Bayesian meta-analysis assuming that the effect size \(d\) varies across studies with standard deviation \(\tau\) (i.e., a random-effects model).

```
meta_random(
y,
SE,
labels,
data,
d = prior("cauchy", c(location = 0, scale = 0.707)),
tau = prior("invgamma", c(shape = 1, scale = 0.15)),
rscale_contin = 0.5,
rscale_discrete = 0.707,
centering = TRUE,
logml = "integrate",
summarize = "stan",
ci = 0.95,
rel.tol = .Machine$double.eps^0.3,
logml_iter = 5000,
silent_stan = TRUE,
...
)
```

- y
effect size per study. Can be provided as (1) a numeric vector, (2) the quoted or unquoted name of the variable in

`data`

, or (3) a`formula`

to include discrete or continuous moderator variables.- SE
standard error of effect size for each study. Can be a numeric vector or the quoted or unquoted name of the variable in

`data`

- labels
optional: character values with study labels. Can be a character vector or the quoted or unquoted name of the variable in

`data`

- data
data frame containing the variables for effect size

`y`

, standard error`SE`

,`labels`

, and moderators per study.- d
`prior`

distribution on the average effect size`d`

. The prior probability density function is defined via`prior`

.- tau
`prior`

distribution on the between-study heterogeneity`tau`

(i.e., the standard deviation of the study effect sizes`dstudy`

in a random-effects meta-analysis. A (nonnegative) prior probability density function is defined via`prior`

.- rscale_contin
scale parameter of the JZS prior for the continuous covariates.

- rscale_discrete
scale parameter of the JZS prior for discrete moderators.

- centering
whether continuous moderators are centered.

- logml
how to estimate the log-marginal likelihood: either by numerical integration (

`"integrate"`

) or by bridge sampling using MCMC/Stan samples (`"stan"`

). To obtain high precision with`logml="stan"`

, many MCMC samples are required (e.g.,`logml_iter=10000, warmup=1000`

).- summarize
how to estimate parameter summaries (mean, median, SD, etc.): Either by numerical integration (

`summarize = "integrate"`

) or based on MCMC/Stan samples (`summarize = "stan"`

).- ci
probability for the credibility/highest-density intervals.

- rel.tol
relative tolerance used for numerical integration using

`integrate`

. Use`rel.tol=.Machine$double.eps`

for maximal precision (however, this might be slow).- logml_iter
number of iterations (per chain) from the posterior distribution of

`d`

and`tau`

. The samples are used for computing the marginal likelihood of the random-effects model with bridge sampling (if`logml="stan"`

) and for obtaining parameter estimates (if`summarize="stan"`

). Note that the argument`iter=2000`

controls the number of iterations for estimation of the random-effect parameters per study in random-effects meta-analysis.- silent_stan
whether to suppress the Stan progress bar.

- ...
further arguments passed to

`rstan::sampling`

(see`stanmodel-method-sampling`

). Relevant MCMC settings concern the number of warmup samples that are discarded (`warmup=500`

), the total number of iterations per chain (`iter=2000`

), the number of MCMC chains (`chains=4`

), whether multiple cores should be used (`cores=4`

), and control arguments that make the sampling in Stan more robust, for instance:`control=list(adapt_delta=.97)`

.

```
# \donttest{
### Bayesian Random-Effects Meta-Analysis (H1: d>0)
data(towels)
set.seed(123)
mr <- meta_random(logOR, SE, study,
data = towels,
d = prior("norm", c(mean = 0, sd = .3), lower = 0),
tau = prior("invgamma", c(shape = 1, scale = 0.15))
)
mr
#> ### Bayesian Random-Effects Meta-Analysis ###
#> Prior on d: 'norm' (mean=0, sd=0.3) truncated to the interval [0,Inf].
#> Prior on tau: 'invgamma' (shape=1, scale=0.15) with support on the interval [0,Inf].
#>
#> # Bayes factors:
#> (denominator)
#> (numerator) random_H0 random_H1
#> random_H0 1.00 0.262
#> random_H1 3.81 1.000
#>
#> # Posterior summary statistics of random-effects model:
#> mean sd 2.5% 50% 97.5% hpd95_lower hpd95_upper n_eff Rhat
#> d 0.194 0.089 0.028 0.192 0.374 0.013 0.351 5967.4 1.001
#> tau 0.129 0.090 0.033 0.106 0.355 0.020 0.299 5138.6 1.001
plot_posterior(mr)
# }
```