 Research article
 Open Access
 Published:
Using structural equation modeling for network metaanalysis
BMC Medical Research Methodology volume 17, Article number: 104 (2017)
Abstract
Background
Network metaanalysis overcomes the limitations of traditional pairwise metaanalysis by incorporating all available evidence into a general statistical framework for simultaneous comparisons of several treatments. Currently, network metaanalyses are undertaken either within the Bayesian hierarchical linear models or frequentist generalized linear mixed models. Structural equation modeling (SEM) is a statistical method originally developed for modeling causal relations among observed and latent variables. As random effect is explicitly modeled as a latent variable in SEM, it is very flexible for analysts to specify complex random effect structure and to make linear and nonlinear constraints on parameters. The aim of this article is to show how to undertake a network metaanalysis within the statistical framework of SEM.
Methods
We used an example dataset to demonstrate the standard fixed and random effect network metaanalysis models can be easily implemented in SEM. It contains results of 26 studies that directly compared three treatment groups A, B and C for prevention of first bleeding in patients with liver cirrhosis. We also showed that a new approach to network metaanalysis based on the technique of unrestricted weighted least squares (UWLS) method can also be undertaken using SEM.
Results
For both the fixed and random effect network metaanalysis, SEM yielded similar coefficients and confidence intervals to those reported in the previous literature. The point estimates of two UWLS models were identical to those in the fixed effect model but the confidence intervals were greater. This is consistent with results from the traditional pairwise metaanalyses. Comparing to UWLS model with common variance adjusted factor, UWLS model with unique variance adjusted factor has greater confidence intervals when the heterogeneity was larger in the pairwise comparison. The UWLS model with unique variance adjusted factor reflects the difference in heterogeneity within each comparison.
Conclusion
SEM provides a very flexible framework for univariate and multivariate metaanalysis, and its potential as a powerful tool for advanced metaanalysis is still to be explored.
Background
Metaanalysis is a very important methodological tool for evidence synthesis [1]. Traditional metaanalysis compares outcomes of two groups directly using data from studies in which the difference in the results between these two groups were tested. When more than two groups are to be compared, multiple pairwise metaanalyses need to be undertaken. When two of those groups have never been compared directly by any study, it becomes impossible to undertake the traditional metaanalysis for them. Even if each pair of those groups have been compared directly, different pairwise comparisons involve different studies using different evidence bases in their comparisons, and the results may not be consistent. For instance, in three pairwise comparisons for groups A, B, and C, pairwise metaanalyses may show A is better than B, B is better than C, but A is not better than C. The limitations of the traditional approach to comparing multiple groups have been documented extensively [2,3,4,5,6].
One recent development in metaanalysis methodology to resolve those issues is network metaanalysis for comparisons of multiple treatment groups [7,8,9,10,11,12,13,14,15]. Network metaanalysis incorporates all available evidence into a general statistical framework to yield consistent results for comparisons of all available treatments. Whilst the idea of indirect comparisons for treatments that had not been tested directly was first proposed in 1990s [16, 17], Lumley coined the term network metaanalysis and proposed a linear mixed model approach to comparisons of multiple treatments within the same statistical model [7]. Later, Lu and Ades developed a sophisticated Bayesian hierarchical model, providing a flexible statistical framework to take into account the complexity in the data structure within multiarm trials [11]. Their statistical approach, widely known as mixed treatments comparison or Bayesian network metaanalysis, has been a popular approach to comparisons of multiple treatments [5, 12, 18, 19].
Several recent articles looked further into the complexity in modeling multiple treatments comparisons with an attempt to implement Lu and Ades’s approach within the generalized linear mixed modelling framework [8, 12, 19, 20], to make network metaanalysis more accessible to clinicians and metaanalysts who are not familiar with Bayesian statistics. However, it is quite a challenging task to develop a formal statistical model for undertaking multiple treatment comparisons [9, 11, 18, 19, 21,22,23,24,25]. Two specific issues arise from implementing Lu & Ades’s Bayesian model into generalized linear mixed model: First, Lu and Ades’s approach uses the contrast between two treatment groups, such as log odds ratio or differences in means, as the outcome, and consequently, treatment contrasts between any pair of treatments within a multiarm study are not independent; their correlations therefore need to be taken into account in the model [26, 27]. Secondly, as the random effect structure for those treatment contrasts to address the heterogeneity becomes increasingly complex when the number of treatments involved in a network metaanalysis increases, specifying the random effect structure with treatment contrasts as the outcomes is not a simple task [12].
