# Model-Robust Estimation of Multiple-Group Structural Equation Models
## 3 Simulation Study 1: Unmodeled Residual Error Correlation
### Method  
The data-generating model is a two-dimensional factor model involving six manifest variables \(X_{1},\ldots,X_{6}\), and two latent (factor) variables \(\eta_{1}\) and \(\eta_{2}\). The data-generating model is graphically presented in Figure 1. The first three items load on the first factor, while the last three item load on the second factor. Model errors exist by introducing two unmodeled residual correlations of items corresponding to the two factors (i.e., of items \(X_{1}\) and \(X_{4}\), and items \(X_{2}\) and \(X_{5}\)).  
The means of all variables were set to equal in the population. However, the mean structure was left saturated in estimation because it is not of interest in this simulation study. The specified factor loadings \(\lambda\) were all set to 0.7, and all error variances were set to 0.5. The latent correlation \(\phi_{12}\) of the factors \(\eta_{1}\) and \(\eta_{2}\) was set to 0.5 in the simulation. The factor variable \(\eta\), as well as the residuals \(\epsilon\), were multivariate normally distributed.  
We varied the size of both residual error correlations as 0.5 - \(\delta\) with \(\delta=0\), 0.3, and 0.6, indicating moderate-sized and large-sized residual error correlations. The sample size \(N\) was chosen to 500, 1000, or 2500.  
We estimated the two-dimensional factor model with unmodeled residual error correlations. That is, the analysis models were misspecified (except for RegML). The six factor loadings and the six residual variances were freely estimated. The variances of the latent factors were set to 1. We employed ML, ULS, RegML, and RME, utilizing the \(L_{p}\) loss function with powers \(p=0.25\), 0.5, and 1. ML and ULS can be considered non-robust SEM estimation approaches, while RegML and RME are considered robust approaches. In total, six different estimates were compared for each simulated dataset in each condition.  
For RegML, we only imposed the SCAD penalty on nondiagonal entries in the residual covariance matrix \(\mathbf{\Psi}\). RegML was estimated at a grid of regularization parameters between 10 and 0.002 (see replication material on [[https://osf.io/v8f45](https://osf.io/v8f45)]([https://osf.io/v8f45](https://osf.io/v8f45)) (accessed on 27 March 2023) for specification details). In this simulation study, the tuning parameter \(a\) was fixed to 3.7. We chose estimates of the regularization approach using the optimal regularization \(\kappa_{\phi}\) based on the BIC.  
We studied the estimation recovery of the factor correlation \(\phi_{12}\), a factor loading \(\lambda_{11}\), and a residual variance \(\psi_{11}\). We assessed the bias and root mean square error (RMSE) of a parameter \(\hat{\theta}\) of interest. In each of \(R=2000\) replications in a simulation condition, the estimate \(\hat{\theta}_{r}\) (\(r=1,\ldots,R\)) was calculated. The bias was estimated by  
\[\text{Bias}(\hat{\theta})\ =\ \frac{1}{R}\sum_{r=1}^{R}(\hat{\theta}_{r}- \theta). \tag{25}\]  
The RMSE was estimated by  
\[\text{RMSE}(\hat{\theta})\ =\ \frac{1}{R}\sum_{r=1}^{R}(\hat{\theta}_{r}- \theta)^{2}. \tag{26}\]  
To ease the comparability of the RMSE between different methods across sample sizes, we used a relative RMSE in which we divided the RMSE of a particular method by the RMSE  
Figure 1: Simulation Study 1: Data-generating model.  
of the best-performing method in a simulation condition. Hence, a relative RMSE of 100 is the reference value for the best-performing method.  
The statistical software R [60] was employed for all parts of the simulation. All estimators of the SEM were obtained using the mgsem() function in the sirt [61] package. Replication material can be found at [[https://osf.io/v8f45](https://osf.io/v8f45)]([https://osf.io/v8f45](https://osf.io/v8f45)) (accessed on 27 March 2023).