14.1 The Idea behind Meta-Analytic SEM

What is Structural Equation Modeling?

Structural Equation Modeling (SEM) is a statistical technique to model and test hypotheses about the relationship of manifest (observed) and (usually) latent (unobserved or unobservable) variables (Kline 2015). Latent variables, as said, are either not observed or observable (personality, for example, is a latent construct which can only be measured indirectly, for example through different items in a questionnaire). In SEM, the assumed relationship between the manifest and latent variables (the “structure”) can be modeled using the manifest, measured variables while taking their measurement error into account.

SEM analysis is somewhat different to “conventional” statistical tests (e.g., \(t\)-tests). Usually, for example in a \(t\)-test, the researcher tests against a null hypthesis, such as \(H_0:\mu_1 = \mu_2\) (where \(\mu_1\) and \(\mu_2\) are the means of two groups). The researchers “aims” to reject the null hypothesis to conclude that the two groups differ. In SEM, a specific model is proposed beforehand instead, and the researcher “aims” to accept this model if the goodness of fit is sufficient (Cheung 2015a).

14.1.1 Model Specification

Usually, SEM are specified and represented mathematically as a series of matrices. You can think of a matrix as a simple table containing rows and columns, much like a data.frame object in R (in fact, most data.frames can be easily converted to a matrix using the as.matrix() function). Visually, SEM can be represented as path diagrams. Such path diagrams are often very intuitive, so we will start specifying a SEM using this approach first, and then move on to the matrix notation.

14.1.1.1 Path Diagrams

Path diagrams represent the mathematical model of our SEM graphically. There is no full consensus on how path diagrams should be drawn, yet there are a few conventions. Here are the main components of path diagrams, and how they are represented.

Symbol Name Description
\(\square\) Rectangle Manifest/observed variables
\(\circ\) Circle Latent/unobserved variables
\(\triangle\) Triangle Intercept (fixed vector of 1s)
\(\rightarrow\) Arrow Prediction. The variable at the start of the arrow predicts the variable at the end of the arrow: Predictor \(\rightarrow\) Target.
\(\leftrightarrow\) Double Arrow (Co-)Variance. If a double arrow connects two variables (rectangles/circles), it signifies the covariance/correlation between the two variables. If a double arrow forms a loop on one single variable, it signifies the variance of the variable.

As an illustration, let us create path diagram for a simple linear (non-meta-analytic) regression model, in which we want to predict \(y\) with \(x\). The model formula looks like this:

\[y_i = \beta_0 + \beta_1x_i + e_i\] In this model, \(x_i\) and \(y_i\) are the observed variables. There are no unobserved variables. The parameter \(\mu_x\) is the population mean of \(x\), while the population mean of \(y\) is the regression intercept, \(\beta_0\). The variance of our observed data for \(x\) is \(\sigma^{2}_{x}\). Provided that \(x\) is not a perfect predictor of \(y\), there will be some amount of error variance \(\sigma^{2}_{e_y}\) in our predictions associated with \(y\). There are two regression coefficients: \(\beta_0\), the intercept, and \(\beta_1\), the slope coefficient of \(x\).

Using these components, we can build a graphical model for a simple univariate linear regression:

We can also use this graphical model as a starting point to reassemble the regression model equation. From the model, we can infer that \(y\) is influenced by two components: \(x \times \beta_1\) and \(1 \times \beta_0\). If we add these two parts together, we again arrive at the formula for \(\hat{y}\) from before.

14.1.1.2 Matrix Representation

There are several common ways to represent SEM mathematically through matrices (Jöreskog and Sörbom 2006; Muthén and Muthén 2012; McArdle and McDonald 1984). Here, we will focus on the Reticular Action Model formulation, or RAM (McArdle and McDonald 1984), because this formulation is used by the metaSEM package we will be introducing later on. In the RAM, four matrices are specified: \(\mathbf{F}\), \(\mathbf{A}\), \(\mathbf{S}\) and \(\mathbf{M}\). Because the \(\mathbf{M}\) matrix is not necessary to fit the meta-analytic SEM we will present later, we omit it here (see Cheung (2015a) for a more extensive introduction). We will now specify the remaining three matrices for our linear regression from before. The three matrices all have the same number of rows and columns, corresponding with the (manifest and latent) variables we have in our model: \(x\) and \(y\). The generic structure in our regression example for all matrices therefore looks like this:

The \(\mathbf{A}\) Matrix: Single Arrows

The \(\mathbf{A}\) matrix represents the asymmetrical (single) arrows in our path model. The way to fill this matrix is to search for the matrix column entry of the variable in which the arrow starts (\(x\)), and then for the matrix row entry of the variable in which the arrow ends (\(y\)). We put the value of our arrow, \(\beta_1\), where the selected column and row intersect in the matrix (\(i_{y,x}\)). Given that there are no more paths between the variables in our matrix, we fill all other fields with \(0\). The \(\mathbf{A}\) matrix for our example therefore looks like this:

The \(\mathbf{S}\) Matrix: Double Arrows/Variances

The \(\mathbf{S}\) matrix represents the (co)variances we want to estimate for the included variables. For \(x\), our predictor, we want to estimate the variance \(\sigma^{2}_{x}\). For our estimated regression target \(\hat{y}\), we want to know the error variance \(\sigma^{2}_{e_y}\). We therefore specify \(\mathbf{S}\) like this:

The \(\mathbf{F}\) Matrix: Observed Variables

The \(\mathbf{F}\) matrix allows us to specify the observed (vs. latent) variables in our model. To specify that a variable has been observed, we simply insert \(1\) in the respective diagonal field of the matrix. Given that both \(x\) and \(y\) are observed in our model, we put \(1\) in all diagonal fields of the matrix:

Once these matrices are set, it is possible to estimate the parameters in our SEM, and to derive how good the specified model fits the data using matrix algebra and Maximum-Likelihood estimation. We omit how these steps are performed here. If you are interested in understanding the details behind this step, you can have a look at Cheung (2015a), McArdle and McDonald (1984), or this blog post.

