Robust Bayesian meta-analysis model-averages over the presence or absence of an effect, of heterogeneity and of publication bias, reporting inclusion Bayes factors and a bias-adjusted pooled effect.
A conventional meta-analysis commits to a fixed- or random-effects model and, at most, tests for publication bias as an afterthought. The robust Bayesian version turns each of those choices into a question and averages across them. It builds an ensemble by crossing three components: an effect (μ ~ Normal(0, 1) vs. μ = 0), heterogeneity (τ ~ Inverse-Gamma vs. τ = 0), and publication bias (none, vs. selection weight-functions on the study p-values, vs. PET / PEESE small-study-effect regressions). Every model's marginal likelihood is computed by deterministic quadrature, and the models are combined by their posterior probabilities.
Each of the three questions gets an INCLUSION Bayes factor — the posterior odds that the component is present divided by the prior odds (½ vs. ½, so 1). BF_effect answers 'is there a non-zero pooled effect?', BF_heterogeneity answers 'do the true effects vary across studies?', and BF_pubbias answers 'is there evidence of publication bias / small-study effects?'.
The model-averaged pooled effect and its credible interval already account for uncertainty about heterogeneity and bias: when the data show funnel asymmetry, the selection and PET-PEESE models gain posterior weight and pull the estimate toward its bias-corrected value; when the data are symmetric, the no-bias models dominate and the estimate matches an ordinary Bayesian meta-analysis. Restricting the ensemble to the no-bias models reproduces a fixed-effect (τ = 0) or random-effects Bayesian meta-analysis exactly.
Two numeric columns: effect size yᵢ and its standard error seᵢ (+ optional study labels), with adjustable effect and heterogeneity prior widths.
Three inclusion Bayes factors (effect / heterogeneity / publication bias) with posterior inclusion probabilities, a model-averaged pooled effect with 95% credible interval (plus the effect-present conditional estimate), and a per-model ensemble table (relative log marginal likelihood + posterior probability).
BF_effect > 1 favours a non-zero pooled effect; BF_heterogeneity > 1 favours between-study variation; BF_pubbias > 1 favours the presence of publication bias. Read each on the Jeffreys scale (3 / 10 / 100).
A publication-bias BF well above 1 means the naive pooled estimate is likely inflated — the model-averaged estimate reported here already discounts it.
The ensemble table shows which combinations of assumptions the data support; a single dominant row means the averaging is effectively selecting one model.