Prior specification for model-averaging

The RoBMA function extends the prior specification described in prior_specification by allowing mixture priors that define competing hypotheses for Bayesian model-averaging. This enables multimodel inference via the product space method (Carlin and Chib 1995; Lodewyckx et al. 2011) , where a single MCMC chain explores a joint model space containing all competing models.

Arguments

prior_effect

prior distribution(s) for the alternative effect component(s).

prior_heterogeneity

prior distribution(s) for the alternative heterogeneity component(s).

prior_mods

prior distribution(s) for alternative moderator components. A single prior applies to all terms; a named list can specify term-specific components.

prior_scale

prior distribution(s) for alternative scale-regression components. A single prior applies to all terms; a named list can specify term-specific components.

prior_heterogeneity_allocation

prior distribution(s) for the alternative cluster-level heterogeneity allocation component(s).

prior_bias

prior distribution(s) for alternative publication-bias component(s), such as weight functions, PET, or PEESE.

Alternative prior arguments can be:

A single prior distribution object (creates a mixture with one alternative)
A list of prior distributions (creates a mixture with multiple alternatives)
NULL or FALSE (omits the alternative hypothesis component)

See publication_bias_prior_specification for details on specifying publication-bias priors and prior_specification for details on specifying meta-analytic parameter priors.

prior_effect_null

prior distribution(s) for the null effect component(s).

prior_heterogeneity_null

prior distribution(s) for the null heterogeneity component(s).

prior_mods_null

prior distribution(s) for null moderator components. A single prior applies to all terms; a named list can specify term-specific components.

prior_scale_null

prior distribution(s) for null scale-regression components. A single prior applies to all terms; a named list can specify term-specific components.

prior_heterogeneity_allocation_null

prior distribution(s) for the null cluster-level heterogeneity allocation component(s).

prior_bias_null

prior distribution(s) for null publication-bias component(s), usually prior_none(). See publication_bias_prior_specification.

Null prior arguments can be:

A single prior distribution object (creates a mixture with one null)
A list of prior distributions (creates a mixture with multiple nulls)
NULL or FALSE (omits the null hypothesis component)

Defaults to a point mass (spike) at zero for effect and heterogeneity parameters.

model_type

character string specifying predefined publication-bias model ensembles. One of:

"PSMA" (default): Full RoBMA-PSMA ensemble with 6 weight functions + PET + PEESE
"6w": Six weight function models
"2w": Two weight function models
"PP": PET-PEESE models only

Custom prior_bias replaces the preset alternative bias components. If prior_bias is omitted, model_type determines the default alternatives even when prior_bias_null is customized or omitted.

Details

The product space method for model-averaging

RoBMA implements Bayesian model-averaging using the product space method (Carlin and Chib 1995; Lodewyckx et al. 2011) . Rather than fitting each model separately and combining results after the fitting is complete, this approach embeds all competing models within a single joint model space. A discrete model indicator variable selects which model component is "active" at each MCMC iteration, and the posterior probability of each model is estimated from the proportion of iterations spent in that model's subspace.

This is achieved by specifying mixture priors that combine:

Null hypothesis component(s): Typically point masses (spikes) representing the absence of an effect, heterogeneity, or publication bias
Alternative hypothesis component(s): Continuous distributions representing the presence of the parameter

The mixture structure enables direct computation of:

Inclusion Bayes factors: Evidence for the presence vs. absence of each parameter (e.g., effect, heterogeneity, bias)
Model-averaged estimates: Posterior estimates that account for model uncertainty by weighting across all models
Conditional estimates: Posterior estimates given that a particular hypothesis is true

Default mixture prior structure

By default, RoBMA creates the following mixture priors:

Parameter	Null component	Alternative component
Effect (\(\mu\))	Spike(0)	Normal(0, UISD-scaled)
Heterogeneity (\(\tau\))	Spike(0)	Normal+(0, UISD-scaled)
Publication bias	No bias (prior_none)	Weight functions + PET + PEESE
Allocation (\(\rho\))	(none by default)	Beta(1, 1)

See prior_specification for details on how the UISD scaling is determined.

Specifying custom mixture priors

Single prior vs. list of priors

Each prior_* and prior_*_null argument accepts either a single prior or a list:

Single prior: Creates one component in the mixture
List of priors: Creates multiple components, enabling model-averaging across different prior specifications (e.g., different effect size scales)

When multiple priors are specified in a list, they are all treated as alternative (or null) hypothesis components for the main inclusion Bayes factor. The inclusion Bayes factor tests the combined evidence for all alternative components against all null components. The detailed summary output shows posterior probabilities and inclusion Bayes factors for each individual component separately, allowing finer-grained inference about which specific prior specification is most supported by the data.

Omitting hypothesis components

For top-level mixture components, setting a prior argument to NULL or FALSE removes that hypothesis component:

prior_effect_null = NULL: No null hypothesis for effect (assumes effect exists)
prior_effect = NULL: No alternative hypothesis (assumes no effect)
prior_bias_null = NULL or FALSE: No "no bias" model (assumes some bias mechanism)
prior_bias = NULL or FALSE: No bias model (assumes no bias mechanism)

For moderator and scale regression terms, an omitted or whole-argument NULL prior_mods / prior_scale means "use defaults". To omit a component for a specific term, use a named list entry such as prior_mods_null = list(x1 = NULL).

