Package 'metaBMA' reference manual

Title:	Bayesian Model Averaging for Random and Fixed Effects Meta-Analysis
Description:	Computes the posterior model probabilities for standard meta-analysis models (null model vs. alternative model assuming either fixed- or random-effects, respectively). These posterior probabilities are used to estimate the overall mean effect size as the weighted average of the mean effect size estimates of the random- and fixed-effect model as proposed by Gronau, Van Erp, Heck, Cesario, Jonas, & Wagenmakers (2017, <doi:10.1080/23743603.2017.1326760>). The user can define a wide range of non-informative or informative priors for the mean effect size and the heterogeneity coefficient. Moreover, using pre-compiled Stan models, meta-analysis with continuous and discrete moderators with Jeffreys-Zellner-Siow (JZS) priors can be fitted and tested. This allows to compute Bayes factors and perform Bayesian model averaging across random- and fixed-effects meta-analysis with and without moderators. For a primer on Bayesian model-averaged meta-analysis, see Gronau, Heck, Berkhout, Haaf, & Wagenmakers (2021, <doi:10.1177/25152459211031256>).
Authors:	Daniel W. Heck [aut, cre] , Quentin F. Gronau [ctb], Eric-Jan Wagenmakers [ctb], Indrajeet Patil [ctb]
Maintainer:	Daniel W. Heck <[email protected]>
License:	GPL-3
Version:	0.6.9
Built:	2024-10-26 05:34:32 UTC
Source:	https://github.com/danheck/metabma

metaBMA: Bayesian Model Averaging for Random and Fixed Effects Meta-Analysis

Description

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 $\tau$ (see prior). 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, both tests can be performed by integrating over the other model. This allows to test whether an effect exists while accounting for uncertainty whether study heterogeneity exists (so-called inclusion Bayes factors). For a primer on Bayesian model-averaged meta-analysis, see Gronau, Heck, Berkhout, Haaf, and Wagenmakers (2020).

Details

The most general functions in metaBMA is meta_bma, which fits 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). Prior distributions can be specified and plotted using the function prior.

Moreover, meta_fixed and meta_random fit a single meta-analysis models. The model-specific posteriors for $d$ can be averaged by bma and inclusion Bayes factors be computed by inclusion.

Results can be visualized with the functions plot_posterior, which compares the prior and posterior density for a fitted meta-analysis, and plot_forest, which plots study and overall effect sizes.

For more details how to use the package, see the vignette: vignette("metaBMA").

Funding

Funding for this research was provided by the Berkeley Initiative for Transparency in the Social Sciences, a program of the Center for Effective Global Action (CEGA), Laura and John Arnold Foundation, and by the German Research Foundation (grant GRK-2277: Statistical Modeling in Psychology).

Author(s)

Heck, D. W. & Gronau, Q. F.

References

Gronau, Q. F., Erp, S. V., Heck, D. W., Cesario, J., Jonas, K. J., & Wagenmakers, E.-J. (2017). A Bayesian model-averaged meta-analysis of the power pose effect with informed and default priors: the case of felt power. Comprehensive Results in Social Psychology, 2(1), 123-138. doi:10.1080/23743603.2017.1326760

Gronau, Q. F., Heck, D. W., Berkhout, S. W., Haaf, J. M., & Wagenmakers, E.-J. (2021). A primer on Bayesian model-averaged meta-analysis. Advances in Methods and Practices in Psychological Science, 4(3), 1–19. doi:10.1177/25152459211031256

Heck, D. W., Gronau, Q. F., & Wagenmakers, E.-J. (2019). metaBMA: Bayesian model averaging for random and fixed effects meta-analysis. https://CRAN.R-project.org/package=metaBMA

Bayesian Model Averaging

Description

Model averaging for different meta-analysis models (e.g., random-effects or fixed-effects with different priors) based on the posterior model probability.

Usage

bma(
  meta,
  prior = 1,
  parameter = "d",
  summarize = "integrate",
  ci = 0.95,
  rel.tol = .Machine$double.eps^0.5
)
bma(
  meta,
  prior = 1,
  parameter = "d",
  summarize = "integrate",
  ci = 0.95,
  rel.tol = .Machine$double.eps^0.5
)

Arguments

`meta`	list of meta-analysis models (fitted via `meta_random` or `meta_fixed`)
`prior`	prior probabilities over models (possibly unnormalized). For instance, if the first model is as likely as models 2, 3 and 4 together: `prior = c(3,1,1,1)`. The default is a discrete uniform distribution over models.
`parameter`	either the mean effect `"d"` or the heterogeneity `"tau"` (i.e., the across-study standard deviation of population effect sizes).
`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).