14.1.2 Meta-Analysis from a SEM perspective

We will now combine our knowledge about meta-analysis models and SEM to formulate the random/fixed-effect model as a structural equation model (Cheung 2008).

To begin, let us return to the formula of the random-effects model first. Previously, we already described that the random-effects model follows a multilevel structure, which looks like this:

Level 1:

\[ \hat\theta_k = \theta_k + \epsilon_k \]

Level 2:

\[ \theta_k = \mu + \zeta_k \]

On the first level, we assume that the effect size \(\hat{\theta}_k\) we observe for some study \(k\) in our meta-analysis is an estimator for the true effect size of \(k\), \(\theta_k\). The observed effect size \(\hat{\theta}_k\) deviates from the true effect size \(\theta_k\) because of the sampling error \(\epsilon_k\), the variance of which is \(Var(\epsilon_k)=v_i\).

In a random-effects model, we assume that even the true effect size for \(k\) is only drawn from a “super-population” of true effect sizes at level 2. The mean of this “super-population” \(\mu\) is what we want to estimate as the pooled effect in a random-effects model, along with the variance of the “super-population”, \(Var(\zeta_k) = \tau^2\): the between-study heterogeneity. The fixed-effect model is only a special case of the random-effects model where we assume that \(\tau^2\) is zero.

It is quite straightforward to represent this model as a SEM graph if we use the variables on level 1 as latent variables to “explain” how the effect sizes we observed came into being (Cheung 2015a):

In this graphical model, it becomes clear that the observed effect size \(\hat{\theta}_k\) in the \(k\)th study is “influenced” by two arms: by the sampling error \(\epsilon_k\) with variance \(v_k\), and the true effect size \(\theta_k\) with variance \(\tau^2\).

14.1.3 The Two-Stage Meta-Analytic SEM approach

Above, we defined the (random-effects) meta-analysis model from a SEM perspective. Although this is interesting from a theoretical standpoint, the model above is still not more or less capable than the meta-analysis techniques we learned before: it simply pools effect sizes assuming a random-effects model.

To really use the versatility of meta-analytic SEM, an approach involving two steps is required (Tang and Cheung 2016; Cheung 2015a). In Two-Stage Structural Equation Modeling (TSSEM), we first pool the effect sizes from each study (usually correlations between variables we want to use for modeling). This allows for evaluating the heterogeneity of the pooled effect sizes, and if a random-effects model or subgroup analyses should be used. Using the maximum-likelihood-based approach used by the metaSEM package we will introduce in the following, even studies with missing data can be included.

In the second step, weighted least squares (WLS) estimation is used to fit the structural equation model we specified. The function for the specified model \(\mathbf{\rho}(\mathbf{\theta})\) is (Cheung and Chan 2009; Cheung 2015a):

\[ F_{WLS}(\mathbf{\theta}) = (\mathbf{r} - \mathbf{\rho}(\mathbf{\theta}))^\top \mathbf{V}^{-1} (\mathbf{r} - \mathbf{\rho}(\mathbf{\theta}))\]

Where \(\mathbf{r}\) is a transformation of the pooled correlation matrix. The important part in this formula is \(\mathbf{V}^{-1}\), the matrix containing the covariances of \(\mathbf{r}\), from which the inverse is taken. This approach is quite similar to the inverse-variance principle (see Chapter 4) traditionally used in meta-analysis. It gives effects with lower variance (i.e., greater precision/\(N\)) a larger weight in the estimation process. This is also a good way to account for the uncertainty in our estimates which may have been introduced by missing data. Importantly, the formula for this second step is the same whether we assume a random or fixed-effect model, because the between study-heterogeneity, if existant, is already taken care of in step 1.

References

Kline, Rex B. 2015. Principles and Practice of Structural Equation Modeling. Guilford publications.

Cheung, Mike W-L. 2015a. Meta-Analysis: A Structural Equation Modeling Approach. John Wiley & Sons.

Jöreskog, Karl G, and Dag Sörbom. 2006. “LISREL 8.80.” Chicago: Scientific Software International.

Muthén, Linda K, and Bengt O Muthén. 2012. “MPlus: Statistical Analysis with Latent Variables–User’s Guide.” Citeseer.

McArdle, J Jack, and Roderick P McDonald. 1984. “Some Algebraic Properties of the Reticular Action Model for Moment Structures.” British Journal of Mathematical and Statistical Psychology 37 (2). Wiley Online Library: 234–51.

Cheung, Mike W-L. 2008. “A Model for Integrating Fixed-, Random-, and Mixed-Effects Meta-Analyses into Structural Equation Modeling.” Psychological Methods 13 (3). American Psychological Association: 182.

Tang, Ryan W, and Mike W-L Cheung. 2016. “Testing Ib Theories with Meta-Analytic Structural Equation Modeling: The Tssem Approach and the Univariate-R Approach.” Review of International Business and Strategy 26 (4). Emerald Group Publishing Limited: 472–92.

Cheung, Mike WL, and Wai Chan. 2009. “A Two-Stage Approach to Synthesizing Covariance Matrices in Meta-Analytic Structural Equation Modeling.” Structural Equation Modeling: A Multidisciplinary Journal 16 (1). Taylor & Francis: 28–53.

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