartung-Knapp-Sidik-Jonkman (HKSJ) adjustment (test = 'knha'):
res_hksj <- rma(yi = log_or, vi = log_or_se^2, data = studies,
method ='REML', test ='knha')# HKSJ for k<5
HKSJ uses a t-distribution with k-1 degrees of freedom and adjusts SE via the Q-statistic — well-calibrated even at k=3. For k < 3 a meta-analysis is not advisable; report individual study effects in a forest plot without a pooled summary.
Also: I² is uninterpretable below k = 5 (Borenstein 2017 Res Synth Methods 8:5); the point estimate has wide CI dominated by k itself, not heterogeneity. Do not report I² for k < 5.
Funnel Plot and Egger Test
Goal: Diagnose publication bias by visual asymmetry of effect size vs precision.
Approach: Plot effect (x) vs SE (inverted y); under no bias, points form a symmetric inverted funnel with the pooled estimate at the apex. Asymmetry suggests missing small-N null-result studies. Egger's regression test (Egger 1997 BMJ 315:629) formalizes the asymmetry.
funnel(res,
xlab ='log(OR)',
refline = res$b)# Egger's test
regtest(res, model ='lm', predictor ='sei')# significant p indicates asymmetry; suggests publication bias# Trim-and-fill (Duval-Tweedie 2000) -- adjusts for asymmetry
res_tf <- trimfill(res)
forest(res_tf)
funnel(res_tf)
Contour-enhanced funnel plot (Peters 2008 J Clin Epidemiol 61:991) overlays significance contours (p < 0.10, < 0.05, < 0.01); asymmetry concentrated in "non-significant" regions indicates publication bias more specifically than generic asymmetry.
Pooling under high heterogeneity without explanation
Trigger: Random-effects meta-analysis pooled with I² > 75%; no subgroup or meta-regression.
Mechanism: Pooled estimate is a weighted average across substantively different effects; biologically meaningless.
Symptom: Pooled OR = 1.5 with 95% CI (1.2-1.8) but per-study effects range 0.3-5.0.
Fix: Run subgroup analysis or meta-regression to explain heterogeneity; report I², τ², and prediction interval; do NOT report a single pooled effect as the answer.
Fixed-effect when studies are heterogeneous
Trigger: Default fixed-effect model on multi-population data.
Bowden J, Davey Smith G, Burgess S. 2015. Mendelian randomization with invalid instruments: effect estimation and bias detection through Egger regression. Int J Epidemiol 44(2):512-525.
Brookes ST, Whitley E, Peters TJ, et al. 2001. Subgroup analyses in randomised controlled trials: quantifying the risks of false-positives and false-negatives. Health Technol Assess 5(33):1-56.
DerSimonian R, Laird N. 1986. Meta-analysis in clinical trials. Control Clin Trials 7(3):177-188.
Duval S, Tweedie R. 2000. Trim and fill: a simple funnel-plot–based method of testing and adjusting for publication bias in meta-analysis. Biometrics 56:455-463.
Egger M, Davey Smith G, Schneider M, Minder C. 1997. Bias in meta-analysis detected by a simple, graphical test. BMJ 315(7109):629-634.
Higgins JPT, Thompson SG. 2002. Quantifying heterogeneity in a meta-analysis. Stat Med 21(11):1539-1558.
Higgins JPT, Thompson SG, Spiegelhalter DJ. 2009. A re-evaluation of random-effects meta-analysis. JRSS-A 172(1):137-159.
Peters JL, Sutton AJ, Jones DR, Abrams KR, Rushton L. 2008. Contour-enhanced meta-analysis funnel plots help distinguish publication bias from other causes of asymmetry. J Clin Epidemiol 61(10):991-996.
Sterne JAC, Sutton AJ, Ioannidis JPA, et al. 2011. Recommendations for examining and interpreting funnel plot asymmetry in meta-analyses of randomised controlled trials. BMJ 343:d4002.
Viechtbauer W. 2010. Conducting meta-analyses in R with the metafor package. J Stat Softw 36(3):1-48.
Hartung J, Knapp G. 2001. A refined method for the meta-analysis of controlled clinical trials with binary outcome. Stat Med 20(24):3875-3889.
IntHout J, Ioannidis JPA, Borm GF. 2014. The Hartung-Knapp-Sidik-Jonkman method for random effects meta-analysis is straightforward and considerably outperforms the standard DerSimonian-Laird method. BMC Med Res Methodol 14:25.
Borenstein M, Higgins JPT, Hedges LV, Rothstein HR. 2017. Basics of meta-analysis: I² is not an absolute measure of heterogeneity. Res Synth Methods 8(1):5-18.
Higgins JPT, Thomas J, Chandler J, et al (editors). Cochrane Handbook for Systematic Reviews of Interventions (current version). Section 10.10.2 — interpretation tiers for I².
Related Skills
clinical-biostatistics/effect-measures - HR / OR / RR / NNT definitions
clinical-biostatistics/subgroup-analysis - Interaction tests for subgroup HRs