Examples


# model averaging for fixed and random effects
data(towels)
fixed <- meta_fixed(logOR, SE, study, towels)
random <- meta_random(logOR, SE, study, towels)

averaged <- bma(list("fixed" = fixed, "random" = random))
averaged
plot_posterior(averaged)
plot_forest(averaged, mar = c(4.5, 20, 4, .3))

# model averaging for fixed and random effects
data(towels)
fixed <- meta_fixed(logOR, SE, study, towels)
random <- meta_random(logOR, SE, study, towels)

averaged <- bma(list("fixed" = fixed, "random" = random))
averaged
plot_posterior(averaged)
plot_forest(averaged, mar = c(4.5, 20, 4, .3))

Data Set: Facial Feedback

Description

Preregistered replication (Wagenmakers et al., 2016) that investigated the facial feedback hypothesis (Strack, Martin, & Stepper, 1988).

Usage

facial_feedback
facial_feedback

Format

A data frame with three variables:

study: Authors of original study (see Wagenmakers et. al, 2016)
d: Measure of effect size: Cohen's d (difference between smile vs. pout condition)
SE: Measure of precision: standard error of Cohen's d

Details

The facial-feedback hypothesis states that people's affective responses can be influenced by their own facial expression (e.g., smiling, pouting), even when their expression did not result from their emotional experiences (Strack, Martin, & Stepper, 1988).

References

Strack, F., Martin, L. L., & Stepper, S. (1988). Inhibiting and facilitating conditions of the human smile: A nonobtrusive test of the facial feedback hypothesis. Journal of Personality and Social Psychology, 54, 768–777. doi:10.1037/0022-3514.54.5.768

Wagenmakers, E.-J., Beek, T., Dijkhoff, L., Gronau, Q. F., Acosta, A., Adams, R. B., ... Zwaan, R. A. (2016). Registered replication report: Strack, Martin, & Stepper (1988). Perspectives on Psychological Science, 11, 917-928. doi:10.1177/1745691616674458

Examples

data(facial_feedback)
head(facial_feedback)
mf <- meta_fixed(d, SE, study, facial_feedback)
mf
plot_posterior(mf)
data(facial_feedback)
head(facial_feedback)
mf <- meta_fixed(d, SE, study, facial_feedback)
mf
plot_posterior(mf)

Inclusion Bayes Factor

Description

Computes the inclusion Bayes factor for two sets of models (e.g., A={M1,M2} vs. B={M3,M4}).

Usage

inclusion(logml, include = 1, prior = 1)
inclusion(logml, include = 1, prior = 1)

Arguments

`logml`	a vector with log-marginal likelihoods. Alternatively, a list with meta-analysis models (fitted via `meta_random` or `meta_fixed`).
`include`	integer vector which models to include in inclusion Bayes factor/posterior probability. If only two marginal likelihoods/meta-analyses are supplied, the inclusion Bayes factor is identical to the usual Bayes factor BF_{M1,M2}. One can include models depending on the names of the models (such as `"random_H1"`) by providing a character value, for instance: `include="H1"` (all H1 vs. all H0 models) or `include="random"` (all random- vs. all fixed-effects models).
`prior`	prior probabilities over models (possibly unnormalized). For instance, if the first model is as likely as models 2, 3 and 4 together: `prior = c(3,1,1,1)`. The default is a discrete uniform distribution over models.

Examples

#### Example with simple Normal-distribution models
# generate data:
x <- rnorm(50)

# Model 1: x ~ Normal(0,1)
logm1 <- sum(dnorm(x, log = TRUE))
# Model 2: x ~ Normal(.2, 1)
logm2 <- sum(dnorm(x, mean = .2, log = TRUE))
# Model 3: x ~ Student-t(df=2)
logm3 <- sum(dt(x, df = 2, log = TRUE))

# BF: Correct (Model 1) vs. misspecified (2 & 3)
inclusion(c(logm1, logm2, logm3), include = 1)
#### Example with simple Normal-distribution models
# generate data:
x <- rnorm(50)

# Model 1: x ~ Normal(0,1)
logm1 <- sum(dnorm(x, log = TRUE))
# Model 2: x ~ Normal(.2, 1)
logm2 <- sum(dnorm(x, mean = .2, log = TRUE))
# Model 3: x ~ Student-t(df=2)
logm3 <- sum(dt(x, df = 2, log = TRUE))

# BF: Correct (Model 1) vs. misspecified (2 & 3)
inclusion(c(logm1, logm2, logm3), include = 1)

Model Averaging for Meta-Analysis

Description

Fits random- and fixed-effects meta-analyses and performs Bayesian model averaging for H1 (d != 0) vs. H0 (d = 0).

Usage

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

Arguments

`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.
`prior`	prior probabilities over models (possibly unnormalized) in the order `c(fixed_H0, fixed_H1, random_H0, random_H1)`. For instance, if we expect fixed effects to be two times as likely as random effects and H0 and H1 to be equally likely: `prior = c(2,2,1,1)`.
`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)`.

