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.

To fit a meta-analysis model, prior distributions on the average effect $d$ and
the heterogeneity $\tau$ are required. The package `metaBMA`

leaves the user the
freedom to choose from several predefined distributions or even define an owen
prior density function. The function `prior`

facilitates the construction and
visual inspection of prior distributions to check whether they meet the prior
knowledge about the field of interest.

# load package library("metaBMA") # load data set data(towels) # Half-normal (truncated to > 0) p1 <- prior("norm", c(mean=0, sd=.3), lower = 0) p1 p1(1:3) plot(p1) # custom prior p1 <- prior("custom", function(x) x^3-2*x+3, lower = 0, upper = 1) plot(p1, -.5, 1.5)

The functions `meta_fixed()`

and `meta_random()`

fit Bayesian meta-analysis
models. The model-specific posteriors for $d$ can then be averaged by `bma()`

and inclusion Bayes factors be computed by `inclusion()`

.

The fixed-effects meta-analysis assumes that the effect size is identical across studies. This model requires only one prior distribution for the overall effect $d$:

# Fixed-effects progres <- capture.output( # suppress Stan progress for vignette mf <- meta_fixed(logOR, SE, study, towels, d = prior("norm", c(mean=0, sd=.3), lower=0)) ) mf # plot posterior distribution plot_posterior(mf)

In contrast, the random-effects meta-analysis assumes that the effect size varies across studies. Specifically, it is assumed that study effect sizes follow a normal distribution with mean $d$ and standard deviation $\tau$. This model requires two prior distributions for both parameters:

# Random-effects progres <- capture.output( # suppress Stan progress for vignette mr <- meta_random(logOR, SE, study, towels, d = prior("norm", c(mean=0, sd=.3), lower=0), tau = prior("t", c(location=0, scale=.3, nu=1), lower=0), iter = 1500, logml_iter = 2000, rel.tol = .1) ) mr # plot posterior distribution plot_posterior(mr, main = "Average effect size d") plot_posterior(mr, "tau", main = "Heterogeneity tau")

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).

mb <- meta_bma(logOR, SE, study, towels, d = prior("norm", c(mean=0, sd=.3), lower=0), tau = prior("t", c(location=0, scale=.3, nu=1), lower=0), iter = 1500, logml_iter = 2000, rel.tol = .1) mb plot_posterior(mb, "d", -.1, 1.4) plot_forest(mb)

Often, it is of interest to judge how much additional evidence future studies can contribute to the present knowledge. Conditional on the outcome of the model averaging for meta-analysis, the function `predicted_bf()`

samples new data sets from the posterior and performs model selection for each replication. Thereby, a distribution of predicted Bayes factors is obtain that represents the expected evidence one expects when running a new study. The following example is not executed since it requires time-intensive computations:

mp <- predicted_bf(mb, SE = .2, sample = 30) plot(mp)

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