inst/help/BayesianMetaAnalysis.md

In jasp-stats/jaspMetaAnalysis: Meta-Analysis Module for JASP

Bayesian Meta-Analysis

Bayesian meta-analysis allows the user to estimate an overall effect size estimate from multiple studies and test whether the effect size estimate deviates from zero.

Input

Data

• Effect Size: Effect size of the studies.
• Effect Size Standard Error: Standard errors of the effect sizes per study. Must always be positive. A 95% CI for the effect sizes can be provided instead.
• 95% CI Lower and Upper Bound: 95% CI lower and upper bound of the effect sizes per study. Must be separated into two columns. The standard errors of the effect sizes can be provided instead.
• Study Labels: Optional. Will add study labels to 'Effect sizes per study' table and the (cumulative) forest plot.

Model

• Fixed effects: performs a fixed effects Bayesian meta-analysis.
• Random effects: performs a random effects Bayesian meta-analysis.
• Model averaged: performs a model averaged Bayesian meta-analysis. Averages over the fixed and random effects models. Fixed and random effects results are also given.
• Constrained random effects
• All postive: performs a constrained random effects Bayesian meta-analysis, where the effect size estimates are constrained to be postive.
• All negative: performs a constrained random effects Bayesian meta-analysis, where the effect size estimates are constrained to be negative.

Bayes Factor

• BF10: By selecting this option, the Bayes factor will show evidence for the alternative hypothesis relative to the null hypothesis. This option is selected by default.
• BF01: By selecting this option, the Bayes factor will show evidence for the null hypothesis relative to the alternative hypothesis. This is equal to 1/BF10.
• Log(BF10): By selecting this option, the natural logarithm of BF10, BFm, BFInclusion, BF10, U will be displayed in the output.

Table

• Model probabilities: Displays the prior and posterior model probabilities.
• Effect sizes per study: Displays the observed and estimated effect sizes per study. For the fixed effects model, there are no estimated effect sizes per study.

Prior

Effect size

• Cauchy: Scale of the Cauchy prior distribution on the effect size under the alternative hypothesis; the default is 0.707.
• Normal: Mean and standard deviation.
• Student's t: Scale, Location and degrees of freedom (df).

Heterogeneity

• Inverse gamma: Shape and scale.
• Student's half t: Scale and degrees of freedom (df). Location is always to zero.

Truncation

• Lower bound: adds lower truncation to the prior distribution.
• Upper bound: adds upper truncation to the prior distribution.

Plot prior(s)

Displays the prior density function(s). This option is also available before assigning the data.

Plots

Forest plot

• Observed: Displays a forest plot with the observed effect sizes and the estimated overall effect size(s).
• Estimated: Displays a forest plot with the estimated effect sizes and the estimated overall effect size(s).
• Both: Displays a forest plot with both the observed and estimated effect sizes and the estimated overall effect size(s).
• Order
• Ascending: Displays the effect sizes in the forest plot in ascending order.
• Descending: Displays the effect sizes in the forest plot in descending order.
• Row order: Displays the effect sizes in the forest plot in the same order as in the provided data.

Cumulative forest plot

Displays the cumulative forest plot.

Prior and posterior

Displays the prior and posterior distribution of the effect size/heterogeneity under the alternative hypothesis.

• Additional info:
• Effect size: Adds the Bayes factor computed with the user-defined prior; adds a probability wheel depicting the odds of the data under the null hypothesis vs. an alternative hypothesis; adds the mean and the 95% credible interval of the posterior distribution of the effect size.
• Heterogeneity: Adds the Bayes factor computed with the user-defined prior; adds a probability wheel depicting the odds of the data under random effects H1 (& H0 for model averaging) vs. fixed effects H1 (& H0 for model averaging); adds the mean and the 95% credible interval of the posterior distribution of the effect size.
• Add fixed and random effects posterior: Adds the posterior density functions of the fixed and random effects models.
• Shade 95% CI: Colour the 95% CI in the posterior density function.

Sequential plot

• Bayes factors: Displays the development of the Bayes factor as the data come in using the user-defined prior.
• Posterior model probabilities: Displays the development of posterior model probabilities as the data come in using the user-defined prior model probabilities.

Advanced

Prior model probability

• Fixed effects H0: Sets the prior model probability for the null hypothesis of the fixed effects model.
• Fixed effects H1: Sets the prior model probability for the alternative hypothesis of the fixed effects model.
• Random effects H0: Sets the prior model probability for the null hypothesis of the random effects model.
• Random effects H1: Sets the prior model probability for the alternative hypothesis of the random effects model.

