Meta-Analysis R Package for SEM (MetaSEM

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MetaSEM: An R Package for Meta-Analysis using Structural Equation
Modelling
Dr. Nancy Agens, Head,
Technical Operations, Pubrica
In Brief
SEM are used meta-analytical model
formulated for conducting Meta-analysis
which is used to analyse structural
relationships. SEM can be univariate,
multivariate, and three-level meta-
analysis. Structural equation model
(SEM) in general optimized and fit by
using OpenMx package. The routine
analysis of batch mode either
interactively or noninteractively can be
analysed by R package. Using the
graphical interface like R studio is
convenient method for users to interfere
the analysis.
I. INTRODUCTION
A methodological tool used for
comparing the data of the studies obtained
between two groups was Meta- analysis.
Multiple pairwise meta-analyses were used
for comparing the data obtained from more
than two groups. In recent, network meta-
analysis was developed for analysing the
data obtained from multiple group. SEM is
one such method used for analysing
longitudinal data. A collection of functions
via., R statistical platform accessed by
OpenMx package for conducting meta-
analysis using SEM is the
metaSEM package. Correlation and
covariance matrices can be obtained using
the SEM in this approach. Meta-analysis
can be conducted by various unrelated
programs for performing research in
scientific and social studies. But the data
analysis in computational studies using R
is considered as a popular source for
conducting statistical analysis. R is an
open source platform compromises several
packages. Among which MetaSEM along
with OpenMx package is used for
conducting meta-analysis in univariant,
Multivariant and three level strategies.
MetaSEM also uses two stage structural
equation modelling (TSSEM) for
correlating and studying covariance
matrices in meta-analytic structural
modelling.
II. SEM (STRUCTURAL EQUATION
MODELLING)
The relation among measured
variables and latent constructs in the aspect
of structural can be analysed using SEM. It
also possesses the techniques like path and
factor analysis, regression and latent
growth curve modelling for solving linear
equations. It is a single analysis technique
used for estimating interrelated
dependence and multiple factors. Endo and
exogenous variables can be used
simultaneously in SEM. The approach of
analysing relation between correlated
traits, makes SEM a popular tool for
analysing complex genetic traits. It also
associates multiple variance and
correlation between genetic phenotypes of
interest. There are various software for
analysing independent observations using
SEM framework.
III. STRUCTURAL EQUATION
MODELLING BASED META-
ANALYSIS
A hypothesis can be tested and fit
with the multivariate technique using the
SEM model. It is used for continuously
analyse the multivariate distribution using
the variable number observed. It is
postulated that the model for the first
which includes the vector of parameters

that can be regression coefficients, error
variances, factor loadings, and factor
variances. The model is:
μ=μ(θ) and Σ=Σ(θ)
where μ and Σ are the vector of
mean population and covariance matrix.
The most common method for estimating
method in SEM is Maximum likelihood
(ML) estimation method. The −2*log-
likelihood (−2LL) for the ith case is,
−2LLi(θ;yi)ML=pilog(2π)+log|Σi(θ)| + (yi
−μi(θ))⊤Σi(θ)−1(yi−μi(θ))
here filtered variable numerical is
represented by pi, the mean vector implied
model are ith case, μi(θ) and Σi(θ) and
covariance matric implied in the model can
be represented as ith case, respectively.
The i in subscript represents the variation
in mean vector and covariance matrix
implied in the mean implied. It has the
potential for handling the incomplete data
in the log-likelihood function using
estimation method used for maximum
likelihood (M. W.-L. Cheung, 2015).
The minimized data of total sum of
all −2LLi represents the estimation of
parameters. The inverse of Hessian matrix
gives the details of estimates of parameter
sampling covariance matrix of asymptotic
sampling while being convergent. The
square root element diagonal to covariance
matrix of covariance matrix can be
considered as the standard error. Nested
model can be compared using the statistics
of the likelihood ratio. The model fit and
the significance of individual parameters
can be tested (M. Cheung, 2015).
MetaSEM can use various model for
analysing the data.
IV. UNIVARIATE FIXED-EFFECTS
MODEL
One effect size yi used meta-
analysis in the ith study, yi can be any
effect size, such as the odds ratio, raw
mean difference, standardized mean
difference, correlation coefficient, or its
Fisher's z transformed score. yi can be
assumed to be normally distributed with a
variance of vi while the sample sizes in the
primary studies are reasonably large (M.
Cheung, 2015).
Fig. 1 Univariate Fixed-Effects Model
V. UNIVARIATE RANDOM-EFFECTS
MODEL
The own specific study effect can
be selected for the random-effects model
in case of the variation in the expected
population size. The model for the ith
study is:
yi=βR+ui+ei,
where βR is the average population
effect under the random-effects model, and
Var(ui) = τ2
is the heterogeneity variance
that has to be estimated. To fit the model