Details

Bayesian model averaging for four meta-analysis models: Fixed- vs. random-effects and H0 ( $d=0$ ) vs. H1 (e.g., $d>0$ ). For a primer on Bayesian model-averaged meta-analysis, see Gronau, Heck, Berkhout, Haaf, and Wagenmakers (2020).

By default, the log-marginal likelihood is computed by numerical integration (logml="integrate"). This is relatively fast and gives precise, reproducible results. However, for extreme priors or data (e.g., very small standard errors), numerical integration is not robust and might provide incorrect results. As an alternative, the log-marginal likelihood can be estimated using MCMC/Stan samples and bridge sampling (logml="stan").

To obtain posterior summary statistics for the average effect size d and the heterogeneity parameter tau, one can also choose between numerical integration (summarize="integrate") or MCMC sampling in Stan (summarize="stan"). If any moderators are included in a model, both the marginal likelihood and posterior summary statistics can only be computed using Stan.

References

Berkhout, S. W., Haaf, J. M., Gronau, Q. F., Heck, D. W., & Wagenmakers, E.-J. (2023). A tutorial on Bayesian model-averaged meta-analysis in JASP. Behavior Research Methods.

Examples


### Bayesian Model-Averaged Meta-Analysis (H1: d>0)
data(towels)
set.seed(123)
mb <- meta_bma(logOR, SE, study, towels,
  d = prior("norm", c(mean = 0, sd = .3), lower = 0),
  tau = prior("invgamma", c(shape = 1, scale = 0.15))
)
mb
plot_posterior(mb, "d")

### Bayesian Model-Averaged Meta-Analysis (H1: d>0)
data(towels)
set.seed(123)
mb <- meta_bma(logOR, SE, study, towels,
  d = prior("norm", c(mean = 0, sd = .3), lower = 0),
  tau = prior("invgamma", c(shape = 1, scale = 0.15))
)
mb
plot_posterior(mb, "d")

Defaults for Model Averaging in Meta-Analysis

Description

Wrapper with default prior for Bayesian meta-analysis. Since version 0.6.6, the default priors for Cohen's d have been changed from a normal distribution with scale=0.3 to a Cauchy distribution with scale=0.707. Moreover, scale adjustments were implemented when using Fisher's z or log odds-ratios.

Usage

meta_default(y, SE, labels, data, field = "psychology", effect = "d", ...)
meta_default(y, SE, labels, data, field = "psychology", effect = "d", ...)

Arguments

`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.
`field`	either`"psychology"` or `"medicine"`
`effect`	the type of effect size used in the meta-analysis: either Cohen's d (`"d"`), Fisher's z-transformed correlation (`"z"`), or log odds ratios (`"logOR"`).
`...`	further arguments passed to `meta_bma`

Details

The prior distribution depends on the scale of the effect size that is used in the meta-analysis (Cohen's d, Fisher's z, or log odds ratio). To ensure that the results are comparable when transforming between different effect sizes (e.g., using the function transform_es), it is necessary to adjust the prior distributions. The present adjustments merely use a linear re-scaling of the priors to achieve approximately invariant results when using different types of effect sizes.

The distribution of Fisher's z is approximately half as wide as the distribution of Cohen's d and hence the prior scale parameter is divided by two.
The distribution of the log odds ratio is approximately pi / sqrt(3) = 1.81 times as wide as the distribution of Cohen's d. Hence, the prior scale parameter is doubled by this factor.

For field = "psychology", this results in the following defaults:

effect = "d" (Cohen's d): Cauchy distribution with scale=0.707 on the overall effect size (parameter d) and inverse gamma distribution with shape=1 and scale=0.15 on the standard deviation of effect sizes across studies (parameter tau).
effect = "z" (Fisher's z): Cauchy distribution with scale=0.354 on d and inverse gamma with shape=1 and scale=0.075 on tau.
effect = "logOR" (log odds ratio): Cauchy distribution with scale=1.283 on d and inverse gamma with shape=1 and scale=0.272 on tau.

Currently, the same priors are used when specifying field = "medicine".

Default prior distributions can be plotted using plot_default.

References

Borenstein, M., Hedges, L. V., Higgins, J. P. T., & Rothstein, H. R. (2009). Converting among effect sizes. In Introduction to Meta-Analysis (pp. 45–49). John Wiley & Sons, Ltd. doi:10.1002/9780470743386.ch7

Examples


data(towels)
set.seed(123)
md <- meta_default(logOR, SE, study, towels,
  field = "psychology", effect = "logOR"
)
md
plot_forest(md)


data(towels)
set.seed(123)
md <- meta_default(logOR, SE, study, towels,
  field = "psychology", effect = "logOR"
)
md
plot_forest(md)

Bayesian Fixed-Effects Meta-Analysis

Description

Runs a Bayesian meta-analysis assuming that the mean effect $d$ in each study is identical (i.e., a fixed-effects analysis).