Structural equation modeling (SEM) is a statistical method originally developed for modeling causal relations among observed and latent variables. It can also be used to analyze longitudinal data and its results have been shown to be equivalent to those from multilevel modeling. Recent developments in SEM extend its application to multilevel data and noncontinuous dependent variables. Consequently, generalized linear mixed modeling can now be undertaken within SEM framework. As random effect is explicitly modeled as a latent variable in SEM, it is very flexible for analysts to specify complex random effect structure and to make linear and nonlinear constraints on parameters. Those advantages have been shown to be very useful for undertaking multivariate metaanalysis within SEM [28,29,30]. In our previous studies, we have shown how to undertake network metaanalysis by means of generalized linear mixed modelling [25, 31,32,33]. In this article, we attempt to develop a SEM approach to network metaanalysis based on the Lu & Ades’s model. This article is organized as follows: we first briefly review the Lu & Ades model and show how it can be implemented within generalized linear mixed models using treatment contrasts as the outcome. We then use an example to show how SEM can be used to undertake a network metaanalysis for the fixed and random effect network metaanalysis and how the weighting for each study can be taken into account. Finally, we demonstrate how a new approach to network metaanalysis, namely the unrestricted weight least squares (UWLS) method, can be implemented in SEM.
Methods
There are two models for network metaanalysis: fixed effect model assumes that treatment effects are common across studies, and random effect model assumes that treatment effects are heterogeneous across studies.
Fixed effect model for network metaanalysis
The fixed effect network metaanalysis for multiple treatment comparisons based on Lu and Ades’s approach can be specified as:
where treatments are coded as A, B, C,..., K, and K is the number of treatments to be compared within the network. The \( {\hat{y}}_{k\cdot j} \) is the expected value for y _{ k }._{ j }, which is the observed outcome for treatment k in study j. The g() is a link function for the model to transform the \( {\hat{y}}_{k\cdot j} \) to η _{ k }._{ j }, which is the expected value given by the model for arm k in study j, and μ _{ b }._{ j } is the baseline treatment effect in trial j. The difference between the other treatment k and treatment b in the same trial will be estimated by expressing them in terms of effects relative to the treatment A, which is the global baseline treatment within the whole network. Due to identification reason and its interpretation as the effect of treatment A compared to itself, d _{ AA } is fixed at 0, and Lu and Ades called d _{ AB } to d _{ Ak } the basic parameters. The advantage of expressing all treatment comparisons as the relations between basic parameters is that the number of pairwise comparisons to be estimated for a network metaanalysis involving k treatments is reduced to k – 1 for the fixed effect [34].
Random effect model for network metaanalysis
For the random effect network metaanalysis, d _{ bk } in Eq. (1) is replaced by δ _{ kb }._{ j }, the trialspecific effect of treatment k relative to trialspecific baseline treatment b, and the equation is given as:
These trialspecific effects are then drawn from a normal distribution: \( {\delta}_{bk\cdot j} \sim N\left({d}_{bk},{\tau}_{bk}^2\right) \). Then, d _{ bk } is expressed in terms of the basic parameters: d _{ bk } = d _{ Ak } − d _{ Ab }, with d _{ AA } being fixed at 0 [9, 12, 35]. Note that although the model in Eq. (2) uses data from each treatment arm of a study, it selects one treatment within each study as the trialspecific baseline treatment to estimate the treatment contrast between this baseline treatment and other treatments within the same study. When a study consists of more than two treatment arms, it will contribute more than one treatment contrast, and these treatment contrasts are not independent. Therefore, δ _{ bk }._{ j } will follow a multivariate normal distribution. For instance, suppose study 1 compares treatment B, C and D, and δ _{ BC } _{.1} and δ _{ BD } _{.1} in Eq. (2) for this study will then follow bivariate normal distribution:
Where cv is the covariance between \( {\tau}_{BC}^2 \) and \( {\tau}_{BD}^2 \). In the Lu & Ades approach, all the random effect variances are constrained to be equal, i.e. \( {\tau}_{BC}^2={\tau}_{BD}^2={\tau}^2 \), and cv is \( \frac{1}{2}{\tau}^2 \), i.e. the correlation between random effects is 0.5 [11].
