The metaSEM package
Mike Cheung
23 April, 2016
Introduction
The
metaSEMpackage provides functions to conducting univariate and multivariate meta-analysis using a structural equation modeling approach via the OpenMx package. It also implemented the two-stage structural equation modeling (TSSEM) approach to conducting fixed- and random-effects meta-analytic structural equation modeling (MASEM) on correlation/covariance matrices.- The
metaSEMpackage is based on the following papers:- Cheung, M.W.-L. (2008). A model for integrating fixed-, random-, and mixed-effects meta-analyses into structural equation modeling. Psychological Methods, 13, 182-202.
- Cheung, M.W.-L. (2009). Constructing approximate confidence intervals for parameters with structural equation models. Structural Equation Modeling, 16, 267–294.
- Cheung, M.W.-L. (2010). Fixed-effects meta-analyses as multiple-group structural equation models. Structural Equation Modeling, 17, 481-509.
- Cheung, M.W.-L. (2013). Implementing restricted maximum likelihood estimation in structural equation models. Structural Equation Modeling, 20, 157-167.
- Cheung, M.W.-L. (2013). Multivariate meta-analysis as structural equation models. Structural Equation Modeling, 20, 429-454.
- Cheung, M.W.-L. (2014). MetaSEM: an R package for meta-analysis using structural equation modeling. Frontiers in Psychology, 5 (1521).
- Cheung, M.W.-L. (2014). Fixed- and random-effects meta-analytic structural equation modeling: Examples and analyses in R. Behavior Research Methods, 46, 29-40.
- Cheung, M.W.-L. (2014). Modeling dependent effect sizes with three-level meta-analyses: A structural equation modeling approach. Psychological Methods, 19, 211-229.
- Cheung, M.W.-L. (2015). Meta-Analysis: A Structural Equation Modeling Approach. Wiley.
- Cheung, M.W.-L., & Chan, W. (2004). Testing dependent correlation coefficients via structural equation modeling. Organizational Research Methods, 7, 206–223.
- Cheung, M.W.-L., & Chan, W. (2005). Meta-analytic structural equation modeling: A two-stage approach. Psychological Methods, 10, 40-64.
- Cheung, M.W.-L., & Chan, W. (2009). A two-stage approach to synthesizing covariance matrices in meta-analytic structural equation modeling. Structural Equation Modeling, 6, 28-53.
Examples
Univariate meta-analysis
Random-effects model
## Load the library
library(metaSEM)
## Show the first few studies of the data set
head(Becker83) study di vi percentage items
1 1 -0.33 0.03 25 2
2 2 0.07 0.03 25 2
3 3 -0.30 0.02 50 2
4 4 0.35 0.02 100 38
5 5 0.69 0.07 100 30
6 6 0.81 0.22 100 45## Random-effects meta-analysis with ML
summary( meta(y=di, v=vi, data=Becker83) )
Call:
meta(y = di, v = vi, data = Becker83)
95% confidence intervals: z statistic approximation
Coefficients:
Estimate Std.Error lbound ubound z value Pr(>|z|)
Intercept1 0.174734 0.113378 -0.047482 0.396950 1.5412 0.1233
Tau2_1_1 0.077376 0.054108 -0.028674 0.183426 1.4300 0.1527
Q statistic on the homogeneity of effect sizes: 30.64949
Degrees of freedom of the Q statistic: 9
P value of the Q statistic: 0.0003399239
Heterogeneity indices (based on the estimated Tau2):
Estimate
Intercept1: I2 (Q statistic) 0.6718
Number of studies (or clusters): 10
Number of observed statistics: 10
Number of estimated parameters: 2
Degrees of freedom: 8
-2 log likelihood: 7.928307
OpenMx status1: 0 ("0" or "1": The optimization is considered fine.
Other values may indicate problems.)Fixed-effects model
## Fixed-effects meta-analysis by fixiing the heterogeneity variance at 0
summary( meta(y=di, v=vi, data=Becker83, RE.constraints=0) )
Call:
meta(y = di, v = vi, data = Becker83, RE.constraints = 0)
95% confidence intervals: z statistic approximation
Coefficients:
Estimate Std.Error lbound ubound z value Pr(>|z|)
Intercept1 0.100640 0.060510 -0.017957 0.219237 1.6632 0.09627 .
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Q statistic on the homogeneity of effect sizes: 30.64949
Degrees of freedom of the Q statistic: 9
P value of the Q statistic: 0.0003399239
Heterogeneity indices (based on the estimated Tau2):
Estimate
Intercept1: I2 (Q statistic) 0
Number of studies (or clusters): 10
Number of observed statistics: 10
Number of estimated parameters: 1
Degrees of freedom: 9
-2 log likelihood: 17.86043
OpenMx status1: 0 ("0" or "1": The optimization is considered fine.