Usage

meta_fixed(
  y,
  SE,
  labels,
  data,
  d = prior("cauchy", c(location = 0, scale = 0.707)),
  rscale_contin = 1/2,
  rscale_discrete = 0.707,
  centering = TRUE,
  logml = "integrate",
  summarize = "integrate",
  ci = 0.95,
  rel.tol = .Machine$double.eps^0.3,
  silent_stan = TRUE,
  ...
)
meta_fixed(
  y,
  SE,
  labels,
  data,
  d = prior("cauchy", c(location = 0, scale = 0.707)),
  rscale_contin = 1/2,
  rscale_discrete = 0.707,
  centering = TRUE,
  logml = "integrate",
  summarize = "integrate",
  ci = 0.95,
  rel.tol = .Machine$double.eps^0.3,
  silent_stan = TRUE,
  ...
)

Arguments

`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`.
`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).
`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)`.

Examples

### Bayesian Fixed-Effects Meta-Analysis (H1: d>0)
data(towels)
mf <- meta_fixed(logOR, SE, study,
  data = towels,
  d = prior("norm", c(mean = 0, sd = .3), lower = 0)
)
mf
plot_posterior(mf)
plot_forest(mf)
### Bayesian Fixed-Effects Meta-Analysis (H1: d>0)
data(towels)
mf <- meta_fixed(logOR, SE, study,
  data = towels,
  d = prior("norm", c(mean = 0, sd = .3), lower = 0)
)
mf
plot_posterior(mf)
plot_forest(mf)

Meta-Analysis with Order-Constrained Study Effects

Description

Computes the Bayes factor for the hypothesis that the true study effects in a random-effects meta-analysis are all positive or negative.

Usage

meta_ordered(
  y,
  SE,
  labels,
  data,
  d = prior("norm", c(mean = 0, sd = 0.3), lower = 0),
  tau = prior("invgamma", c(shape = 1, scale = 0.15)),
  prior = c(1, 1, 1, 1),
  logml = "integrate",
  summarize = "stan",
  ci = 0.95,
  rel.tol = .Machine$double.eps^0.3,
  logml_iter = 5000,
  iter = 5000,
  silent_stan = TRUE,
  ...
)
meta_ordered(
  y,
  SE,
  labels,
  data,
  d = prior("norm", c(mean = 0, sd = 0.3), lower = 0),
  tau = prior("invgamma", c(shape = 1, scale = 0.15)),
  prior = c(1, 1, 1, 1),
  logml = "integrate",
  summarize = "stan",
  ci = 0.95,
  rel.tol = .Machine$double.eps^0.3,
  logml_iter = 5000,
  iter = 5000,
  silent_stan = TRUE,
  ...
)

Arguments

`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`.
`prior`	prior probabilities over models (possibly unnormalized) in the order `c(fixed_H0, fixed_H1, ordered_H1, random_H1)`. Note that the model `random_H0` is not included in the comparison.
`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.
`iter`	number of MCMC iterations for the random-effects meta-analysis. Needs to be larger than usual to estimate the probability of all random effects being ordered (i.e., positive or negative).
`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)`.

Details

Usually, in random-effects meta-analysis,the study-specific random-effects are allowed to be both negative or positive even when the prior on the overall effect size d is truncated to be positive). In contrast, the function meta_ordered fits and tests a model in which the random effects are forced to be either all positive or all negative. The direction of the study-specific random-effects is defined via the prior on the mode of the truncated normal distribution d. For instance, d=prior("norm", c(0,.5), lower=0) means that all random-effects are positive (not just the overall mean effect size).

The posterior summary statistics of the overall effect size in the model ordered refer to the the average/mean of the study-specific effect sizes (as implied by the fitted truncated normal distribution) and not to the location parameter d of the truncated normal distribution (which is only the mode, not the expected value of a truncated normal distribution).

The Bayes factor for the order-constrained model is computed using the encompassing Bayes factor. Since many posterior samples are required for this approach, the default number of MCMC iterations for meta_ordered is iter=5000 per chain.

References

Haaf, J. M., & Rouder, J. N. (2018). Some do and some don’t? Accounting for variability of individual difference structures. Psychonomic Bulletin & Review, 26, 772–789. doi:10.3758/s13423-018-1522-x

Examples


### Bayesian Meta-Analysis with Order Constraints (H1: d>0)
data(towels)
set.seed(123)
mo <- meta_ordered(logOR, SE, study, towels,
  d = prior("norm", c(mean = 0, sd = .3), lower = 0)
)
mo
plot_posterior(mo)

### Bayesian Meta-Analysis with Order Constraints (H1: d>0)
data(towels)
set.seed(123)
mo <- meta_ordered(logOR, SE, study, towels,
  d = prior("norm", c(mean = 0, sd = .3), lower = 0)
)
mo
plot_posterior(mo)

Bayesian Random-Effects Meta-Analysis

Description

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

Usage

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

Arguments

`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)`.

