Prior specification

Prior distributions can be supplied for all model parameters. When omitted, the fitting functions construct defaults from the effect-size measure, sample sizes, or informed-prior settings when available.

This typically includes the pooled effect \(\mu\) and between-study heterogeneity \(\tau\). In the case of meta-regression, the pooled effect \(\mu\) corresponds to the intercept, and additional prior distributions for the regression coefficients are required. In the case of a location-scale model, the between-study heterogeneity corresponds to the intercept of the scale regression, and additional prior distributions for the scale regression coefficients are required.

Arguments

prior_effect: prior distribution for the effect size (\(\mu\)) parameter (the intercept). If omitted, a default prior is constructed. In single-model functions, explicit NULL or FALSE sets a spike at zero.
prior_heterogeneity: prior distribution for the heterogeneity (\(\tau\)) parameter. If omitted, a default prior is constructed. In single-model functions, explicit NULL or FALSE sets a spike at zero.
prior_mods: prior distribution for the moderators (\(\beta\)) parameters. A single prior applies to all terms; a named list can specify term-specific priors. If omitted or NULL, default priors are used.
prior_scale: prior distribution for the scale (\(\delta\)) parameters. A single prior applies to all terms; a named list can specify term-specific priors. If omitted or NULL, default priors are used.
prior_heterogeneity_allocation: prior distribution for the fraction of heterogeneity allocated to the cluster-level component in multilevel models (\(\rho\)). If omitted or NULL, defaults to Beta(1, 1).
prior_baserate: prior distribution for the estimate-specific midpoint base-rate probability in binomial GLMM models. If omitted or NULL, defaults to independent Beta(1, 1) priors.
prior_lograte: prior distribution for the estimate-specific midpoint log-rate in Poisson GLMM models. If omitted or NULL, a data-based unit-information normal prior is used independently for each estimate.
prior_unit_information_sd: numeric. The unit information standard deviation (\(\sigma_{unit}\)). Cannot be used together with prior_informed_field.
rescale_priors: numeric. A scaling factor for supported prior distributions. Point and none priors are unchanged. For constructors with publication-bias prior distributions, rescale_priors does not rescale them except for the default PEESE prior's UISD adjustment. Defaults to 1.
standardize_continuous_predictors: logical. Whether to standardize continuous predictors. Defaults to TRUE.
set_contrast_factor_predictors: character. How to set contrast for factor predictors. Defaults are constructor-specific and shown in each function usage; single-model constructors use "treatment", while model-averaging constructors use "meandif".
prior_informed_field: character. The field of the informed prior distributions. Omit to use the standard default prior specification; explicit NULL is invalid.
prior_informed_subfield: character. The subfield of the informed prior distributions. Omit to use the field-specific default, such as "Cochrane" for prior_informed_field = "medicine"; explicit NULL is invalid.

Details

There are several ways to specify the prior distributions:

via a standardized effect size measure with known unit information standard deviation,
by estimating unit information standard deviation using sample sizes ni,
by manually setting prior_unit_information_sd,
by specifying informed empirical prior distributions via prior_informed_field and prior_informed_subfield,
or via fully custom specification using the prior_effect, prior_heterogeneity, prior_mods, prior_scale, and prior_heterogeneity_allocation arguments.

In all cases, the prior behavior can be further modified by the rescale_priors, standardize_continuous_predictors, and set_contrast_factor_predictors arguments.

(1) Specifying prior distributions via standardized effect size measures with known

unit information standard deviation This is the easiest way to specify prior distributions. The width of prior distributions is based on a fraction of the known unit information standard deviation (UISD) (Röver et al. 2021) . The default prior distributions for the parameters are set as follows:

effect size:	Normal(0, \(\frac{1}{2}\) `UISD`)
heterogeneity:	Normal+(0, \(\frac{1}{4}\) `UISD`)
effect moderation:	Normal(0, \(\frac{1}{4}\) `UISD`)
heterogeneity moderation:	Normal(0, \(\frac{1}{2}\))

The heterogeneity moderation parameters are multiplicative, as such they are independent of UISD.

The default fraction of the UISD can be changed via RoBMA.options() using one of the following arguments: "default_UISD.effect", "default_UISD.heterogeneity", "default_UISD.mods", "default_UISD.scale".

