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.
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) aformulato 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 errorSE,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) = 1.81times 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
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
See also
Examples
# \donttest{
data(towels)
set.seed(123)
md <- meta_default(logOR, SE, study, towels,
field = "psychology", effect = "logOR"
)
#> Warning: There were 1 divergent transitions after warmup. See
#> https://mc-stan.org/misc/warnings.html#divergent-transitions-after-warmup
#> to find out why this is a problem and how to eliminate them.
#> Warning: Examine the pairs() plot to diagnose sampling problems
md
#> ### Meta-Analysis with Bayesian Model Averaging ###
#> Fixed H0: d = 0
#> Fixed H1: d ~ 't' (location=0, scale=1.283, nu=1) with support on the interval [-Inf,Inf].
#> Random H0: d = 0,
#> tau ~ 'invgamma' (shape=1, scale=0.272) with support on the interval [0,Inf].
#> Random H1: d ~ 't' (location=0, scale=1.283, nu=1) with support on the interval [-Inf,Inf].
#> tau ~ 'invgamma' (shape=1, scale=0.272) with support on the interval [0,Inf].
#>
#> # Bayes factors:
#> (denominator)
#> (numerator) fixed_H0 fixed_H1 random_H0 random_H1
#> fixed_H0 1.000 0.343 0.413 1.29
#> fixed_H1 2.913 1.000 1.202 3.76
#> random_H0 2.423 0.832 1.000 3.12
#> random_H1 0.776 0.266 0.320 1.00
#>
#> # Bayesian Model Averaging
#> Comparison: (fixed_H1 & random_H1) vs. (fixed_H0 & random_H0)
#> Inclusion Bayes factor: 1.077
#> Inclusion posterior probability: 0.519
#>
#> # Model posterior probabilities:
#> prior posterior logml
#> fixed_H0 0.25 0.141 -5.58
#> fixed_H1 0.25 0.410 -4.51
#> random_H0 0.25 0.341 -4.69
#> random_H1 0.25 0.109 -5.83
#>
#> # Posterior summary statistics of average effect size:
#> mean sd 2.5% 50% 97.5% hpd95_lower hpd95_upper n_eff Rhat
#> averaged 0.217 0.093 0.027 0.220 0.387 0.037 0.394 NA NA
#> fixed 0.227 0.078 0.074 0.227 0.378 0.076 0.379 3159.9 1.000
#> random 0.186 0.129 -0.095 0.193 0.421 -0.077 0.436 4152.1 1.001
plot_forest(md)
# }