Examples


### 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
plot_posterior(mr)

### 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
plot_posterior(mr)

Sensitivity Analysis for Bayesian Meta-Analysis

Description

Sensitivity analysis assuming different prior distributions for the two main parameters of a Bayesian meta-analysis (i.e., the overall effect and the heterogeneity of effect sizes across studies).

Usage

meta_sensitivity(
  y,
  SE,
  labels,
  data,
  d_list,
  tau_list,
  analysis = "bma",
  combine_priors = "crossed",
  ...
)
meta_sensitivity(
  y,
  SE,
  labels,
  data,
  d_list,
  tau_list,
  analysis = "bma",
  combine_priors = "crossed",
  ...
)

Arguments

`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_list`	a `list` of prior distributions specified via `prior()` for the overall effect size (mean) across studies
`tau_list`	a `list` of prior distributions specified via `prior()` for the heterogeneity (SD) of effect sizes across studies
`analysis`	which type of meta-analysis should be performed for analysis? Can be one of the following: `"fixed"` for fixed-effects model, see `meta_fixed()` `"random"` for random-effects model, see `meta_random()` `"bma"` for model averaging, see `meta_bma()`
`combine_priors`	either `"matched"`, in which case the analysis includes the matched pairwise combinations of the prior distributions specified in `d_list` and `tau_list`, or `crossed`, in which case the analysis uses all possible pairwise combinations of priors.
`...`	further arguments passed to the function specified in `analysis`.

Value

an object of the S3 class meta_sensitivity, that is, a list of fitted meta-analysis models. Results can be printed or plotted using plot.meta_sensitivity().

Examples


data(towels)
sensitivity <- meta_sensitivity(
  y = logOR, SE = SE, labels = study, data = towels,
  d_list = list(prior("cauchy", c(0, .707)),
                prior("norm", c(0, .5)),
                prior("norm", c(.5, .3))),
  tau_list = list(prior("invgamma", c(1, 0.15), label = "tau"),
                  prior("gamma", c(1.5, 3), label = "tau")),
  analysis = "random",
  combine_priors = "crossed")

print(sensitivity, digits = 2)

par(mfrow = c(1,2))
plot(sensitivity, "d", "prior")
plot(sensitivity, "d", "posterior")

plot(sensitivity, "tau", "prior")
plot(sensitivity, "tau", "posterior")


data(towels)
sensitivity <- meta_sensitivity(
  y = logOR, SE = SE, labels = study, data = towels,
  d_list = list(prior("cauchy", c(0, .707)),
                prior("norm", c(0, .5)),
                prior("norm", c(.5, .3))),
  tau_list = list(prior("invgamma", c(1, 0.15), label = "tau"),
                  prior("gamma", c(1.5, 3), label = "tau")),
  analysis = "random",
  combine_priors = "crossed")

print(sensitivity, digits = 2)

par(mfrow = c(1,2))
plot(sensitivity, "d", "prior")
plot(sensitivity, "d", "posterior")

plot(sensitivity, "tau", "prior")
plot(sensitivity, "tau", "posterior")

Plot Default Priors

Description

Plots default priors for the mean effect $d$ and the standard deviation of effects $tau$ .

Usage

plot_default(field = "psychology", effect = "d", ...)
plot_default(field = "psychology", effect = "d", ...)

Arguments

`field`	either`"psychology"` or `"medicine"`
`effect`	the type of effect size used in the meta-analysis: either Cohen's d (`"d"`), Fisher's z-transformed correlation (`"z"`), or log odds ratios (`"logOR"`).
`...`	further arguments passed to `plot` (e.g., `from`, `to`)

Examples

plot_default(field = "psychology", effect = "d")
plot_default(field = "psychology", effect = "d")

Forest Plot for Meta-Analysis

Description

Plots estimated effect sizes for all studies.