Other values may indicate problems.)Mixed-effects model
## Mixed-effects meta-analysis with "log(items)" as the predictor
summary( meta(y=di, v=vi, x=log(items), data=Becker83) )
Call:
meta(y = di, v = vi, x = log(items), data = Becker83)
95% confidence intervals: z statistic approximation
Coefficients:
Estimate Std.Error lbound ubound z value
Intercept1 -3.2015e-01 1.0981e-01 -5.3539e-01 -1.0492e-01 -2.9154
Slope1_1 2.1088e-01 4.5084e-02 1.2251e-01 2.9924e-01 4.6774
Tau2_1_1 1.0000e-10 2.0095e-02 -3.9386e-02 3.9386e-02 0.0000
Pr(>|z|)
Intercept1 0.003552 **
Slope1_1 2.905e-06 ***
Tau2_1_1 1.000000
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Q statistic on the homogeneity of effect sizes: 30.64949
Degrees of freedom of the Q statistic: 9
P value of the Q statistic: 0.0003399239
Explained variances (R2):
y1
Tau2 (no predictor) 0.0774
Tau2 (with predictors) 0.0000
R2 1.0000
Number of studies (or clusters): 10
Number of observed statistics: 10
Number of estimated parameters: 3
Degrees of freedom: 7
-2 log likelihood: -4.208024
OpenMx status1: 0 ("0" or "1": The optimization is considered fine.
Other values may indicate problems.)Multivariate meta-analysis
Random-effects model
## Show the data set
Berkey98 trial pub_year no_of_patients PD AL var_PD cov_PD_AL var_AL
1 1 1983 14 0.47 -0.32 0.0075 0.0030 0.0077
2 2 1982 15 0.20 -0.60 0.0057 0.0009 0.0008
3 3 1979 78 0.40 -0.12 0.0021 0.0007 0.0014
4 4 1987 89 0.26 -0.31 0.0029 0.0009 0.0015
5 5 1988 16 0.56 -0.39 0.0148 0.0072 0.0304## Multivariate meta-analysis with a random-effects model
mult1 <- meta(y=cbind(PD, AL), v=cbind(var_PD, cov_PD_AL, var_AL), data=Berkey98)
summary(mult1)
Call:
meta(y = cbind(PD, AL), v = cbind(var_PD, cov_PD_AL, var_AL),
data = Berkey98)
95% confidence intervals: z statistic approximation
Coefficients:
Estimate Std.Error lbound ubound z value Pr(>|z|)
Intercept1 0.3448392 0.0536312 0.2397239 0.4499544 6.4298 1.278e-10
Intercept2 -0.3379381 0.0812480 -0.4971812 -0.1786951 -4.1593 3.192e-05
Tau2_1_1 0.0070020 0.0090497 -0.0107351 0.0247391 0.7737 0.4391
Tau2_2_1 0.0094607 0.0099698 -0.0100797 0.0290010 0.9489 0.3427
Tau2_2_2 0.0261445 0.0177409 -0.0086270 0.0609161 1.4737 0.1406
Intercept1 ***
Intercept2 ***
Tau2_1_1
Tau2_2_1
Tau2_2_2
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Q statistic on the homogeneity of effect sizes: 128.2267
Degrees of freedom of the Q statistic: 8
P value of the Q statistic: 0
Heterogeneity indices (based on the estimated Tau2):
Estimate
Intercept1: I2 (Q statistic) 0.6021
Intercept2: I2 (Q statistic) 0.9250
Number of studies (or clusters): 5
Number of observed statistics: 10
Number of estimated parameters: 5
Degrees of freedom: 5
-2 log likelihood: -11.68131
OpenMx status1: 0 ("0" or "1": The optimization is considered fine.
Other values may indicate problems.)## Plot the effect sizes
plot(mult1)
## Plot the effect sizes with the forest plots
## Load the library for forest plots
library("metafor")
## Create extra panels for the forest plots
plot(mult1, diag.panel=TRUE, main="Multivariate meta-analysis",
axis.label=c("PD", "AL"))
## Forest plot for PD
forest( rma(yi=PD, vi=var_PD, data=Berkey98) )
title("Forest plot of PD")
## Forest plot for AL
forest( rma(yi=AL, vi=var_AL, data=Berkey98) )
title("Forest plot of AL")
Fixed-effects model
## Fixed-effects meta-analysis by fixiing the heterogeneity variance component at
## a 2x2 matrix of 0.
summary( meta(y=cbind(PD, AL), v=cbind(var_PD, cov_PD_AL, var_AL), data=Berkey98,
RE.constraints=matrix(0, nrow=2, ncol=2)) )
Call:
meta(y = cbind(PD, AL), v = cbind(var_PD, cov_PD_AL, var_AL),
data = Berkey98, RE.constraints = matrix(0, nrow = 2, ncol = 2))
95% confidence intervals: z statistic approximation
Coefficients:
Estimate Std.Error lbound ubound z value Pr(>|z|)
Intercept1 0.307219 0.028575 0.251212 0.363225 10.751 < 2.2e-16 ***
Intercept2 -0.394377 0.018649 -0.430929 -0.357825 -21.147 < 2.2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Q statistic on the homogeneity of effect sizes: 128.2267
Degrees of freedom of the Q statistic: 8
P value of the Q statistic: 0
Heterogeneity indices (based on the estimated Tau2):
Estimate
Intercept1: I2 (Q statistic) 0
Intercept2: I2 (Q statistic) 0
Number of studies (or clusters): 5
Number of observed statistics: 10
Number of estimated parameters: 2
Degrees of freedom: 8
-2 log likelihood: 90.88326
OpenMx status1: 0 ("0" or "1": The optimization is considered fine.
