The goal of blsmeta is to provide a user-friendly interface for Bayesian meta-analysis, including fixed-effects, two-level, and three-level (for dependent effect sizes) random-effects models.
Additionally, a key feature of blsmeta is “scale” modeling, which allows for predicting the variance components with moderators (e.g., perhaps between-study variance is not constant across studies). As a result, heterogeneity statistics and prediction intervals are then a function of those same moderators, thereby opening the door to better understanding heterogeneity in meta-analysis.
Version 1 is forthcoming, say, by the end of June 2021.
You can install the development version from GitHub with:
# install.packages("devtools")
devtools::install_github("donaldRwilliams/blsmeta")Note that the development version will be fully usable, in so far as it will have been tested, documented, etc. It will gradually include more functions over the coming weeks (merged from other branches), culminating in the official release.
Below, there are some simple examples demonstrating how to use blsmeta. In the future, there will be several examples showcasing the utility of scale modeling in meta-analysis.
- Installing JAGS
- Fixed-Effects Model
- Two-Level Model
- Three-Level Model
- MCMC metafor
- Custom Priors
- Student Led Projects
blsmeta uses the popular Bayesian software JAGS to estimate the models. It must be downloaded from the following link: https://sourceforge.net/projects/mcmc-jags/files/
# install for data
if (!require('psymetadata')){
install.packages('psymetadata')
}
library(psymetadata)
library(blsmeta)
# fit model
fit_fe <- blsmeta(yi = yi, vi = vi,
data = gnambs2020)
# results
fit_fe
#> Model: Fixed-Effects
#> Studies: 67
#> Samples: 20000 (4 chains)
#> Formula: ~ 1
#> ------
#> Location:
#> Post.mean Post.sd Cred.lb Cred.ub Rhat
#> (Intercept) -0.07 0.03 -0.12 -0.02 1.00
#>
#> ------
#> Date: Mon Jun 07 12:03:56 2021
There is an important difference from the metafor package, where, by default, a random-effects model is fitted. This is not the case in blsmeta, where, by default, a fixed-effects model will be estimated if the level two variable is not provided.
fit_fe <- blsmeta(yi = yi, vi = vi,
mods = ~ 0 + color,
data = gnambs2020)
# results
fit_fe
#> Model: Fixed-Effects
#> Studies: 67
#> Samples: 20000 (4 chains)
#> Formula: ~ 0 + color
#> ------
#> Location:
#> Post.mean Post.sd Cred.lb Cred.ub Rhat
#> colorblack -0.04 0.13 -0.30 0.22 1.00
#> colorblue -0.04 0.07 -0.18 0.10 1.00
#> colorgray -0.12 0.05 -0.22 -0.01 1.00
#> colorgreen -0.06 0.03 -0.13 0.00 1.00
#> colorwhite 0.00 0.12 -0.23 0.22 1.00
#> ------
#> Date: Mon Jun 07 12:21:07 2021
In the not too distant future (this was written on 6/7/21), it will be
possible to compare those effects (e.g., colorgreen - colorwhite).
A two-level random-effects meta-analysis is implemented with
fit_re <- blsmeta(yi = yi, vi = vi,
es_id = es_id,
data = gnambs2020)
# results
fit_re
#> Model: Two-Level
#> Studies: 67
#> Samples: 20000 (4 chains)
#> Location Formula: ~ 1
#> Scale Formula: ~ 1
#> Note: 'Scale' on standard deviation scale
#> ------
#> Scale:
#> Post.mean Post.sd Cred.lb Cred.ub Rhat
#> sd(Intercept) 0.10 0.06 0.02 0.22 1.00
#>
#> Location:
#> Post.mean Post.sd Cred.lb Cred.ub Rhat
#> (Intercept) -0.08 0.03 -0.14 -0.02 1.00
#>
#> ------
#> Date: Mon Jun 07 12:26:24 2021
Notice the argument es_id, which corresponds to the effect size id
(1:k, where k is the number of studies).
