Author: František Bartoš

# Informed Bayesian Model-Averaged Meta-Analysis with Binary Outcomes

#### František Bartoš

#### 13th of June 2023 (updated: 4th of May 2026)

This vignette accompanies the manuscript Empirical Prior Distributions for Bayesian Meta-Analyses of Binary and Time-to-Event Outcomes preprinted at arXiv (Bartoš et al., 2023).

Bayesian model-averaged meta-analysis can be specified using the binomial likelihood and applied to data with dichotomous outcomes. This vignette illustrates how to do this with an example from Bartoš et al. (2023), who implemented a binomial-normal Bayesian model-averaged meta-analytic model and developed informed prior distributions for meta-analyses of binary and time-to-event outcomes based on the Cochrane database of systematic reviews. See Bartoš et al. (2021) for informed prior distributions for meta-analyses of continuous outcomes, highlighted in the Informed Bayesian Model-Averaged Meta-Analysis in Medicine vignette.

## Binomial-Normal Bayesian Model-Averaged Meta-Analysis

We illustrate how to fit the binomial-normal Bayesian model-averaged meta-analysis using the `RoBMA` R package. For this purpose, we reproduce the example of adverse effects of honey in treating acute cough in children from Bartoš et al. (2023), who reanalyzed two studies with adverse events of nervousness, insomnia, or hyperactivity in the honey vs. no treatment condition that were subjected to a meta-analysis by Oduwole et al. (2018).

We load the `RoBMA` R package and specify the number of adverse events and sample sizes in each arm as described on p. 73 of Oduwole et al. (2018).

    library(RoBMA)

    events_experimental        <- c(5, 2)
    events_control             <- c(0, 0)
    observations_experimental  <- c(35, 40)
    observations_control       <- c(39, 40)
    study_names                <- c("Paul 2007", "Shadkam 2010")

Notice that both studies reported no adverse events in the control group. Using a normal-normal meta-analytic model with log odds ratios would require a continuity correction, which might result in bias. Binomial-normal models allow us to circumvent the issue by modeling the observed proportions directly (see Bartoš et al. (2023) for more details).

We fit the binomial-normal Bayesian model-averaged meta-analysis using informed prior distributions based on the `Acute Respiratory Infections` subfield. We use the `BMA.glmm()` function and pass in the per-arm event counts (`ai`, `ci`) and sample sizes (`n1i`, `n2i`); `measure = "OR"` selects the binomial-normal likelihood. The `prior_informed_field` and `prior_informed_subfield` arguments request the informed prior distributions for the individual medical subfields automatically.

    fit_BMA <- BMA.glmm(
      ai = events_experimental, n1i = observations_experimental,
      ci = events_control, n2i = observations_control,
      slab = study_names, measure = "OR",
      prior_informed_field    = "medicine",
      prior_informed_subfield = "Acute Respiratory Infections",
      seed = 1, parallel = TRUE
    )

with `prior_informed_field = "medicine"` and `prior_informed_subfield = "Acute Respiratory Infections"` corresponding to the \(\mu \sim T(0,0.48,3)\) and \(\tau \sim InvGamma(1.67, 0.45)\) informed prior distributions on the log-odds-ratio scale (see `?prior_informed` for more details regarding the informed prior distributions).

We obtain the output with the `summary()` function. Adding the `conditional = TRUE` argument allows us to inspect the conditional estimates, i.e., the effect size estimate assuming that the models specifying the presence of the effect are true, and the heterogeneity estimates assuming that the models specifying the presence of heterogeneity are true. The summary reports parameter estimates on the log-odds-ratio scale; we use `pooled_effect()` with `transform = "EXP"` to obtain the estimates on the odds-ratio scale.