Contrastbased model
To implement the treatment contrasts model in Eqs. (1) and (2) into general or generalized linear mixed model, we can either use the contrastbased approach [36], where treatment contrasts are derived from each study before undertaking network analysis, or use the armbased approach [25, 37], where data from each arm is used directly. For the contrastbased approach, the dependency of treatment contrasts within a multiarm trial needs to be taken into account in the model. As taking into account this dependency is not straightforward in most software packages, data transformation using some matrix algebra techniques can be used to create an independent dataset [25, 28, 31]. In the contrastbased fixed effect model shown in Eq. (1), effect size summary odds ratio or risk ratio, needs to be transformed into natural log odds ratio or risk ratio, which behaves approximately as a normal, and the model can now be written as:
where Δ _{ i }._{ j } is the effect size summary of the i ^{th} treatment contrast in study j such as difference in means or log odds ratio, t _{Ak } is the contrast coding dummy variable for treatment contrast A versus k for k = B to K, b _{AB} to b _{AK} are regression coefficients for treatment contrasts A versus B to A versus K in the network, and \( {\sigma}_{i\cdot j}^2 \) is the known variance of Δ _{ i }._{ j }. The vector b for regression coefficients can be obtained by [38]:
where the matrix X contains all the covariates t _{AB}, t _{AC},…, and t _{AK}, X ^{T} is the transposed X, Δ is the vector of Δ _{ i }._{ j }, and V ^{−1}is the inverse of the blockdiagonal matrix V:
The diagonal elements in V are V _{ j }, j = 1 to J, the variancecovariance matrix of v _{ i }._{ j } in Eq. (3). V _{ j } is a scalar if study j is a twoarm study and a matrix if study j is a multiarm study. Cheung proposed to use Cholesky decomposition to decompose V ^{−1} = LL ^{T}, where L is a lower triangular matrix and L ^{T} is the transpose of L [28]. We can premultiply X and Δ by L ^{T} to obtain the transformed matrix \( \overset{\sim }{\mathbf{X}}={\mathbf{L}}^{\mathbf{T}}\mathbf{X} \) and the transformed vector \( \overset{\sim }{\boldsymbol{\Delta}}={\mathbf{L}}^{\mathbf{T}}\boldsymbol{\Delta} \). So Eq. (4) can be rewritten as:
Under this transformation, the impact of v _{ i }._{ j } in Eq. (3) has been absorbed into \( \overset{\sim }{\mathbf{X}} \) and \( \overset{\sim }{\boldsymbol{\Delta}} \), so Eq. (3) can be rewritten as an ordinary least squares model:
where \( {\overset{\sim }{\varDelta}}_{i\cdot j} \) is the transformed Δ _{ i }._{ j }, and x _{Ak } is the transformed t _{Ak } in Eq. (3).
Example data: sclerotherapy
The example dataset contains results of 26 studies that directly compared three treatment groups A, B and C for prevention of first bleeding in patients with liver cirrhosis [39]: A was the control group, B was sclerotherapy, and C was the use of betablocker. The whole dataset can be found in the Additional file 1. Among the 26 study, two are threearm trials, and seven compared A to C and 17 compared A to B. Throughout the analysis in this article, treatment A was chosen as the global baseline treatment.
