Fixed-effects meta-analyses assume that the effect size d is identical in all studies. In contrast, random-effects meta-analyses assume that effects vary according to a normal distribution with mean d and standard deviation τ. When assuming prior distributions for d and τ, both models can be compared using Bayes factors. Alternatively, posterior model probabilities can be used to compare the evidence for or against an effect (i.e., whether d = 0) and the evidence for or against random effects (i.e., whether τ = 0). By using Bayesian model averaging (BMA), both types of tests can be performed by marginalizing over the other question. Most importantly, this allows to test whether an effect exists while accounting for uncertainty whether study heterogeneity exists or not.

Installing metaBMA

To install the latest stable release of metaBMA from CRAN, run:


The latest developer version of metaBMA can be installed from GitHub via:

### install dependencies if necessary:
# install.packages(c("rstan", "rstantools", "bridgesampling",
#                    "LaplacesDemon", "logspline", "mvtnorm",
#                    "coda", "knitr", "methods"))

if (!require("devtools"))

Note that metaBMA requires the software Stan. In case of issues with using Stan, information how to install the R package rstan is available here:

Getting Started

The most general functions in metaBMA are meta_bma and meta_default, which fit random- and fixed-effects models, compute the inclusion Bayes factor for the presence of an effect and the averaged posterior distribution of the mean effect d (which accounts for uncertainty regarding study heterogeneity).

Moreover, meta_fixed() and meta_random() fit standard meta-analysis models with fixed-effects and random-effects, respectively. The model-specific posteriors for the parameter d can be averaged with bma() and inclusion Bayes factors be computed with inclusion().

The function prior() facilitates the construction and visual inspection of prior distributions. Sensitivity analysis can be performed with the function meta_sensitivity().

For an overview, see:


If you use metaBMA, please cite the software as follows:

Heck, D. W., Gronau, Q. F., & Wagenmakers, E.-J. (2019). metaBMA: Bayesian model averaging for random and fixed effects meta-analysis.

An (open-access) introduction to Bayesian meta-analysis with model averaging is available at:

Gronau, Q. F., Heck, D. W., Berkhout, S. W., Haaf, J. M., & Wagenmakers, E.-J. (2021). A primer on Bayesian model-averaged meta-analysis. Advances in Methods and Practices in Psychological Science, 4, 1–19.