A key feature of blsmeta is scale modeling that allows for predicting the between-study variance (or “scale”) with moderators (just like for the effect size or “location”). In this following example, heterogeneity is predicted study size.
fit_re <- blsmeta(yi = yi, vi = vi,
es_id = es_id,
mods_scale2 = ~ n,
data = gnambs2020)
# results
fit_re
#> Model: Two-Level
#> Studies: 67
#> Samples: 20000 (4 chains)
#> Location Formula: ~ 1
#> Scale Formula: ~ n
#> Note: 'Scale' on standard deviation scale
#> ------
#> Scale:
#> Post.mean Post.sd Cred.lb Cred.ub Rhat
#> (Intercept) 0.01 0.42 -0.77 0.88 1.03
#> n -0.03 0.01 -0.05 -0.01 1.03
#>
#> Location:
#> Post.mean Post.sd Cred.lb Cred.ub Rhat
#> (Intercept) -0.06 0.03 -0.11 0.00 1.00
#>
#> ------
#> Date: Mon Jun 07 12:30:18 2021
Notice that the n parameter is negative, implying that studies with
larger sample sizes are more consistent (i.e., less heterogeneity). That
effect is on the log-scale, which is far from intuitive.
To make sense of the scale model, it is possible to obtain predicted values of between-study heterogeneity at particular values of the moderator, that is,
tau2(fit_re, type = "sd",
newdata_scale2 = data.frame(n = seq(20, 200, 20)))
#> Post.mean Post.sd Cred.lb Cred.ub
#> 1 0.634 0.165 0.357 1.003
#> 2 0.378 0.083 0.226 0.553
#> 3 0.234 0.072 0.096 0.372
#> 4 0.149 0.063 0.037 0.273
#> 5 0.098 0.053 0.014 0.207
#> 6 0.065 0.043 0.005 0.161
#> 7 0.045 0.035 0.002 0.126
#> 8 0.031 0.028 0.001 0.100
#> 9 0.022 0.022 0.000 0.080
#> 10 0.015 0.018 0.000 0.063
Notice type = "sd", which ensures we are on the standard deviation
scale (easier to interpret). The results indicate that there is quite a
bit of heterogeneity in small studies, but it goes to practically zero
as study size increases.
Three-level location-scale meta-analysis is fully implemented as well.
The key is providing the study_id argument, which is the higher level
grouping variable that the effect sizes are nested within. This
accommodates dependent effect sizes.
fit <- blsmeta(yi = yi,
vi = vi,
es_id = es_id,
study_id = study_id,
data = gnambs2020)
fit
#> Model: Three-Level
#> Studies2: 67
#> Studies3: 22
#> Samples: 20000 (4 chains)
#> Location Formula: ~ 1
#> Scale2 Formula: ~ 1
#> Scale3 Formula: ~ 1
#> Note: 'Scale' on standard deviation scale
#> ------
#> Scale2:
#> Post.mean Post.sd Cred.lb Cred.ub Rhat
#> sd(Intercept) 0.06 0.04 0.01 0.15 1.00
#>
#> Scale3:
#> Post.mean Post.sd Cred.lb Cred.ub Rhat
#> sd(Intercept) 0.20 0.06 0.09 0.34 1.01
#>
#> Location:
#> Post.mean Post.sd Cred.lb Cred.ub Rhat
#> (Intercept) -0.12 0.06 -0.25 -0.02 1.00
#>
#> ------
#> Date: Sun Jun 13 11:27:08 2021
One question might be whether one variance component is larger, which
can be tested with the linear_hypothesis function.
linear_hypothesis(obj = fit,
cred = 0.90,
lin_comb = "scale3_Intercept > scale2_Intercept",
sub_model = "scale")
#> Hypotheses:
#> C1: scale3_Intercept > scale2_Intercept
#> ------
#> Posterior Summary:
#>
#> Post.mean Post.sd Cred.lb Cred.ub Pr.less Pr.greater
#> C1 1.27 0.73 0.14 2.51 0.03 0.97
#> ------
#> Note:
#> Pr.less: Posterior probability less than zero
#> Pr.greater: Posterior probability greater than zero
These estimates are on the log-scale, and there is a 0.97 posterior probability that the level three variance component is larger than the level two variance component.