Other values may indicate problems.)Mixed-effects model
## Multivariate meta-analysis with "publication year-1979" as a predictor
summary( meta(y=cbind(PD, AL), v=cbind(var_PD, cov_PD_AL, var_AL), data=Berkey98,
x=scale(pub_year, center=1979)) )
Call:
meta(y = cbind(PD, AL), v = cbind(var_PD, cov_PD_AL, var_AL),
x = scale(pub_year, center = 1979), data = Berkey98)
95% confidence intervals: z statistic approximation
Coefficients:
Estimate Std.Error lbound ubound z value Pr(>|z|)
Intercept1 0.3440001 0.0857659 0.1759020 0.5120982 4.0109 6.048e-05
Intercept2 -0.2918175 0.1312797 -0.5491209 -0.0345140 -2.2229 0.02622
Slope1_1 0.0063540 0.1078235 -0.2049762 0.2176842 0.0589 0.95301
Slope2_1 -0.0705888 0.1620966 -0.3882922 0.2471147 -0.4355 0.66322
Tau2_1_1 0.0080405 0.0101206 -0.0117955 0.0278766 0.7945 0.42692
Tau2_2_1 0.0093413 0.0105515 -0.0113392 0.0300218 0.8853 0.37599
Tau2_2_2 0.0250135 0.0170788 -0.0084603 0.0584873 1.4646 0.14303
Intercept1 ***
Intercept2 *
Slope1_1
Slope2_1
Tau2_1_1
Tau2_2_1
Tau2_2_2
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Q statistic on the homogeneity of effect sizes: 128.2267
Degrees of freedom of the Q statistic: 8
P value of the Q statistic: 0
Explained variances (R2):
y1 y2
Tau2 (no predictor) 0.0070020 0.0261
Tau2 (with predictors) 0.0080405 0.0250
R2 0.0000000 0.0433
Number of studies (or clusters): 5
Number of observed statistics: 10
Number of estimated parameters: 7
Degrees of freedom: 3
-2 log likelihood: -12.00859
OpenMx status1: 0 ("0" or "1": The optimization is considered fine.
Other values may indicate problems.)Installation and help
- First of most important, you need R to run it. You can install the
metaSEMpackage by running the following command inside anRsession:
install.packages("metaSEM")- The developmental version can be installed from GitHub by running:
library(devtools)
install_github("mikewlcheung/metasem")Potential issues with the OpenMx available at CRAN
- The OpenMx available at CRAN includes only the open source
SLSQPoptimizer. - If the
SLSQPoptimizer does not work well for you, e.g., there are many error codes, you may try to rerun the analysis. For example,
random1 <- meta(y=yi, v=vi, data=Hox02)
random1 <- rerun(random1)
summary(random1)- If you still prefer the non-free
NPSOLoptimizer, you may install it from the OpenMx website and call it by running:
mxOption(NULL, "Default optimizer", "NPSOL")Help
- Reference manual
- Vignette
- If you need help,
OpenMx discussion forum: A discussion forum for the
metaSEMpackage inOpenMx. You may post technical questions related tometaSEMthere. Please include information on the session. It will be helpful if you can include a reproducible example. You may save a copy of your data, saymy.df, and attach the content ofmyData.Rin the post by usingsessionInfo() dump(c("my.df"), file="myData.R")Meta-analysis resources: A discussion forum related to the conceptual issues of meta-analytic structural equation modeling.
The files are based on the following versions of R and R packages:
sessionInfo()R version 3.2.5 (2016-04-14)
Platform: x86_64-pc-linux-gnu (64-bit)
Running under: Ubuntu 14.04.3 LTS
locale:
[1] LC_CTYPE=en_SG.UTF-8 LC_NUMERIC=C
[3] LC_TIME=en_SG.UTF-8 LC_COLLATE=en_SG.UTF-8
[5] LC_MONETARY=en_SG.UTF-8 LC_MESSAGES=en_SG.UTF-8
[7] LC_PAPER=en_SG.UTF-8 LC_NAME=C
[9] LC_ADDRESS=C LC_TELEPHONE=C
[11] LC_MEASUREMENT=en_SG.UTF-8 LC_IDENTIFICATION=C
attached base packages:
[1] parallel stats graphics grDevices utils datasets methods
[8] base
other attached packages:
[1] metafor_1.9-8 metaSEM_0.9.8 OpenMx_2.5.2 Rcpp_0.12.4
[5] Matrix_1.2-5 MASS_7.3-45 digest_0.6.9 knitr_1.12.3
[9] rmarkdown_0.9.5
loaded via a namespace (and not attached):
[1] lattice_0.20-33 grid_3.2.5 formatR_1.3 magrittr_1.5
[5] ellipse_0.3-8 evaluate_0.8.3 stringi_1.0-1 tools_3.2.5
[9] stringr_1.0.0 yaml_2.1.13 htmltools_0.3.5