Publication Bias & the Funnel Plot (2D)

The Experiment Pool

True effect size δ 0.20

Studies simulated 150

File-drawer strength 85%

Significance threshold α 0.05

Max study size (n) 800

Show unpublished (file-drawer) studies

Meta-Analysis Readout

True effect

0.20

Naive pooled (published only)

—

Bias (naive − true)

—

Egger's intercept

—

PET bias-corrected estimate

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Published / total studies

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Published — significant (p<α) Published — non-significant, got through anyway Unpublished — buried in the file drawer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 < α (your significance threshold). 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. Sample size n is drawn log-uniformly between 15 and your "max study size" so both tiny underpowered studies and large well-funded ones appear.

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

The funnel plot (left) puts each study's effect size on the horizontal axis and its precision (1/SE) on the vertical axis — drag to pan, scroll to zoom. With no bias, points scatter symmetrically around the true effect, narrowing to a point at high precision. With publication bias, small noisy studies that landed on the "wrong" side of zero get filtered out, leaving a lopsided funnel.

Egger's regression (top right) regresses z = effect/SE against precision (1/SE); its intercept is the classic asymmetry test — a non-zero value flags a funnel that isn't symmetric around zero. Algebraically, effect = intercept·SE + slope, so the regression's slope term is exactly the effect size extrapolated to SE→0 — the PET bias-corrected estimate, a real meta-analysis correction technique (Stanley & Doucouliagos' precision-effect test) surfaced here from the same fit, not a separate approximation.

The running estimate (bottom right) re-pools only the studies published so far, in the order they were simulated, showing how an early, small, biased literature can look confidently wrong before enough studies accumulate to correct it — the actual mechanism behind real reproducibility-crisis findings.

Funnel plot: effect vs. precision

Hover a point for details.
Drag to pan · scroll to zoom.

X = effect size · Y = precision (1/SE)

Egger's regression: z vs. precision

Running pooled estimate vs. study order