Estimation settings (MCMC)

• Iterations: Sets the number of iterations for the MCMC estimation.
• Chains: Sets the number of chains for the MCMC estimation.

Repeatability

• Set seed: Gives the option to set a seed for your analysis. Setting a seed will exclude random processes influencing an analysis. Note, however, that the seed may not reproduce the same results across operating systems.

Bayes factor computation

• Integration: Use integration to compute the Bayes factor.
• Bridge sampling: Use bridge sampling to compute the Bayes factor.

Output

Posterior Estimates per Model

• The first column contains the model(s) included in the analysis.
• The second column indicates the parameter.
• $\mu$: Group-level effect size.
• $\tau$: Group-level standard deviation.
• Mean: This column contains the estimated overall effect size/heterogeneity.
• SD: This column containts the estimated standard deviations for the overall effect size/heterogeneity.
• 95% Credible Interval: This columns contain the 95% credible interval lower and upper bounds for the overall effect size/heterogeneity.
• BF10: This column contains the Bayes factor that quantifies evidence for:
• Effect size: The alternative hypothesis relative to the null hypothesis/null model.
• Heterogeneity: The random effects H1 (& H0 for model averaging) relative to the fixed effects H1 (& H0 for model averaging).
• BF01: This column contains the Bayes factor that quantifies evidence for:
• Effect size: The null hypothesis/null model relative to the alternative hypothesis.
• Heterogeneity: The fixed effects H1 (& H0 for model averaging) relative to the random effects H1 (& H0 for model averaging).

Tables

• Model probabilities
• Prior: This column contains the (user-defined) prior model probabilites (must sum to 1, if user-defined probabilities do not sum to 1 the probabilities are scaled so they do sum to 1: $\frac{prior\,model\,probabilities}{\sum{prior\, model\,probabilities}}$).
• Posterior: This column contains the posterior model probabilities.
• Effect sizes per study
• Observed: This column contains the observed effect sizes per study.
• Estimated: This column contains the estimated effect sizes per study.

Plots

Prior

Displays the prior density function of the effect size/heterogeneity under the alternative hypothesis.

Forest plot

Displays the observed/estimated effect sizes per study above the estimated overall effect size, with on the y-axis the studies and on the x-axis the effect size.

Cumulative forest plot

Displays the development of the overall effect size estimate as a function of the number of studies.

Prior and posterior

Displays the prior (dashed line) and posterior (solid line) distribution of the overall effect size/heterogeneity under the alternative hypothesis.

• Additional info:
• Effect size: Displays the Bayes factor computed with the user-defined prior; displays a probability wheel depicting the odds of the data under the null hypothesis vs. the alternative hypothesis; displays the mean and 95% credible interval of the posterior distribution.
• Heterogeneity: Displays the Bayes factor computed with the user-defined prior; displays a probability wheel depicting the odds of the data under random effects H1 (& H0 for model averaging) vs. fixed effects H1 (& H0 for model averaging); displays the mean and 95% credible interval of the posterior distribution.
• Add fixed and random effects posterior: Displays the posterior density functions of the fixed and random effects models under the alternative hypothesis.
• Shade 95% CI: Displays the 95% CI as a shade in the posterior density function.

Sequential plot

• Bayes factors: Displays the development of the Bayes factor as a function of the number of studies using the user-defined prior; displays the Bayes factor computed with the user-defined prior; displays the mean and 95% credible interval of the posterior distribution; shows the decisiveness of the evidence in terms of Jeffreys' (1961) evidence categories.
• Effect size: Displays a probability wheel depicting the odds of the data under the null hypothesis/hypotheses vs. the alternative hypothesis/hypotheses.
• Heterogeneity: Displays a probability wheel depicting the odds of the data under random effects H1 (& H0 for model averaging) vs. fixed effects H1 (& H0 for model averaging).
• Posterior model probabilities: Displays the development of posterior model probabilities as a function of the number of studies using the user-defined prior model probabilities.

References

• Heck, D. W., Gronau, Q. F., & Wagenmakers, E.-J. (2019). metaBMA: Bayesian model averaging for random and fixed effects meta-analysis. Retrieved from https://CRAN.R-project.org/package=metaBMA
• Gronau, Q. F., Van Erp, S., Heck, D. W., Cesario, J., Jonas, K. J., & Wagenmakers, E. J. (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(1), 123-138.
• Rouder, J. N., Haaf, J. M., Davis-Stober, C. P., & Hilgard, J. (2019). Beyond overall effects: A Bayesian approach to finding constraints in meta-analysis. Psychological methods.

R-packages

• metaBMA

jasp-stats/jaspMetaAnalysis documentation built on April 5, 2025, 4:03 p.m.