Publication Bias & the Funnel Plot

The Experiment Pool

True effect size δ 0.20

Studies simulated 150

File-drawer strength 85%

Show unpublished (file-drawer) studies

Meta-Analysis Readout

True effect

0.20

Naive pooled (published only)

—

Bias (naive − true)

—

Egger's intercept

—

Published / total studies

—

Published — significant (p<.05) Published — non-significant, got through anyway Unpublished — buried in the file drawer

How it works

Every study draws a sample of size n, measures a noisy estimate of the same true effect δ, and reports it with a standard error:

observed = δ + noise,  noise ~ N(0, σ²/n)
SE = σ / √n
z  = observed / SE
p  = 2·(1 − Φ(|z|))   (two-tailed, vs. the null δ=0)

A study is always published if p < 0.05. If it is not significant, it only survives with probability (1 − bias) — the "file-drawer strength" slider. At 100% bias, only significant results ever see daylight.

pooled effect = Σ(wᵢ·effectᵢ) / Σwᵢ,   wᵢ = 1/SEᵢ²   (inverse-variance weights)

The funnel plot puts each study's effect size on the horizontal axis and its precision (1/SE, i.e. how large/reliable the study was) on the vertical axis. With no bias, points scatter symmetrically around the true effect, narrowing to a point at high precision — a funnel shape. With publication bias, the low-precision (small, noisy) studies that landed on the "wrong" side of zero get filtered out, leaving a lopsided funnel and a pooled estimate that overstates the true effect.

Egger's regression intercept is the classic asymmetry test: regress z = effect/SE against precision (1/SE); a non-zero intercept flags a funnel that isn't symmetric around zero, exactly the signature publication bias leaves behind. This is the mechanism behind real reproducibility-crisis findings — meta-analyses of small, biased literatures routinely overstate true effects until large, pre-registered replications correct them.

Drag to orbit · scroll to zoom.
X = effect size · Y = precision (1/SE)
Funnel lines mark the 95% significance boundary.

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