Usage

plot_forest(
  meta,
  from,
  to,
  shrinked = "random",
  summary = c("mean", "hpd"),
  mar = c(4.5, 12, 4, 0.3),
  cex.axis = 1,
  ...
)
plot_forest(
  meta,
  from,
  to,
  shrinked = "random",
  summary = c("mean", "hpd"),
  mar = c(4.5, 12, 4, 0.3),
  cex.axis = 1,
  ...
)

Arguments

`meta`	fitted meta-analysis model
`from`	lower limit of the x-axis
`to`	upper limit of the x-axis
`shrinked`	which meta-analysis model should be used to show (shrinked) estimates of the study effect sizes. The name must match the corresponding name in the list `meta`. Can be suppressed by `shrinked = ""`
`summary`	character vector with two values: first, either `"mean"` or `"50%"`; and second, either highest-probability-density interval `"hpd"` or the Bayesian credibility interval `"bci"`.
`mar`	margin of the plot in the order `c(bottom, left, top, right)` (see `par`)
`cex.axis`	size of the y-axis annotation for the labels of studies.
`...`	arguments passed to `plot` (e.g., `from`, `to`)

Examples

data(towels)
mf <- meta_fixed(logOR, SE, study, towels)
plot_forest(mf, mar = c(4.5, 20, 4, .2), xlab = "Log Odds Ratio")
data(towels)
mf <- meta_fixed(logOR, SE, study, towels)
plot_forest(mf, mar = c(4.5, 20, 4, .2), xlab = "Log Odds Ratio")

Plot Posterior Distribution

Description

Plot Posterior Distribution

Usage

plot_posterior(
  meta,
  parameter = "d",
  from,
  to,
  summary = c("mean", "hpd"),
  ...
)
plot_posterior(
  meta,
  parameter = "d",
  from,
  to,
  summary = c("mean", "hpd"),
  ...
)

Arguments

`meta`	fitted meta-analysis model
`parameter`	only for random-effects model: whether to plot `"d"` or `"tau"`
`from`	lower limit of the x-axis
`to`	upper limit of the x-axis
`summary`	character vector with two values: first, either `"mean"` or `"50%"`; and second, either highest-probability-density interval `"hpd"` or the Bayesian credibility interval `"bci"`.
`...`	arguments passed to `plot`

Plot Predicted Bayes Factors

Description

Plot Predicted Bayes Factors

Usage

## S3 method for class 'meta_pred'
plot(x, which = "BF.inclusion", scale = "BF", ...)
## S3 method for class 'meta_pred'
plot(x, which = "BF.inclusion", scale = "BF", ...)

Arguments

`x`	an object of the class `"prediction"` which contains observed and predicted Bayes factors
`which`	a character value defining which Bayes factor to plot. Some options are: `"BF.inclusion"` = model averaging across fixed- and random-effects models `"BF.fixed_H1.fixed_H0"` = consider only fixed-effects model `"BF.random_H1.random_H0"` = consider only fixed-effects model
`scale`	either plot Bayes factors (`"BF"`), inverse Bayes factors (`"1/BF"`), log Bayes factors (`"log"`), or the log-inverse Bayes factor (`"1/log"`)
`...`	arguments passed to `plot`

Plot Sensitivity Analysis for Meta-Analysis

Description

Plot prior or posterior distributions of multiple analyses performed with meta_sensitivity().

Usage

## S3 method for class 'meta_sensitivity'
plot(
  x,
  parameter = "d",
  distribution = "posterior",
  from,
  to,
  n = 101,
  legend = TRUE,
  ...
)
## S3 method for class 'meta_sensitivity'
plot(
  x,
  parameter = "d",
  distribution = "posterior",
  from,
  to,
  n = 101,
  legend = TRUE,
  ...
)

Arguments

`x`	prior probability density function defined via `prior`.
`parameter`	which parameter should be plotted: `"d"` or `"tau"`.
`distribution`	which distribution should be plotted: `"prior"` or `"posterior"`.
`from`	lower boundary
`to`	upper boundary
`n`	integer; the number of x values at which to evaluate.
`legend`	whether to print all prior specifications and plot a corresponding legend.
`...`	further arguments passed to `plot`

Details

For meta-analysis with model averaging via meta_bma(), plotting the model-averaged posterior of tau is not yet supported. Instead, the posterior distributions for the random effects models will be plotted.

Plot Prior Distribution

Description

Plot the probability density function of a prior distribution.

Usage

## S3 method for class 'prior'
plot(x, from, to, ...)
## S3 method for class 'prior'
plot(x, from, to, ...)

Arguments

`x`	prior probability density function defined via `prior`.
`from`	lower boundary
`to`	upper boundary
`...`	further arguments passed to `plot`

Examples

p1 <- prior("t", c(location = 0, scale = 0.707, nu = 1), 0, 3)
plot(p1, 0, 2)

# define custom prior pdf up to a constant:
p2 <- prior("custom", function(x) x^.5, 0, .5)
plot(p2)
p1 <- prior("t", c(location = 0, scale = 0.707, nu = 1), 0, 3)
plot(p1, 0, 2)

# define custom prior pdf up to a constant:
p2 <- prior("custom", function(x) x^.5, 0, .5)
plot(p2)

Data Set: Power Pose Effect

Description

Includes six pre-registered replication studies testing whether participants feel more powerful if they adopt expansive as opposed to constrictive body postures. In the data set power_pose_unfamiliar, only those participants are included who were unfamiliar with the power pose effect.

