Default Prior Distributions
Basic example
The simplest way to use the defaults is to tell the
RoBMA R package what effect-size measure is being analyzed;
the package then specifies prior distributions on the matching scale. We
start with brma() because it fits one meta-analytic model
and makes the prior distributions easiest to inspect.
The BCG vaccine dataset from the metadat package
contains two-by-two tables. We compute log risk ratios and their
standard errors, then use brma() to show the default prior
distributions for a simple random-effects meta-analysis.
data("dat.bcg", package = "metadat")
bcg <- metafor::escalc(
ai = tpos, bi = tneg, ci = cpos, di = cneg,
measure = "RR", data = dat.bcg
)bcg_priors <- brma(
yi = yi, vi = vi, measure = "RR",
data = bcg, only_priors = TRUE
)
plot_prior(bcg_priors, parameter = "mu")
The first plot shows the prior distribution for the average effect,
\(\mu\). The second plot shows the
prior distribution for the between-study heterogeneity, \(\tau\). For measure = "RR" the
package places a Normal(0, 1) prior distribution on the log
risk ratio and a positive Normal(0, 0.5) prior distribution
on heterogeneity. The next subsection explains where these scales come
from.
Effect-size measures and their scales
Different effect-size measures live on different scales: while a
standardized mean difference of 1 is large, a Fisher’s \(z\) of 1 is enormous. As such, a single
fixed prior such as Normal(0, 1) would not be a sensible
default across different effect sizes.
The RoBMA R package deals with this problem by relying
on the unit-information standard deviation (UISD) concept (Röver et al.,
2021). Standardized effect sizes have known UISD values, and
for non-standardized inputs (measure = "GEN") the UISD can
be estimated from the per-study standard errors and sample sizes (or
specified manually):
| Effect-size measure | measure |
UISD used by default |
|---|---|---|
| Standardized mean difference (Cohen’s \(d\), Hedges’ \(g\)) | "SMD" |
sqrt(2) |
| Fisher’s \(z\)-transformed correlation | "ZCOR" |
1 |
| Log risk ratio | "RR" |
sqrt(4) |
| Log odds ratio | "OR" |
sqrt(4) |
| Log hazard ratio | "HR" |
sqrt(4) |
| Log incidence rate ratio | "IRR" |
sqrt(4) |
| Generic / non-standardized | "GEN" |
estimated from sei and ni, or
set manually |
The default prior distributions then use fractions of the UISD to specify the scale of each prior distribution.
| Parameter | Meaning | Default prior distribution |
|---|---|---|
mu |
average effect | Normal(0, 0.5 * UISD) |
tau |
heterogeneity | positive Normal(0, 0.25 * UISD) |
mods |
meta-regression coefficients | Normal(0, 0.25 * UISD) |
scale |
heterogeneity-regression coefficients | Normal(0, 0.5) |
rho |
heterogeneity allocation in multilevel models | Beta(1, 1) |
The scale regression coefficients are not on the
effect-size scale: they describe changes in log heterogeneity, a
multiplicative effect, and are therefore scale-free. The notation for
the meta-regression and multilevel parameters follows the
RoBMA R package meta-regression and multilevel papers (Bartoš et al.,
2025, 2026).
The default prior distributions can be made wider or narrower with
rescale_priors. Values larger than one make the prior
distributions wider; values smaller than one make them narrower.
bcg_wider_priors <- brma(
yi = yi, vi = vi, measure = "RR",
rescale_priors = 2,
data = bcg, only_priors = TRUE
)
print_prior(bcg_wider_priors)
#> mu:
#> Normal(0, 2)
#> tau:
#> Normal(0, 1)[0, Inf]The effect prior distribution changed from Normal(0, 1)
to Normal(0, 2), and the heterogeneity prior distribution
from positive Normal(0, 0.5) to positive
Normal(0, 1).
Standardized effect-size inputs
Several common effect-size measures live on familiar but inconvenient
scales. Correlations are bounded in \((-1,
1)\) with non-linear behaviour near the boundaries, and their
sampling variance depends on the effect size itself. Odds ratios and
risk ratios are positive and asymmetric around the no-effect value of 1.
For each of these, the standard practice is to transform the inputs to a
related scale on which the sampling distribution is approximately
normal: Fisher’s \(z\) for
correlations, log odds ratio for odds ratios, log risk ratio for risk
ratios. metafor::escalc() performs all these
transformations, and the RoBMA R package is set up to
operate on the transformed scale. Use measure = "ZCOR" for
Fisher’s \(z\),
measure = "OR" for log odds ratios,
measure = "RR" for log risk ratios, and so on. The default
prior distributions then use the UISD that matches the transformed
scale.