As the outcome is a binary variable, the difference in the outcome between any two treatments may be expressed as odds ratio or risk ratio, but to undertake a trialbased approach, we need to take a natural log transformation of odds ratio or risk ratio. Here, we used log odds ratio as the effect size measure. For the threearm trials, we calculated two treatment contrasts, A vs B and A vs C, and the covariance between the two correlated treatment contrasts is the variance of log odds ratio for treatment A. The regression model for the fixed effect network metaanalysis is therefore written as:
where lnOR _{ i }._{ j } is the i ^{th} log odds ratio for study j, \( {\sigma}_{i\cdot j}^2 \) is the variance of lnOR _{ i }._{ j }, t _{AB} is a dummy variable where treatment contrast for A versus B is denoted 1 and contrast for A versus C denoted 0, and t _{AC} a dummy variable where treatment contrast for A versus C is denoted 1 and contrast for A versus B denoted 0. Note that if there are trials that compared B to C, t _{AB} would coded −1 and t _{AC} coded 1 for those trials [25]. The regression coefficient b _{AB} and b _{AC} in Eq. (6) cannot be directly estimated in SEM, because lnOR _{ i }._{ j } are not independent in the threearm trials; but b _{AB} and b _{AC} can be obtained by transforming lnOR _{ i }._{ j }, t _{AB} and t _{AC} using the procedure described in the previous section. For the random effect model, b _{ AB . j } and b _{AC . j } in Eq. (6) are replaced with \( {b}_{AB. j}^{\ast } \) and \( {b}_{AC. j}^{\ast } \), which are assumed to follow a bivariate normal distribution:
where the β _{AB} and β _{AC} are the average treatment effect difference between A and B and between A and C, respectively; and τ ^{2} is the treatment effect variability across studies.
Contrastbased SEM network metaanalysis
SEM is a multivariate statistical analysis technique that is a combination of factor analysis and multiple regression analysis [40]. Many traditional statistical methods such as analysis of variance, regression analysis, and factor analysis can therefore be considered as special models of SEM. Traditional SEM requires that the outcome variables and the latent constructs have to be continuous, but with new development of SEM theory and software packages, these are no longer limitations of SEM. As a result, generalized linear mixed models and SEM can now be considered generalized latent variable models [41]. The main difference between SEM and generalized linear mixed models is that random effects are explicitly specified as latent variables in SEM and relationships between observed/latent variable are explicitly specified as causal or noncausal. A comprehensive overview of SEM is beyond the scope of this article, and readers can find an indepth discussion of applications of SEM to univariate and multivariate metaanalyses in a series of articles and a textbook [28,29,30, 42,43,44,45].
Although network metaanalysis can now be undertaken within the statistical framework of generalized linear mixed models, we feel integrating network metaanalysis into SEM framework has several advantages: first, network metaanalysis can be visualized in SEM, and this can be useful for understanding the complexity of the model, especially when analysts wish to look into the role of potential effect modifiers or moderators in the comparisons of multiple treatments by undertaking metaregression [46]. Secondly, SEM software packages are more flexible in making constraints on model parameters such as regression coefficients, variances and covariances, because random effects are explicitly modelled as latent variables. Thirdly, SEM is a primary research tool for social scientists, but they are less familiar with network metaanalysis, which is becoming more and more popular in biological and medical research. Therefore, integrating network metaanalysis into SEM framework will bring network metaanalysis to attentions of greater audiences [45].
UWLS for metaanalysis
Recently, a new approach has been proposed for metaanalysis, which differs from the standard fixed or random effect models [47, 48]. The standard fixed effect metaanalysis for pairwise comparisons is just weight least squares regression and can be written as:
where Δ _{ j } may be the log odds ratio or difference in means between two treatments, v _{ j } is the standard error of Δ _{ j } and \( {v}_j \sim N\left(0,{\sigma}_j^2\right) \), where \( {\sigma}_j^2 \) is the variance of Δ _{ j }. In the (UWLS approach, the variance of v _{ j } is in proportion to the variance of Δ _{ j }, i.e. \( {v}_j \sim N\left(0,\phi {\sigma}_j^2\right) \). The introduction of variance adjustment factor ϕ to Eq. (8) will not affect the point estimate for μ, but its standard error will be affected: when ϕ is larger than 1, the confidence interval for μ will be greater than that given by the standard fixed effect model, but in contrast, when ϕ is smaller than 1, the confidence interval for μ will become smaller. According to recent studies [47, 49], UWLS approach provides satisfactory estimates and confidence intervals that are comparable to random effects when there is no publication bias and identical to fixedeffect metaanalysis when there is no heterogeneity.