Usage

power_pose

power_pose_unfamiliar
power_pose

power_pose_unfamiliar

Format

A data frame with three variables:

study: Authors of original study
n_high_power: number of participants in high-power condition
n_low_power: number of participants in low-power condition
mean_high_power: mean rating in high-power condition on a 5-point Likert scale
mean_low_power: mean rating in low-power condition on a 5-point Likert scale
sd_high_power: standard deviation of ratings in high-power condition
sd_low_power: standard deviation of ratings in low-power condition
t_value: t-value for two-sample t-test
df: degrees of freedom for two-sample t-test
two_sided_p_value: two-sided p-value of two-sample t-test
one_sided_p_value: one-sided p-value of two-sample t-test
effectSize: Cohen's d, the standardized effect size (high vs. low power)
SE: Standard error of Cohen's d

Data frame with 6 rows and 13 variables

An object of class data.frame with 6 rows and 13 columns.

Details

See Carney, Cuddy, and Yap (2010) for more details.

References

Carney, D. R., Cuddy, A. J. C., & Yap, A. J. (2010). Power posing: Brief nonverbal displays affect neuroendocrine levels and risk tolerance. Psychological Science, 21, 1363–1368.

Examples

data(power_pose)
head(power_pose)

# Simple fixed-effects meta-analysis
mfix <- meta_fixed(effectSize, SE, study,
  data = power_pose
)
mfix
plot_posterior(mfix)
data(power_pose)
head(power_pose)

# Simple fixed-effects meta-analysis
mfix <- meta_fixed(effectSize, SE, study,
  data = power_pose
)
mfix
plot_posterior(mfix)

Predicted Bayes Factors for a New Study

Description

How much can be learned by an additional study? To judge this, this function samples the distribution of predicted Bayes factors for a new study given the current evidence.

Usage

predicted_bf(meta, SE, sample = 100, ...)
predicted_bf(meta, SE, sample = 100, ...)

Arguments

`meta`	model-averaged meta-analysis (fitted with `meta_bma`).
`SE`	a scalar: the expected standard error of future study. For instance, SE = 1/sqrt(N) for standardized effect sizes and N = sample size)
`sample`	number of simulated Bayes factors
`...`	further arguments passed to rstan::sampling to draw posterior samples for d and tau.

Prior Distribution

Description

Defines a prior distribution/probability density function for the average effect size $d$ or for the heterogeneity of effect sizes $\tau$ .

Usage

prior(
  family,
  param,
  lower,
  upper,
  label = "d",
  rel.tol = .Machine$double.eps^0.5
)
prior(
  family,
  param,
  lower,
  upper,
  label = "d",
  rel.tol = .Machine$double.eps^0.5
)

Arguments

`family`	a character value defining the distribution family.
`param`	numeric parameters for the distribution. See details for the definition of the parameters of each family.
`lower`	lower boundary for truncatation of prior density. If `family="beta"`, the interval `⁠[0,1]⁠` is rescaled to the interval `⁠[lower,upper]⁠`. Must be specified if `family = "custom"`.
`upper`	See `lower`.
`label`	optional: parameter label.
`rel.tol`	relative tolerance used for integrating the density of `family="custom"`.

Details

The following prior distributions are currently implemented:

"norm": Normal distribution with param = c(mean, sd) (see Normal).
"t": Student's t-distribution with param = c(location, scale, nu) where nu are the degrees of freedom (see dist.Student.t).
"cauchy": Cauchy distribution with param = c(location, scale). The Cauchy distribution is a special case of the t-distribution with degrees of freedom nu=1.
"gamma": Gamma distribution with param = c(shape, rate) with rate parameter equal to the inverse scale (see GammaDist).
"invgamma": Inverse gamma distribution with param = c(shape, scale) (see dist.Inverse.Gamma).
"beta": (Scaled) beta distribution with param = c(shape1, shape2) (see Beta).
"custom": User-specified prior density function defined by param (see examples; the density must be nonnegative and vectorized, but is normalized internally). Integration is performed from (-Inf, Inf), which requires that the function returns zeros (and not NAs) for values not in the support of the distribution.

Value

an object of the class prior: a density function with the arguments x (parameter values) and log (whether to return density or log-density).

