Reproducing Bayesian Model-Averaged Meta-Analysis

Reproducing Bayesian Model-Averaged Meta-Analysis (BMA)

We illustrate how to fit a classical BMA (not adjusting for publication bias) using RoBMA. For this purpose, we reproduce a meta-analysis of registered reports on Power posing by Gronau et al. (2017). We focus only on the analysis of all reported results using a Cauchy prior distribution with scale \(1/\sqrt{2}\) for the effect size estimation (half-Cauchy for testing) and inverse-gamma distribution with shape = 1 and scale = 0.15 for the heterogeneity parameter. You can find the figure from the original publication here and the paper’s supplementary materials at https://osf.io/fxg32/.

First, we load the power posing data provided within the metaBMA package and reproduce the analysis performed by Gronau et al. (2017).

data("power_pose", package = "metaBMA")
power_pose[,c("study", "effectSize", "SE")]
#>                study effectSize        SE
#> 1      Bailey et al.  0.2507640 0.2071399
#> 2       Ronay et al.  0.2275180 0.1931046
#> 3 Klaschinski et al.  0.3186069 0.1423228
#> 4     Bombari et al.  0.2832082 0.1421356
#> 5        Latu et al.  0.1463949 0.1416107
#> 6      Keller et al.  0.1509773 0.1221166

fit_BMA_test <- metaBMA::meta_bma(y   = power_pose$effectSize, SE = power_pose$SE,
                                  d   = metaBMA::prior(family = "halfcauchy", param = 1/sqrt(2)),
                                  tau = metaBMA::prior(family = "invgamma", param = c(1, .15)))
 
fit_BMA_est  <- metaBMA::meta_bma(y   = power_pose$effectSize, SE = power_pose$SE,
                                  d   = metaBMA::prior(family = "cauchy", param = c(0, 1/sqrt(2))),
                                  tau = metaBMA::prior(family = "invgamma", param = c(1, .15)))

fit_BMA_test$inclusion
#> ### Inclusion Bayes factor ###
#>       Model Prior Posterior included
#> 1  fixed_H0  0.25   0.00868         
#> 2  fixed_H1  0.25   0.77745        x
#> 3 random_H0  0.25   0.02061         
#> 4 random_H1  0.25   0.19325        x
#> 
#>   Inclusion posterior probability: 0.971 
#>   Inclusion Bayes factor: 33.136

round(fit_BMA_est$estimates,2)
#>          mean   sd 2.5%  50% 97.5% hpd95_lower hpd95_upper  n_eff Rhat
#> averaged 0.22 0.06 0.09 0.22  0.34        0.09        0.34     NA   NA
#> fixed    0.22 0.06 0.10 0.22  0.34        0.10        0.34 3026.5    1
#> random   0.22 0.08 0.07 0.22  0.37        0.07        0.37 6600.4    1

From the output, we can see the inclusion Bayes factor for the effect size was \(BF_{10} = 33.14\) and the effect size estimate 0.22, 95% HDI [0.09, 0.34], which matches the reported results. Please note that the metaBMA package model-averages only across the \(H_{1}\) models, whereas the RoBMA package model-averages across all models (assuming the presence and absence of the effect).

Using RoBMA

Now we reproduce the analysis with RoBMA. We set the corresponding prior distributions for effect sizes (\(\mu\)) and heterogeneity (\(\tau\)), and remove the alternative prior distributions for the publication bias by setting priors_bias = NULL. To specify the half-Cauchy prior distribution with the RoBMA::prior() function we use a regular Cauchy distribution and truncate it at zero (note that both metaBMA and RoBMA export their own prior() functions that will clash when loading both packages simultaneously). The inverse-gamma prior distribution for the heterogeneity parameter is the default option (we specify it for completeness). We omit the specifications for the null prior distributions for the effect size, heterogeneity (both of which are set to a spike at 0 by default), and publication bias (which is set to no publication bias by default). Note that starting from version 3.1, the package includes the NoBMA() function, which allows users to skip publication bias adjustment directly.

Since metaBMA model-averages the effect size estimates only across the models assuming presence of the effect, we remove the models assuming absence of the effect from the estimation ensemble with priors_effect_null = NULL. Finally, we set transformation = "cohens_d" to estimate the models on Cohen’s d scale. RoBMA uses Fisher’s z scale by default and transforms the estimated coefficients back to the scale that is used for specifying the prior distributions. We speed up the computation by setting parallel = TRUE, and set a seed for reproducibility.

