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 tau. Both models can be compared in a Bayesian framework by assuming specific prior distribution for d and tau. Given the posterior model probabilities, the evidence for or against an effect (i.e., whether *d* = 0) and the evidence for or against random effects can be evaluated (i.e., whether tau = 0). By using Bayesian model averaging (i.e., inclusion Bayes factors), 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.

## The Package metaBMA

The R package `metaBMA`

is available on GitHub at https://github.com/danheck/metaBMA

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 a single meta-analysis models. The model-specific posteriors for d can be averaged by `bma`

and inclusion Bayes factors be computed by `inclusion`

. Finally, the function `prior`

facilitates the construction and visual inspection of prior distributions.

## Installing metaBMA

`metaBMA`

requires the software JAGS. To install the latest stable release of `metaBMA`

from CRAN, run:

```
install.packages("metaBMA")
```

To install the latest developer version from GitHub, use:

```
### Dependencies:
# install.packages("devtools", "mvtnorm", "runjags", "LaplacesDemon", "logspline")
library(devtools)
install_github("danheck/metaBMA", build_vignettes = TRUE)
```

## Citation

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

- Heck, D. W., & Gronau, Q. F. (2017). metaBMA: Bayesian model averaging for random- and fixed-effects meta-analysis.

[BibTeX] [https://github.com/danheck/metaBMA]`@book{heck2017metabma, title = {{{metaBMA}}: {{Bayesian}} Model Averaging for Random- and Fixed-Effects Meta-Analysis}, url = {https://github.com/danheck/metaBMA}, date = {2017}, author = {Heck, Daniel W and Gronau, Quentin F}, groups = {dwh} }`

The package was used in:

- Gronau, Q. F., Erp, S. V., Heck, D. W., Cesario, J., Jonas, K. J., & Wagenmakers, E. (2017). A Bayesian model-averaged meta-analysis of the power pose effect with informed and default priors: the case of felt power. Comprehensive Results in Social Psychology, 2, 123-138. doi:10.1080/23743603.2017.1326760

[BibTeX] [Abstract]

Earlier work found that – compared to participants who adopted constrictive body postures – participants who adopted expansive body postures reported feeling more powerful, showed an increase in testosterone and a decrease in cortisol, and displayed an increased tolerance for risk. However, these power pose effects have recently come under considerable scrutiny. Here, we present a Bayesian meta-analysis of six preregistered studies from this special issue, focusing on the effect of power posing on felt power. Our analysis improves on standard classical meta-analyses in several ways. First and foremost, we considered only preregistered studies, eliminating concerns about publication bias. Second, the Bayesian approach enables us to quantify evidence for both the alternative and the null hypothesis. Third, we use Bayesian model-averaging to account for the uncertainty with respect to the choice for a fixed-effect model or a random-effect model. Fourth, based on a literature review, we obtained an empirically informed prior distribution for the between-study heterogeneity of effect sizes. This empirically informed prior can serve as a default choice not only for the investigation of the power pose effect but for effects in the field of psychology more generally. For effect size, we considered a default and an informed prior. Our meta-analysis yields very strong evidence for an effect of power posing on felt power. However, when the analysis is restricted to participants unfamiliar with the effect, the meta-analysis yields evidence that is only moderate.

`@article{gronau2017bayesian, title = {A {{Bayesian}} Model-Averaged Meta-Analysis of the Power Pose Effect with Informed and Default Priors: The Case of Felt Power}, volume = {2}, doi = {10.1080/23743603.2017.1326760}, shorttitle = {A {{Bayesian}} Model-Averaged Meta-Analysis of the Power Pose Effect with Informed and Default Priors}, abstract = {Earlier work found that – compared to participants who adopted constrictive body postures – participants who adopted expansive body postures reported feeling more powerful, showed an increase in testosterone and a decrease in cortisol, and displayed an increased tolerance for risk. However, these power pose effects have recently come under considerable scrutiny. Here, we present a Bayesian meta-analysis of six preregistered studies from this special issue, focusing on the effect of power posing on felt power. Our analysis improves on standard classical meta-analyses in several ways. First and foremost, we considered only preregistered studies, eliminating concerns about publication bias. Second, the Bayesian approach enables us to quantify evidence for both the alternative and the null hypothesis. Third, we use Bayesian model-averaging to account for the uncertainty with respect to the choice for a fixed-effect model or a random-effect model. Fourth, based on a literature review, we obtained an empirically informed prior distribution for the between-study heterogeneity of effect sizes. This empirically informed prior can serve as a default choice not only for the investigation of the power pose effect but for effects in the field of psychology more generally. For effect size, we considered a default and an informed prior. Our meta-analysis yields very strong evidence for an effect of power posing on felt power. However, when the analysis is restricted to participants unfamiliar with the effect, the meta-analysis yields evidence that is only moderate.}, journaltitle = {Comprehensive Results in Social Psychology}, date = {2017}, pages = {123--138}, author = {Gronau, Quentin F. and Erp, Sara Van and Heck, Daniel W and Cesario, Joseph and Jonas, Kai J. and Wagenmakers, Eric-Jan}, groups = {Meta Analysis,dwh} }`