The Kroupova2021 dataset of correlations between student
employment and educational outcomes makes the workflow concrete. We use
escalc() to convert correlations and sample sizes into
Fisher’s \(z\) effect sizes and their
sampling variances:
data("Kroupova2021", package = "RoBMA")
Kroupova2021 <- metafor::escalc(
ri = r, ni = sample_size, measure = "ZCOR",
data = Kroupova2021
)brma() recognises measure = "ZCOR" and uses
the default Fisher’s-\(z\) prior
distributions:
Kroupova2021_priors <- brma(
yi = yi, vi = vi, measure = "ZCOR",
data = Kroupova2021, only_priors = TRUE
)
print_prior(Kroupova2021_priors)
#> mu:
#> Normal(0, 0.5)
#> tau:
#> Normal(0, 0.25)[0, Inf]The same pattern applies to other standardized measures: compute the
appropriate transformation in escalc() and pass the
matching measure to brma().
The model and its prior distributions then operate on the transformed
scale, but posterior summaries can be reported back on the familiar
original scale via the output_measure and
transform arguments of the relevant summary functions
(e.g., pooled_effect(), marginal_means(), and
plot() functions). For correlations fit with
measure = "ZCOR", set output_measure = "COR"
to obtain a pooled correlation. For log-scale measures
("OR", "RR", "HR",
"IRR"), set transform = "EXP" to obtain the
corresponding ratio.
Non-standardized inputs
Some studies report effects on a scale specific to the analysis: a
regression coefficient on raw units, an unstandardized mean difference,
or a measure outside the standardized catalogue. Use
measure = "GEN" for these. Because the package cannot know
what counts as a “moderate” effect on an arbitrary scale, the default
prior scale must be supplied: either directly via
prior_unit_information_sd, or implicitly by passing the
per-study sample sizes in ni, from which the package
estimates the UISD via estimate_unit_information_sd().
The Havrankova2025 dataset reports regression
coefficients estimating the beauty premium across studies on a
non-standardized scale.
data("Havrankova2025", package = "RoBMA")
head(Havrankova2025)
#> y se facing_customer study_id N
#> 1 8.910 2.960133 No 1 1806
#> 2 8.964 5.272941 No 1 1806
#> 3 2.484 3.029268 No 1 1806
#> 4 9.180 4.831579 No 1 1806
#> 5 21.600 6.467066 No 1 1806
#> 6 15.012 10.075168 No 1 1806The simplest specification passes the sample sizes in ni
so that the package estimates the UISD:
Havrankova_priors <- brma(
yi = y, sei = se, ni = N, measure = "GEN",
data = Havrankova2025, only_priors = TRUE
)
print_prior(Havrankova_priors)
#> mu:
#> Normal(0, 25.81)
#> tau:
#> Normal(0, 12.9)[0, Inf]The UISD can also be set directly when sample sizes are unavailable, when an alternative scale is more appropriate, or when performing subgroup analyses (it is sensible to compute UISD based on the full dataset so the prior distributions match across models):
Havrankova_UISD <- estimate_unit_information_sd(
sei = Havrankova2025$se,
ni = Havrankova2025$N
)
Havrankova_UISD
#> [1] 51.61241Havrankova_manual_priors <- brma(
yi = y, sei = se, measure = "GEN",
prior_unit_information_sd = Havrankova_UISD,
data = Havrankova2025, only_priors = TRUE
)
print_prior(Havrankova_manual_priors)
#> mu:
#> Normal(0, 25.81)
#> tau:
#> Normal(0, 12.9)[0, Inf]The manually specified value takes precedence over the automatic estimate.
Meta-regression
Meta-regression introduces moderators to explain part of the between-study variation. The default prior distributions for moderator coefficients reuse the same UISD machinery as the prior distributions on the average effect and heterogeneity, but two specification choices change how those prior distributions are interpreted: the contrast scheme used for categorical moderators, and the standardization applied to continuous moderators.
We illustrate both with Havrankova2025, fitting two
moderators at once: the categorical facing_customer (No /
Some / Direct) and the continuous N (per-study sample
size). Each moderator’s prior can be inspected separately by passing its
name to parameter_mods in print_prior() or
plot_prior().
Categorical moderators. The
set_contrast_factor_predictors argument controls how
categorical predictors are coded, applying a single contrast scheme
jointly to every categorical moderator in the model (per-moderator
overrides are described in Per-moderator overrides below). The
choice affects both how the intercept is interpreted and what the
inclusion Bayes factor for the moderator tests.