    summary(fit_BMA, conditional = TRUE, include_mcmc_diagnostics = FALSE)
    #>
    #> Bayesian Model-Averaged Random-Effects Model (k = 2)
    #>
    #> Component Inclusion
    #>               Prior prob. Post. prob. Inclusion BF
    #> Effect              0.500       0.726        2.643
    #> Heterogeneity       0.500       0.559        1.269
    #>
    #> Estimates
    #>      Mean    SD  0.025   0.5 0.975
    #> mu  0.704 0.838 -0.180 0.506 2.732
    #> tau 0.399 0.734  0.000 0.149 2.487
    #>
    #> Conditional Estimates
    #>      Mean    SD  0.025   0.5 0.975
    #> mu  0.971 0.843 -0.255 0.835 2.933
    #> tau 0.713 0.859  0.096 0.412 3.089

The output from the `summary()` function has three parts. The first one 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 show that the inclusion Bayes factor for the effect is similar to the one reported in Bartoš et al. (2023), \(\text{BF}_{10} = 2.63\) and \(\text{BF}_{\text{rf}} = 1.30\) (minor differences are a consequence of the product-space estimation algorithm in the new version of the package), giving weak/undecided evidence for the presence of the effect and heterogeneity.

The second part under the ‘Model-averaged estimates’ heading displays the parameter estimates model-averaged across all specified models (i.e., including models specifying the effect size to be zero). These estimates are shrunk towards the null hypotheses of no effect or no heterogeneity in accordance with the posterior uncertainty about the presence of the effect or heterogeneity. We find the model-averaged mean effect OR = 3.39, 95% CI [0.84, 15.14], and a heterogeneity estimate \(\tau_\text{logOR} = 0.42\), 95% CI [0.00, 2.59].

The third part under the ‘Conditional estimates’ heading displays the conditional effect size and heterogeneity estimates (i.e., estimates assuming presence of the effect or heterogeneity) corresponding to the ones reported in Bartoš et al. (2023), OR = 4.24, 95% CI [0.78, 17.61], and a heterogeneity estimate \(\tau_\text{logOR} = 0.75\), 95% CI [0.10, 3.23].

The dedicated helper functions can be used to extract the pooled effect and heterogeneity estimates directly on the odds-ratio scale (assuming the presence of the effect and heterogeneity respectively):

    pooled_effect(fit_BMA, conditional = TRUE, transform = "EXP")
    #>
    #> Conditional Pooled Effect Size (odds ratio)
    #>     Mean Median 0.025  0.975
    #> mu 6.241  2.304 0.775 18.780
    #> Effect estimates are summarized on the odds ratio scale using EXP on the log odds ratio measure.
    pooled_heterogeneity(fit_BMA, conditional = TRUE)
    #>
    #> Conditional Pooled Heterogeneity
    #>      Mean Median 0.025 0.975
    #> tau 0.713  0.412 0.096 3.089

We can also visualize the posterior distributions of the effect size and heterogeneity parameters using the `plot()` function. Here, we set the `conditional = TRUE` argument to display the conditional effect size estimate and `prior = TRUE` to include the prior distribution in the plot.

    par(mar = c(4, 4, 0, 4))
    plot(fit_BMA, parameter = "mu", prior = TRUE, conditional = TRUE, lwd = 2)

Additional visualizations and summaries are demonstrated in the Bayesian Meta-Analysis and Informed Bayesian Model-Averaged Meta-Analysis in Medicine vignettes.

## References

Bartoš, F., Gronau, Q. F., Timmers, B., Otte, W. M., Ly, A., & Wagenmakers, E.-J. (2021). Bayesian model-averaged meta-analysis in medicine. Statistics in Medicine, 40(30), 6743–6761. https://doi.org/10.1002/sim.9170
Bartoš, F., Otte, W. M., Gronau, Q. F., Timmers, B., Ly, A., & Wagenmakers, E.-J. (2023). Empirical prior distributions for Bayesian meta-analyses of binary and time-to-event outcomes. https://doi.org/10.48550/arXiv.2306.11468
Oduwole, O., Udoh, E. E., Oyo-Ita, A., & Meremikwu, M. M. (2018). Honey for acute cough in children. Cochrane Database of Systematic Reviews, 4. https://doi.org/10.1002/14651858.CD007094.pub5