in SEM, we may consider the following
model implied moments:
μi(θ)=βR and Σi(θ)=τ2+vi.
Fig. 2 Univariate Random-Effects Model
VI. UNIVARIATE MIXED-EFFECTS
MODEL
The mixed-effects meta-analysis
extends the random-effects meta-analysis
by using study characteristics as
predictors. Assuming that xi is an (m + 1)
× 1 vector of predictors including a
constant of 1 where m is the number
predictors in the ith study, the mixed-
effects model is:
yi=x⊤iβ+ui+ei
where β is a a (m + 1) × 1 vector of
regression coefficients including the
intercept. To fit the model in SEM, we
may use the following model implied
conditional mean and variance:
μi(θ∣∣xi)=x⊤iβ and Σi(θ∣∣xi)=τ2+vi.
Fig. 3 Univariate Mixed-Effects Model

VII. MULTIVARIATE META-
ANALYSIS
To fit the multivariate mixed-
effects meta-analysis in SEM, we use the
following model implied conditional mean
vector and covariance matrix.
Summarizing the primary outcome using
the single effect size inn complicated
research question is not sufficient. Hence,
nmltiple effect sizes are required to
quantify the effect of the studies. A total
of p effect sizes was assumed
with m predictors in k studies. In different
effect sizes, pi can be assumed as effect
sizes in the ith study. The model for the
multivariate mixed-effects meta-analysis
in the ith study is:
yi=Bixi+Ziui+ei,
where yi is a pi × 1 vector of effect
sizes, Bi is a pi × (m + 1) matrix of
regression coefficients including the
intercepts, xi is a (m + 1) × 1 matrix of
predictors including 1 in the first
column, Zi is a pi × p filter matrix selecting
the effect sizes that are present, ui is a p ×
1 study-specific random effects, and ei is
a pi × 1 sampling error.
We assume that Var(ei) = Vi is
known in the ith study and that Var(ui)
= T2
is the variance component of the
between-study heterogeneity that has to be
estimated. Since xi is a design matrix,
missing value is not allowed in xi. When
there are missing values in xi, the whole
study will be deleted before the analysis is
conducted M. Cheung,(2013). The
−2LL of the above model is:
−2LLi(B,T2;yi)ML= pi∗log(2π)+log∣∣ZiT2
Z⊤i+Vi∣∣+ (yi−Bixi)⊤(ZiT2Z⊤i+Vi)−1(yi
−Bixi).
Fig. 4 Multivariate Meta-Analysis

REFERENCES
[1] Cheung, M. (2013). Multivariate Meta-Analysis as
Structural Equation Models. Structural Equation
Modeling: A Multidisciplinary Journal, 20(3),
429–454.
https://doi.org/10.1080/10705511.2013.797827
[2] Cheung, M. (2015). metaSEM: an R package for
meta-analysis using structural equation modeling.
Frontiers in Psychology, 5.
https://doi.org/10.3389/fpsyg.2014.01521
[3] Cheung, M. W.-L. (2015). metaSEM: An R package
for meta-analysis using structural equation
modeling. Frontiers in Psychology, 5, 1521.
Retrieved from
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4
283449/#__ffn_sectitle

Meta-Analysis R Package for SEM (MetaSEM

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