UWLS for network metaanalysis
In network metaanalysis, the numbers of studies involved in pairwise comparisons are usually quite different, and the degree of heterogeneity within each pairwise comparison also varies. Therefore, network metaanalysis usually uses random effect model to take into account the heterogeneity across the whole network. Currently, the Bayesian or nonBayesian network metaanalysis usually assumes a common variance for the random effect estimation; for instance, our analysis of the example data in the previous section assumed that the random effect variances for the comparisons between treatment A and B and between A and C are identical. This assumption effectively reduces the number of parameters to be estimated in the model, rendering it more likely to converge, and saves the computation time. However, it also makes a strong assumption about the distribution of heterogeneity within the network metaanalysis and sometimes may yield ambiguous results. For instance, suppose in a network metaanalysis involving treatment A, B, C, D and E, only one trial that compares A and E was found. If the heterogeneity is large in other parts of the network, the estimated common variance for random effect is likely to be large but the estimated confidence interval for AE comparison would become greater than that reported by the single trial, even if the evidence within the network is consistent. This is because the confidence interval for AE comparison reported by the random effect network metaanalysis is the one given under the assumption that AE comparison has the same degree of heterogeneity as other pairwise comparisons in the network.
The standard random effect network metaanalysis therefore gives rise to a few issues with regard to the assessment of inconsistency between direct and indirect evidence. For treatment contrasts with few headtohead trials, their confidence interval estimated by traditional pairwise metaanalysis is very likely to be smaller than that given by the random effect network metaanalysis assuming a common random effect variance. Consequently, methods for evaluation of inconsistency between direct and indirect evidence may yield different results under different assumptions with regard to the random effect variance [50, 51].
The UWLS approach provides an alternative way to address the heterogeneity. The parameter ϕ in UWLS approach can be interpreted from two perspectives: one is to view ϕ as the dispersion parameter to provide a correction to the known withstudy standard error \( {\sigma}_j^2 \). For a common ϕ, this can be implemented in most statistical packages. However, if ϕ is unique to different treatment contrasts, it will be far more straightforward to fit this type of models in SEM. The other way to interpret ϕ is to consider UWLS as a multiplicative random effect model, while the traditional random effect model is additive in the structure of random effect components. In other words, ϕ can be viewed as the random effect τ ^{2} in Eqs. (6) and (7), where the total variance is ϕ + σ ^{2}, but in UWLS the total variance is ϕσ ^{2}. Consequently, UWLS is to add ϕ into a fixed effect model, making it behave similarly to a random effect model, and a large \( \widehat{\phi} \) indicates large treatment effect heterogeneity.
For different pairwise comparisons within the network metaanalysis, we may estimate different ϕ in Eq. (8) for different pairwise comparisons. We now extend the UWLS approach to network metaanalysis involving treatment A, B, C, …, K with p treatment pairs:
The variable Δ_{ c . j } is the treatment contrast c, c = 1 to p, reported by study j, d _{Ak }, k = B to K, are the basic parameters for the comparison between A and k, \( {\sigma}_{c. j}^2 \) is the variance of Δ_{ c . j } and ϕ _{ c } is the variance adjustment factor for treatment contrast c within the network metaanalysis.
UWLS for SEM network metaanalysis
To implement such a model in SEM requires rearrangement of data. Using the example data for illustration, its UWLS model can be written as:
where Δ _{1 . j } is the log odds ratio reported by study j that compared treatment A to B and Δ _{2 . j } the log odds ratio for study j that compared treatment A to C; d _{AB} is the average treatment difference between A and B; d _{AC} is the average treatment difference between A and C; \( {\sigma}_{1. j}^2 \) is the variance of Δ _{1 . j }; \( {\sigma}_{2. j}^2 \) is the variance of Δ _{2 . j }; and ϕ _{1} and ϕ _{2} are the variance adjustment factors for treatment contrasts AB and AC, respectively.
