Bayesian Egger’s test • bayesma

Introduction

Egger’s regression (Egger et al., 1997) detects funnel plot asymmetry by regressing standardised effect estimates on precision. In a symmetric funnel — the expected pattern under no publication bias — the regression passes through the origin. A non-zero intercept indicates that small-precision (small-sample) studies yield systematically different effects than large-precision (large-sample) studies, a pattern consistent with publication bias or small-study effects.

bayesma implements a fully Bayesian version of Egger’s regression via egger(). The Bayesian formulation provides:

posterior distributions for the Egger intercept and slope, not just p-values
flexible modelling of residual heterogeneity (multiplicative or additive)
coherent quantification of evidence for asymmetry via Bayes factors

Model specification

Following Shi et al. (2020), the Bayesian Egger model is

$\frac{y_i}{s_i} = \alpha \cdot s_i^{-1} + \beta + \varepsilon_i$

where $y_i / s_i$ is the standardised effect (z-score) and $s_i$ is the standard error. Rearranging:

$y_i = \alpha + \beta \cdot s_i + \varepsilon_i$

Here $\alpha$ is the effect at infinite precision (no publication bias) and $\beta$ is the Egger regression coefficient: the rate at which the effect changes as a function of standard error.

Residual heterogeneity

Two models for the error term $\varepsilon_i$ :

Multiplicative heterogeneity (Egger’s original formulation): $\varepsilon_i \sim \mathcal{N}(0, \kappa^2 s_i^2)$ where $\kappa$ is a multiplicative overdispersion factor. $\kappa = 1$ corresponds to the standard fixed-effect Egger model.

Additive heterogeneity: $\varepsilon_i \sim \mathcal{N}(0, s_i^2 + \tau^2)$ where $\tau$ is the between-study heterogeneity, identical to the standard random-effects parameterisation.

Priors:

$\alpha \sim \mathcal{N}(0, 1), \qquad \beta \sim \mathcal{N}(0, 1), \qquad \kappa \sim \text{Half-Normal}(0, 1), \qquad \tau \sim \text{Half-Cauchy}(0, 0.5)$

Fitting Egger’s test

egger_fit <- egger(
  data,
  heterogeneity = "additive"
)

summary(egger_fit)
egger_plot(egger_fit)

heterogeneity = "multiplicative" reproduces the original Egger model. "additive" is preferable when the goal is to estimate the publication-bias adjusted effect $\alpha$ while allowing for genuine heterogeneity.

Interpreting the output

summary() returns:

Parameter	Interpretation
`alpha`	Pooled effect adjusted for publication bias (intercept)
`beta`	Egger slope: change in effect per unit SE
`tau`	Residual heterogeneity (additive model)
`kappa`	Overdispersion (multiplicative model)
`BF_beta`	Bayes factor for $H_1: \beta \neq 0$ vs $H_0: \beta = 0$

A large $|\beta|$ with posterior mass away from zero indicates funnel asymmetry. The Bayes factor quantifies evidence on a continuous scale:

$\text{BF}_{10}$	Interpretation
$< 1$	Evidence against asymmetry
$1$ – $3$	Anecdotal
$3$ – $10$	Moderate
$10$ – $30$	Strong
$> 30$	Very strong

Egger plot

egger_plot() produces a scatter plot of standardised effects vs standard error with the fitted Egger regression line and its 95% credible band.

Limitations

Egger’s test has low power when $k < 10$ . A non-significant result does not rule out publication bias; it only indicates that the data are insufficient to detect asymmetry at the measured precision.

Funnel asymmetry can also arise from causes other than publication bias: genuine small-study effects, heterogeneity correlated with study size, or artefacts in effect size computation. A positive Egger test should prompt further investigation (selection models, PET-PEESE) rather than a reflexive conclusion of publication bias.

For a visual complement to the Egger test, see Funnel Plots.