In the future, it will be possible to compare these models with the Bayes factor.
The package metafor is perhaps the gold-standard for meta-analysis
in R. In blsmeta, it is possible to sample from the posterior
distribution of a model originally estimated with metafor (rma
objects are currently supported).
library(metafor)
fit <- mcmc_rma(rma(yi = yi, vi = vi,
method = "FE", data = gnambs2020),
data = gnambs2020)
# results
fit
#> Model: Fixed-Effects
#> Studies: 67
#> Samples: 20000 (4 chains)
#> Formula: ~ 1
#> ------
#> Location:
#> Post.mean Post.sd Cred.lb Cred.ub Rhat
#> (Intercept) -0.07 0.03 -0.12 -0.02 1.00
#>
#> ------
#> Date: Mon Jun 07 13:27:13 2021
This function works for any kind of model fitted with rma.
User-defined priors can be defined for each parameter. This is
accomplished with the assign_prior function. For example,
prior <-
c(assign_prior(param = "(Intercept)",
prior = "dnorm(0, pow(1, -2))",
dpar = "location"),
assign_prior(param = "n",
prior = "dnorm(0, 1)",
dpar = "location"),
assign_prior(param = "(Intercept)",
prior = "dnorm(-0.5, pow(1.5, -2))",
dpar = "scale", level = "two")
)
The pow(1.5, -2) allows for specifying the standard deviation, whereas
JAGS uses the precision, or the inverse of the variance, which can be
confusing (hence use pow(., -2)). This would then be used in the
prior argument of blsmeta.
For a sanity check, it is possible to verify that the priors made it to the correct parameters as follows
priors <- make_prior(yi = yi,
vi = vi,
mods = ~ n,
prior = prior,
es_id = es_id,
study_id = study_id,
data = gnambs2020)
priors
#> #location priors
#>
#> #(Intercept)
#> beta[1] ~ dnorm(0, pow(1, -2))
#>
#> #n
#> beta[2] ~ dnorm(0, 1)
#>
#> #scale level two priors
#>
#> #(Intercept)
#> gamma[1] ~ dnorm(-0.5, pow(1.5, -2))
#>
#> #scale level three priors
#> #Intercept
#> eta[1] ~ dnorm(-2, 1)
Notice that the “scale” priors are negative. At first, this may not seem correct because the scale refers to the variance. By default, however, a log-linear model is fitted to the variance components. As a result, the priors are on the log-scale which is very flexible on the one hand, but on the other, not very intuitive.
To better understand those priors, use sample_prior.
samps <- sample_prior(priors, iter = 50000)
Then you can plot the prior with
hist(exp(samps$gamma), xlim = c(0, 2), breaks = 10000).
There are a variety of things strategically left out of blsmeta. This is because I am hoping to be a professor. To this end, I am planning to tackle the following with students that join my lab:
-
Bayesian hypothesis testing (with the Bayes factor)
-
Visualization (with ggplot2)
-
Shiny Application
-
Website
Option 1 could be a first year project for a graduate student. I have several other ideas for meta-analysis (to help get the ball rolling, if interested), but these would not likely be a part of blsmeta. Option 2 will result in authorship on the software paper for blsmeta (written once students contribute). Options 3-4 will be ongoing for undergraduate students. For each option, students will learn valuable skills for industry (e.g., data science) and academia (e.g., pursing a PhD).
Note also this “lab” exists only in thought, and will hopefully come to fruition in the fall of 2022 or 2023.
While I intend to save these projects, working with underrepresented students (BIPOC, first-generation, students from developing countries, etc.) takes precedence. If you are interested in the above options, or have an idea of your own, please email (drwwilliams@ucdavis.edu) or DM on Twitter (https://twitter.com/wdonald_1985). I prefer Twitter.
Women of color and Native Americans are especially encouraged to reach out.