We used SEM software package Mplus (version 7.11, Muthen & Muthen, Los Angeles, USA) to undertake all the analyses throughout our study, as Mplus is very flexible in making constraints on parameters estimation. All the data and Mplus codes in this article can be found in the Additional file 1.
Results
Contrastbased SEM network metaanalysis for example data
SEM fits simultaneously a group of regression equations, which specify the relationships between observed and latent variables. Latent variables in SEM represent some hidden constructs that cannot be observed or measured directly but have to be estimated from a group of observed (also known as manifest) variables. One special feature of SEM is that the statistical model can be visualized by using a path diagram, and most SEM software packages allow users to draw their path diagrams and undertake the analysis directly. In a path diagram, observed variables are in squares, while latent variables are in circles. A single arrow represents a prediction or causal relationship, e.g. X → Y dipicts that X predicts Y or X causes Y. A double arrow represents a correlation or covariance, e.g. X ↔ Y depicts that X and Y are correlated. Results show that the log odds ratio for treatment A and B is −0.485 (95% Confidence Interval [CI]: −0.717 to −0.254) and for A and C is −0.600 (95% CI: 0.932 to −0.268).
Figures 1 and 2 show the path diagrams for Eqs. (6) and (7) with fixed and random effects, respectively, demonstrating how to use the multilevel SEM to undertake the random effect network metaanalysis for example data. In the level1 model (the Withinlevel in Fig. 2), y is the transformed log odds ratio, and x _{AB} and x _{AC} are the transformed t _{AB} and t _{AC}, respectively. The filled circle on the arrow from x _{AB} to y represents random slope that is referred to as s _{1} in the level2 model (the Betweenlevel in Fig. 2). The filled circle on the arrow from x _{AC} to y represents random slope that is referred to as s _{2} in the level2 model. The variance of s _{1} and s _{2} is τ ^{2} in Eq. (7), and their covariance is constrained to be \( \frac{1}{2}{\tau}^2 \). Note that in Fig. 2 there is no random intercept, and the intercept of y is fixed at 0. The arrows from the variable in triangle to s _{1} and s _{2} indicate that the means of s _{1} and s _{2} are estimated, which give rise to β _{ AB } and β _{ AC } in Eq. (7). The variance of the residual error term e _{ y } is fixed at unity. Results show that the log odds ratio for treatment A and B is −0.585 (95% CI: 1.087 to −0.082) and for A and C is −0.711 (95% CI: 1.438 to 0.016).
UWLS for SEM network metaanalysis for example data
To estimate UWLS model with common variance adjustment factors ϕ _{1} = ϕ _{2} in Eq. (10), we only need to remove the constraint on the variance of e _{ y } in the fixed effect network metaanalysis model shown in Fig. 1. Table 1 showed results from Mplus for the fixed effect, random effect, and the two UWLS models. Results from Mplus show that ϕ is 3.563, and the log odds ratio for treatment A and B is −0.485 (95% CI: 0.922 to −0.049) and for A and C is −0.600 (95% CI: 1.227 to 0.027). The point estimates are identical to those in the fixed effect model but the confidence intervals are greater. To estimate the UWLS model with unique variance adjustment factors in Eq. (10), we need to create two residual error terms for y: one for studies reporting treatment contrasts AB and the other for those reporting treatment contrast AC. Figure 3 shows the path diagram, where y is the transformed log odds ratios and is regressed on x _{AB} and x _{AC}, which are the transformed variables t _{AB} and t _{AC}, respectively. Variables g _{AB} and g _{AC} are dummy variables for studies reporting treatment contrasts AB and AC, respectively. The filled circle on the arrow from g _{AB} to y represents random slope that is labelled as s _{1}, and the filled circle on the arrow from g _{AC} to y represents random slope that is labelled as s _{2}. The means of s _{1} and s _{2} are fixed at zero, and the variances of s _{1} and s _{2} are ϕ _{1} and ϕ _{2} in Eq. 8, respectively, with their covariance being fixed at zero. In this model, the residual error for y is split into two independent random variables s _{1} and s _{2}, and their variances are estimated separately. Results from Mplus show that the log odds ratio for treatment A and B is −0.484 (95% CI: 0.958 to −0.010) and for A and C is −0.600 (95% CI: 1.075 to −0.125). The point estimates are almost identical to those given by the fixed effect model, but the confidence intervals are greater. Compared to the confidence intervals reported by the UWLS model with common ϕ, the confidence interval for x _{AB} is greater but that for x _{AC} is smaller. This is because the variance adjustment factors ϕ _{1} and ϕ _{2} are 4.288 and 2.031, respectively, indicating a greater degree of heterogeneity within AB headtohead trials. This is consistent with results from the traditional pairwise metaanalyses in which the degree of heterogeneity in studies reporting treatment contrast AB is greater than that of studies reporting contrast AC.