With treatment contrasts ("treatment", the
default in brma() and matching base R), one level is chosen
as a baseline. The intercept is the average effect for the baseline
level, and each remaining coefficient is the difference between its
level and the baseline. In parameter estimation, the regression
coefficients are regularized towards the baseline. In hypothesis
testing, removing the moderator coefficient tests whether each level’s
effect differs from the baseline effect. The choice of reference level
matters for both the prior distribution on the intercept and the
resulting inclusion Bayes factor. Treatment contrasts are most natural
when one level has a clear baseline interpretation (a control condition,
an absence of treatment, a default policy).
With mean-difference contrasts ("meandif", the
default in BMA() and RoBMA()), the levels are
treated as exchangeable. The intercept is the adjusted average effect
(averaged with equal weight across levels), and each coefficient
describes how a given level deviates from the grand mean. In parameter
estimation, the regression coefficients are regularized towards the
average effect. In hypothesis testing, removing the moderator tests
whether any level differs from the average effect, irrespective of which
level was chosen as a label. This is the recommended default for
model-averaged inference because it gives a well-defined “is there
moderation by this factor?” interpretation that does not depend on
relabelling the levels.
The two contrast schemes can be compared on the categorical moderator while the continuous moderator is included in the same model:
Havrankova_treatment <- brma(
yi = y, sei = se, ni = N, measure = "GEN",
mods = ~ facing_customer + N,
set_contrast_factor_predictors = "treatment",
data = Havrankova2025, only_priors = TRUE
)
print_prior(Havrankova_treatment, parameter_mods = "facing_customer")
#> mu_facing_customer:
#> treatment contrast: Normal(0, 12.9)Havrankova_meandif <- brma(
yi = y, sei = se, ni = N, measure = "GEN",
mods = ~ facing_customer + N,
set_contrast_factor_predictors = "meandif",
data = Havrankova2025, only_priors = TRUE
)
print_prior(Havrankova_meandif, parameter_mods = "facing_customer")
#> mu_facing_customer:
#> mean difference contrast: mNormal(0, 12.9)The treatment-contrast version places a Normal(0, 12.9)
prior distribution on each non-reference level’s deviation from the
reference level. The mean-difference version places a multivariate
mNormal(0, 12.9) prior distribution on the deviations of
all levels from the grand mean. The continuous moderator N
is unaffected by the contrast choice and receives the same default prior
distribution in both fits.
Continuous moderators. Continuous predictors are
standardized internally so that the default prior scale applies
regardless of the predictor’s units. The default prior distribution on a
meta-regression coefficient therefore describes a plausible change per
one standard deviation of the predictor, not per one original
unit. By default, plot_prior() displays the prior
distribution on this standardized scale; pass
standardized_coefficients = FALSE to obtain the prior
distribution per one original unit of the moderator.
We inspect the prior distribution on the continuous moderator
N from the same Havrankova_meandif fit:
The first plot shows the prior distribution on the standardized scale
used internally (a plausible change per one standard deviation of
N). The second shows the same prior distribution on the
original scale of N (a plausible change per one unit of
N); because N ranges over orders of magnitude,
the per-unit prior distribution is correspondingly tight.
Per-moderator overrides
The set_contrast_factor_predictors argument is a joint
setting and rescale_priors scales every default prior
distribution by the same factor. Finer control is available through
prior_mods, which accepts a named list keyed by moderator
name. Each entry can be a regular prior() (for continuous
moderators) or a prior_factor() with its own
contrast setting (for categorical moderators), so different
moderators can use different contrast schemes and prior scales within
the same model.
For example, override only the prior distribution on
facing_customer, switching its contrast and tightening its
scale, while leaving N on the default
per-standard-deviation prior distribution:
Havrankova_per_mod <- brma(
yi = y, sei = se, ni = N, measure = "GEN",
mods = ~ facing_customer + N,
prior_mods = list(
facing_customer = prior_factor(
distribution = "mnormal",
parameters = list(mean = 0, sd = 5),
contrast = "meandif"
)
),
data = Havrankova2025, only_priors = TRUE
)
print_prior(Havrankova_per_mod)
#> mu_intercept:
#> Normal(0, 25.81)
#> mu_facing_customer:
#> mean difference contrast: mNormal(0, 5)
#> mu_N:
#> Normal(0, 12.9)
#> tau:
#> Normal(0, 12.9)[0, Inf]The facing_customer prior distribution is now the
user-specified mNormal(0, 5) mean-difference contrast,
while N retains the default per-standard-deviation prior
distribution. The named list extends naturally: with multiple
moderators, each entry can specify its own prior family, scale, and (for
categorical moderators) contrast value.