Funnel plots

Introduction

The funnel plot displays each study’s effect estimate against a measure of its precision (typically the standard error). Under the common-effect assumption with no systematic bias, study effects scatter symmetrically around the pooled estimate, with larger scatter at lower precision. This produces the characteristic inverted-funnel shape.

Asymmetry in the funnel — with studies clustered on one side, especially at low precision — can indicate publication bias, small-study effects, or genuine associations between study size and effect magnitude.

bayesma produces funnel plots via funnel_plot(), with optional contour lines and Shi et al. (2020) pseudo-confidence regions based on latent standard errors.

Basic funnel plot

funnel_plot(fit)

By default this produces a scatter plot of $y_i$ (x-axis) vs $s_i$ (y-axis, inverted), with the y-axis running from 0 at the top to $\max(s_i)$ at the bottom. The pooled effect line is overlaid.

Contour-enhanced funnel plot

Contour lines at $p = 0.10$, $0.05$, and $0.01$ (two-sided) show the statistical significance regions. Studies outside the $p < 0.05$ contour would be significant under the null of no effect.

funnel_plot(fit, contour = TRUE)

A cluster of studies just inside the $p < 0.05$ boundary — with few studies outside it — is consistent with selective reporting of significant results.

Latent standard errors (Shi et al.)

Studies sometimes report effects on scales that mix signal and noise in ways that make the standard error an unreliable precision proxy. Shi et al. (2020) proposed plotting effects against the latent standard error implied by the model, which separates publication-bias-related precision from sampling variation.

funnel_plot(fit, se_type = "latent")

The latent SE funnel plot is particularly informative in the context of the Bayesian Egger model.

Subgroup colouring

When a grouping variable is available, studies can be coloured by subgroup:

funnel_plot(fit, colour_by = "risk_of_bias_domain")

This is useful for investigating whether funnel asymmetry is driven by a particular subgroup.

Interpreting funnel asymmetry

Asymmetry has multiple possible causes:

Cause	Pattern
Publication bias	Missing studies at low precision on one side
Small-study effects	Larger effects in smaller studies regardless of direction
Heterogeneity correlated with study size	Effects vary systematically with precision
Artefact (e.g., intervention fidelity)	Smaller, more controlled studies yield different effects

A funnel plot is a starting point for investigation, not a conclusive test. Formal asymmetry tests (Egger, Bayesian Egger’s test) and model-based adjustments (PET-PEESE, selection models) should follow.

Supported effect measures

funnel_plot() supports the following estimands, with automatic back-transformation for display:

• Log-odds ratio (OR)
• Log-risk ratio (RR)
• Log-hazard ratio (HR)
• Log-incidence rate ratio (IRR)
• Mean difference (MD)
• Standardised mean difference (SMD)

Worked example

fit_re <- bayesma(dat_binary, model_type = "random_effect")

funnel_plot(
  fit_re,
  contour   = TRUE,
  colour_by = "year_quartile",
  label     = FALSE
)