Examples

### Half-Normal Distribution
p1 <- prior("norm", c(mean = 0, sd = .3), lower = 0)
p1
p1(c(-1, 1, 3))
plot(p1, -.1, 1)

### Half-Cauchy Distribution
p2 <- prior("cauchy", c(location = 0, scale = .3), lower = 0)
plot(p2, -.5, 3)

### Custom Prior Distribution
p3 <- prior("custom", function(x) x^2, 0, 1)
plot(p3, -.1, 1.2)

### Half-Normal Distribution
p1 <- prior("norm", c(mean = 0, sd = .3), lower = 0)
p1
p1(c(-1, 1, 3))
plot(p1, -.1, 1)

### Half-Cauchy Distribution
p2 <- prior("cauchy", c(location = 0, scale = .3), lower = 0)
plot(p2, -.5, 3)

### Custom Prior Distribution
p3 <- prior("custom", function(x) x^2, 0, 1)
plot(p3, -.1, 1.2)

Data Set: Reuse of Towels in Hotels

Description

Set of studies that investigated whether people reuse towels in hotels more often if they are provided with a descriptive norm (Scheibehenne, Jamil, & Wagenmakers, 2016).

Usage

towels
towels

Format

A data frame with three variables:

study: Authors of original study (see Scheibehenne et. al, 2016)
logOR: Measure of effect size: log-odds ratio of towel reuse (descriptive-social-norm vs. control)
SE: Measure of precision: standard error of log-odds ratio per study

Details

Two groups of hotel guests received different messages that encouraged them to reuse their towels. One message simply informed the guests about the benefits of environmental protection (the control condition), and the other message indicated that the majority of guests actually reused their towels in the past (the descriptive-social-norm condition). The results suggested that the latter message facilitated towel reuse.

References

Scheibehenne, B., Jamil, T., & Wagenmakers, E.-J. (2016). Bayesian Evidence Synthesis Can Reconcile Seemingly Inconsistent Results: The Case of Hotel Towel Reuse. Psychological Science, 27, 1043–1046. doi:10.1177/0956797616644081

Examples

data(towels)
head(towels)
data(towels)
head(towels)

Transformation of Effect Sizes

Description

Converts between different measures of effect size (i.e., Cohen's d, log odds ratio, Pearson correlation r, and Fisher's z).

Usage

transform_es(y, SE, from, to)
transform_es(y, SE, from, to)

Arguments

`y`	estimate of the effect size (can be vectorized).
`SE`	optional: standard error of the effect-size estimate. Must have the same length as `y`.
`from`	type of effect-size measure provided by the argument `y`. Supported effect sizes are Cohen's d (`"d"`), Fisher's z-transformed correlation (`"z"`), Pearson's correlation (`"r"`), or the log odds ratio (`"logOR"`).
`to`	which type of effect size should be returned (see `from`).

Details

The following chain of transformations is adopted from Borenstein et al. (2009): logOR <--> d <--> r <--> z. The conversion from "d" to "r" assumes equal sample sizes per condition (n1=n2).

Note that in in a Bayesian meta-analysis, the prior distributions need to be adapted to the type of effect size. The function meta_default provides modified default prior distributions for different effect size measures which are approximately transformation-invariant (but results may still differ depending on which type of effect size is used for analysis).

Value

If SE is missing, a vector of transformed effect sizes. Otherwise, a matrix with two columns including effect sizes and standard errors.

References

Examples

# transform a single value of Cohen's
transform_es(y = 0.50, SE = 0.20, from = "d", to = "logOR")

# towels data set:
transform_es(y = towels$logOR, SE = towels$SE, from = "logOR", to = "d")

# transform a single value of Cohen's
transform_es(y = 0.50, SE = 0.20, from = "d", to = "logOR")

# towels data set:
transform_es(y = towels$logOR, SE = towels$SE, from = "logOR", to = "d")

Package 'metaBMA'

Help Index

metaBMA: Bayesian Model Averaging for Random and Fixed Effects Meta-Analysis

Description

Details

Funding

Author(s)

References

See Also

Bayesian Model Averaging

Description

Usage

Arguments

Examples

Data Set: Facial Feedback

Description

Usage

Format

Details

References

Examples

Inclusion Bayes Factor

Description

Usage

Arguments

Examples

Model Averaging for Meta-Analysis

Description

Usage

Arguments

Details

References

See Also

Examples

Defaults for Model Averaging in Meta-Analysis

Description

Usage

Arguments

Details

References

See Also

Examples

Bayesian Fixed-Effects Meta-Analysis

Description

Usage

Arguments

Examples

Meta-Analysis with Order-Constrained Study Effects

Description

Usage

Arguments

Details

References

See Also

Examples

Bayesian Random-Effects Meta-Analysis

Description

Usage

Arguments

Examples

Sensitivity Analysis for Bayesian Meta-Analysis

Description

Usage

Arguments

Value

See Also

Examples

Plot Default Priors

Description

Usage

Arguments

See Also

Examples

Forest Plot for Meta-Analysis

Description

Usage

Arguments

See Also

Examples

Plot Posterior Distribution