Discussion
In this article, we demonstrate how to undertake network metaanalysis within the statistical framework of structural equation modeling. While issues such as the evaluation of inconsistency between direct and indirect evidence are important and can be integrated into SEM framework, it is beyond the scope of the present study to discuss these issues. Standard statistical software packages for generalized linear mixed modeling may be used to analyze the fixed and random effect models discussed in this article, but SEM software packages are more flexible in specifying complex covariance structure and imposing constraints on parameter estimation. Our results are very close to those reported in previous publications using the command mvmeta for the statistical software package Stata [25, 52]. The estimated random effect variance τ ^{2} is 0.877, which is slightly smaller than that given by mvmeta in Stata. Mplus only implements maximum likelihood estimation rather than restricted maximum likelihood estimation [53], but maximum likelihood estimation tends to underestimate the variance component in multilevel models [43]. However, metaSEM package in R has implemented restricted maximum likelihood estimation and can be used to fit multivariate metaanalysis and network metaanalysis [43, 53].
It is quite straightforward to implement UWLS approach to network metaanalysis with heteroscedastic errors in SEM. In the UWLS approach, the between and withinstudy heterogeneities are considered multiplicative (ϕσ ^{2}), and this is different from the traditional random effect model, where they are considered additive (σ ^{2} + τ ^{2}). The additive random effect assumes that the between and withinstudy heterogeneities are independent, while the multiplicative random effect assumes that the between and withinstudy heterogeneities are related. As withinstudy heterogeneity σ ^{2} is a known quantity, it can then be viewed as the weight for the betweenstudy heterogeneity ϕ. Two recent studies compare the performance of additive or multiplicative heterogeneity in traditional pairwise metaanalyses and found that results of these two models tend to agree but multiplicative model produces narrower confidence intervals [48, 49]. Further research is warranted to compare their performance in network metaanalyses.
Conclusion
SEM provides a useful framework for univariate and multivariate metaanalysis, and its potential as a powerful tool for advanced metaanalysis is still to be explored.
Abbreviations
 SEM:

Structural equation modeling
 UWLS:

Unrestricted weighted least squares
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Acknowledgements
We would like to thank the editor and three reviewers for their constructive suggestions and comments, which greatly improve our paper.
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All data generated or analysed during this study are included in the Additional file 1.
Funding
This project was partly funded by a grant from the Ministry of Science & Technology in Taiwan (grant number: MOST 103–2314  B  002  032  MY3).
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YKT conceived the ideas, obtained the data, undertook the analysis and wrote the draft of the manuscript. YCW helped with the statistical analysis, interpretation of results and the revisions of the manuscript. YKT takes the full responsibility for the integrity of this manuscript. All authors read and approved the final manuscript.
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Additional file
Additional file 1:
The dataset and Mplus scripts used for statistical analysis. (DOCX 110 kb)
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Tu, YK., Wu, YC. Using structural equation modeling for network metaanalysis. BMC Med Res Methodol 17, 104 (2017). https://doi.org/10.1186/s1287401703909
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DOI: https://doi.org/10.1186/s1287401703909
Keywords
 Randomized controlled trials
 Network metaanalysis
 Mixed treatments comparisons
 Structural equation modeling
 Generalized linear mixed models
 Multivariate metaanalysis