library(RoBMA)

fit_RoBMA_test <- RoBMA(d = power_pose$effectSize, se = power_pose$SE, study_names = power_pose$study,
                        priors_effect  = prior(
                          distribution = "cauchy",
                          parameters = list(location = 0, scale = 1/sqrt(2)),
                          truncation = list(0, Inf)),
                        priors_heterogeneity = prior(
                          distribution = "invgamma",
                          parameters = list(shape = 1, scale = 0.15)),
                        priors_bias = NULL,
                        transformation = "cohens_d", seed = 1, parallel = TRUE)

fit_RoBMA_est  <- RoBMA(d = power_pose$effectSize, se = power_pose$SE, study_names = power_pose$study,
                        priors_effect  = prior(
                          distribution = "cauchy",
                          parameters = list(location = 0, scale = 1/sqrt(2))),
                        priors_heterogeneity = prior(
                          distribution = "invgamma",
                          parameters = list(shape = 1, scale = 0.15)),
                        priors_bias = NULL,
                        priors_effect_null = NULL,
                        transformation = "cohens_d", seed = 2, parallel = TRUE)

summary(fit_RoBMA_test)
#> Call:
#> RoBMA(d = power_pose$effectSize, se = power_pose$SE, study_names = power_pose$study, 
#>     transformation = "cohens_d", priors_effect = prior(distribution = "cauchy", 
#>         parameters = list(location = 0, scale = 1/sqrt(2)), truncation = list(0, 
#>             Inf)), priors_heterogeneity = prior(distribution = "invgamma", 
#>         parameters = list(shape = 1, scale = 0.15)), priors_bias = NULL, 
#>     parallel = TRUE, seed = 1)
#> 
#> Robust Bayesian meta-analysis
#> Components summary:
#>               Models Prior prob. Post. prob. Inclusion BF
#> Effect           2/4       0.500       0.971       33.112
#> Heterogeneity    2/4       0.500       0.214        0.273
#> 
#> Model-averaged estimates:
#>      Mean Median 0.025 0.975
#> mu  0.213  0.217 0.000 0.348
#> tau 0.022  0.000 0.000 0.178
#> The estimates are summarized on the Cohen's d scale (priors were specified on the Cohen's d scale).

summary(fit_RoBMA_est)
#> Call:
#> RoBMA(d = power_pose$effectSize, se = power_pose$SE, study_names = power_pose$study, 
#>     transformation = "cohens_d", priors_effect = prior(distribution = "cauchy", 
#>         parameters = list(location = 0, scale = 1/sqrt(2))), 
#>     priors_heterogeneity = prior(distribution = "invgamma", parameters = list(shape = 1, 
#>         scale = 0.15)), priors_bias = NULL, priors_effect_null = NULL, 
#>     parallel = TRUE, seed = 2)
#> 
#> Robust Bayesian meta-analysis
#> Components summary:
#>               Models Prior prob. Post. prob. Inclusion BF
#> Effect           2/2       1.000       1.000          Inf
#> Heterogeneity    1/2       0.500       0.200        0.250
#> 
#> Model-averaged estimates:
#>      Mean Median 0.025 0.975
#> mu  0.220  0.220 0.096 0.346
#> tau 0.019  0.000 0.000 0.152
#> The estimates are summarized on the Cohen's d scale (priors were specified on the Cohen's d scale).

The output from the summary.RoBMA() function has 2 parts. The first one under the “Robust Bayesian Meta-Analysis” heading provides a basic summary of the fitted models by component types (presence of the Effect and Heterogeneity). The table summarizes the prior and posterior probabilities and the inclusion Bayes factors of the individual components. The results for the half-Cauchy model specified for testing show that the inclusion BF is nearly identical to the one computed by the metaBMA package, \(\text{BF}_{10} = 33.11\).

The second part under the ‘Model-averaged estimates’ heading displays the parameter estimates. The results for the unrestricted Cauchy model specified for estimation show the effect size estimate \(\mu = 0.22\), 95% CI [0.10, 0.35] that also mirrors the one obtained from metaBMA package.

Visualizing the Results

RoBMA provides extensive options for visualizing the results. Here, we visualize the prior (grey) and posterior (black) distribution for the mean parameter.

plot(fit_RoBMA_est, parameter = "mu", prior = TRUE, xlim = c(-1, 1))

If we visualize the effect size from the model specified for testing, we notice a few more things. The function plots the model-averaged estimates across all models by default, including models assuming the absence of the effect. The arrows represents the probability of a spike, here, at the value 0. The secondary y-axis (right) shows the probability of the value 0 decreased from 0.50, to 0.03 (also obtainable from the “Robust Bayesian Meta-Analysis” field in the summary.RoBMA() function). Furthermore, the continuous prior distributions for the effect size under the alternative hypothesis are truncated to only positive values, reflecting the assumption that the effect size cannot be negative.

plot(fit_RoBMA_test, parameter = "mu", prior = TRUE, xlim = c(-.5, 1))

We can also visualize the estimates from the individual models used in the ensemble. We do that with the plot_models() function, which visualizes the effect size estimates and 95% CI of each of the specified models from the estimation ensemble (Model 1 corresponds to the fixed effect model and Model 2 to the random effect model). The size of the square representing the mean estimate reflects the posterior model probability of the model, which is also displayed in the right-hand side panel. The bottom part of the figure shows the model-averaged estimate that is a combination of the individual model posterior distributions weighted by the posterior model probabilities.

plot_models(fit_RoBMA_est)

The last type of visualization that we show here is the forest plot. It displays both the effect sizes from the original studies and the overall meta-analytic estimate in a single figure. It can be requested by using the forest() function.

forest(fit_RoBMA_est)

For more options provided by the plotting function, see its documentation using ?plot.RoBMA(), ?plot_models(), and ?forest().