The known UISD for standardized effect size measures (measure) are set as follows:

`"SMD"`:	\(\sqrt{2}\)
`"ZCOR"`:	\(1\)
`"RR"`:	\(\sqrt{4}\)
`"OR"`:	\(\sqrt{4}\)
`"HR"`:	\(\sqrt{4}\)
`"IRR"`:	\(\sqrt{4}\)

See Chapter 2.4 in Spiegelhalter et al. (2004) and Chapter 1 in Grieve (2022) .

(2) Estimating unit information standard deviation using sample sizes

When effect sizes are on a non-standardized scale (measure = "GEN") or use a standardized effect size without known UISD, the UISD can be estimated from sample sizes (ni) and standard errors (sei) following Equation 6 in Röver et al. (2021) . The estimated UISD is then used to scale the default prior distributions as described in section (1).

Note that the known UISD for standardized effect size measures (section (1)) is used if available, even when ni is provided.

(3) Manually setting unit information standard deviation

Alternatively, the UISD can be specified directly via the prior_unit_information_sd argument. This is useful when the appropriate scale for prior distributions is known a priori or when multiple analyses are to be performed on subsets of the same data (re-estimating UISD on different subsets of the data can lead to slightly different prior distributions for different subsets; see estimate_unit_information_sd()). The specified prior_unit_information_sd is then used to scale the default prior distributions as described in section (1).

Note that the manually specified prior_unit_information_sd takes precedence over the estimated UISD from ni (section (2)) and the known UISD from measure (section (1)). It cannot be combined with prior_informed_field.

(4) Specifying informed empirical prior distributions

Informed prior distributions can be specified via the prior_informed_field and prior_informed_subfield arguments. Currently, only prior_informed_field = "medicine" with subfields defined in BayesTools::prior_informed_medicine_names is supported, which uses empirically derived prior distributions from medical meta-analyses as described in Bartoš et al. (2021) and Bartoš et al. (2023) .

When prior_informed_field = "medicine", the default prior_informed_subfield is "Cochrane" (i.e., using the whole CDSR database as a reference). The informed prior distributions are available for the following effect size measures:

`"SMD"`:	standardized mean difference
`"OR"`:	log odds ratio
`"RR"`:	log risk ratio
`"RD"`:	risk difference
`"HR"`:	log hazard ratio

Note that informed prior distributions are only available for the effect size (\(\mu\)) and heterogeneity (\(\tau\)) parameters. For effect moderation (prior_mods), the informed effect prior is scaled by a factor of RoBMA.get_option("default_informed_priors.mods"). For heterogeneity moderation (prior_scale), a normal prior with standard deviation specified by RoBMA.get_option("default_informed_priors.scale") is used.

(5) Fully custom prior distributions

Prior distributions can be fully customized by directly specifying the prior_effect, prior_heterogeneity, prior_mods, prior_scale, and prior_heterogeneity_allocation arguments. These should be prior distribution objects created via BayesTools::prior() or related functions (e.g., prior_factor()).

Rescaling prior distributions

The rescale_priors argument allows rescaling supported prior distributions by a multiplicative factor. For example, rescale_priors = 2 doubles the standard deviations/scales of normal, Cauchy, t, and inverse-gamma prior distributions, making them more diffuse. Point and none priors are unchanged. For publication-bias prior distributions, see publication_bias_prior_specification.

References

Bartoš F, Gronau QF, Timmers B, Otte WM, Ly A, Wagenmakers E (2021). “Bayesian model-averaged meta-analysis in medicine.” Statistics in Medicine, 40(30), 6743–6761. doi:10.1002/sim.9170 .

Bartoš F, Otte WM, Gronau QF, Timmers B, Ly A, Wagenmakers E (2023). “Empirical prior distributions for Bayesian meta-analyses of binary and time-to-event outcomes.” doi:10.48550/arXiv.2306.11468 . Preprint available at https://doi.org/10.48550/arXiv.2306.11468.

Grieve AP (2022). Hybrid frequentist/Bayesian power and Bayesian power in planning clinical trials. Chapman and Hall/CRC.

Röver C, Bender R, Dias S, Schmid CH, Schmidli H, Sturtz S, Weber S, Friede T (2021). “On weakly informative prior distributions for the heterogeneity parameter in Bayesian random-effects meta-analysis.” Research Synthesis Methods, 12(4), 448–474. doi:10.1002/jrsm.1475 .

Spiegelhalter DJ, Abrams KR, Myles JP (2004). Bayesian approaches to clinical trials and health-care evaluation. John Wiley and Sons. doi:10.1002/0470092602 .