Parameter-specific customization for moderators

For moderator priors (prior_mods, prior_mods_null), you can specify different priors for different predictors using named lists. A single prior object applies to all moderator terms:


RoBMA(...,
  mods = ~ x1 + x2,
  prior_mods_null = list(x1 = NULL)  # No null for x1, default null for x2
)

This allows testing whether specific moderators have effects while assuming others are always included or excluded.

Mixture priors for different parameter types

Effect size (\(\mu\)) and heterogeneity (\(\tau\))

Effect and heterogeneity priors combine spike-and-slab components:


# Default: spike(0) null + normal alternative
# Custom: multiple alternatives with different scales
RoBMA(...,
  prior_effect = list(
    prior("normal", list(mean = 0, sd = 0.5)),
    prior("normal", list(mean = 0, sd = 1.0))
  )
)

Publication bias

Bias priors combine different publication bias adjustment mechanisms:

No publication bias (null hypothesis)
Weight functions: Selection models with different step functions
PET/PEESE: Regression-based bias adjustment

The model_type argument provides convenient presets:


RoBMA(..., model_type = "PSMA")  # Full ensemble (default)
RoBMA(..., model_type = "PP")    # Only PET-PEESE models

See publication_bias_prior_specification for the bias-prior constructors and preset model spaces.

Heterogeneity allocation (\(\rho\)) for multilevel models

When cluster is specified, \(\rho\) controls the cluster-level variance allocation. The decomposition is \(\tau_{between} = \tau\sqrt{\rho}\) and \(\tau_{within} = \tau\sqrt{1 - \rho}\). By default, only an alternative (Beta(1,1)) is specified, but null hypotheses can be added:


RoBMA(...,
  cluster = study,
  prior_heterogeneity_allocation_null = prior("spike", list(location = 0.5))
)

Moderator and scale regression terms

When mods is specified, the effect prior becomes the intercept prior, and each moderator receives a mixture prior. The intercept inherits the effect mixture structure, while regression coefficients get separate spike-and-slab mixtures.

Similarly, when scale = ~ ... is specified for location-scale models, the heterogeneity prior becomes the scale intercept prior, and each scale predictor receives a mixture prior following the same pattern as moderator terms. Note that the scale intercept (i.e., baseline heterogeneity) is always positive and the multiplicative scale coefficients require non-zero prior distribution on the intercept.

Model weights and prior odds

Each component in a mixture prior has an associated prior weight (prior model probability). User-specified components receive equal weights unless their prior objects set prior_weights. Publication-bias presets use fixed weights: "2w" and "6w" split the alternative bias mass equally across their weight functions, "PP" splits it equally across PET and PEESE, and "PSMA" assigns half of the alternative bias mass to the six weight functions combined and one quarter each to PET and PEESE. Custom weights can be specified via the prior_weights argument in individual prior objects.

The prior odds between null and alternative hypotheses affect the Bayes factor interpretation. With equal prior weights on null and alternative, the posterior odds equal the Bayes factor.

References

Carlin BP, Chib S (1995). “Bayesian model choice via Markov chain Monte Carlo methods.” Journal of the Royal Statistical Society: Series B (Methodological), 57(3), 473–484. doi:10.1111/j.2517-6161.1995.tb02042.x .

Lodewyckx T, Kim W, Lee MD, Tuerlinckx F, Kuppens P, Wagenmakers E (2011). “A tutorial on Bayes factor estimation with the product space method.” Journal of Mathematical Psychology, 55(5), 331–347. doi:10.1016/j.jmp.2011.06.001 .

Examples

if (FALSE) { # \dontrun{
if (requireNamespace("metadat", quietly = TRUE)) {
  data(dat.lehmann2018, package = "metadat")

  # Default mixture priors (spike-and-slab for effect and heterogeneity)
  fit1 <- RoBMA(
    yi      = yi,
    vi      = vi,
    data    = dat.lehmann2018,
    measure = "SMD",
    seed    = 1,
    silent  = TRUE
  )

  # Custom alternative priors (two effect size scales)
  fit2 <- RoBMA(
    yi           = yi,
    vi           = vi,
    data         = dat.lehmann2018,
    measure      = "SMD",
    prior_effect = list(
      prior("normal", list(mean = 0, sd = 0.35)),
      prior("normal", list(mean = 0, sd = 0.70))
    ),
    seed         = 1,
    silent       = TRUE
  )

  # No null hypothesis for effect (assume effect exists)
  fit3 <- RoBMA(
    yi                = yi,
    vi                = vi,
    data              = dat.lehmann2018,
    measure           = "SMD",
    prior_effect_null = NULL,
    seed              = 1,
    silent            = TRUE
  )

  # Only selection model bias adjustment (no PET-PEESE)
  fit4 <- RoBMA(
    yi         = yi,
    vi         = vi,
    data       = dat.lehmann2018,
    measure    = "SMD",
    model_type = "6w",
    seed       = 1,
    silent     = TRUE
  )

  # Meta-regression with different null specifications per moderator
  fit5 <- RoBMA(
    yi              = yi,
    vi              = vi,
    mods            = ~ Preregistered,
    data            = dat.lehmann2018,
    measure         = "SMD",
    prior_mods_null = list(Preregistered = NULL),
    seed            = 1,
    silent          = TRUE
  )
}
} # }