# Implementation Aspects in Regularized Structural Equation Models
## 5 Simulation Study 2: Two-Dimensional Factor Model with Cross-Loadings
### Method  
The data-generating method involves a two-dimensional factor model involving ten manifest variables \(X_{1},\ldots,X_{10}\) (i.e., items), and two latent (factor) variables \(\eta_{1}\) and \(\eta_{2}\). The data-generating model is graphically presented in Figure 3. The first five items load on the first factor, while the last five items load on the second factor. Three cross-loadings for items \(X_{1}\), \(X_{9}\), and \(X_{10}\) were introduced. All variables had zero means and were normally distributed. Furthermore, the latent variables \(\eta_{1}\) and \(\eta_{2}\) were standardized (i.e., they had a true variance of 1). The true factor correlation \(\phi_{12}\) of the two factor variables was set to 0.5. The primary factor loadings of the ten items were 1.000, 0.858, 0.782, 0.877, 0.888, 1.000, 0.815, 0.721, 0.880, and 0.749. The variances of the normally distributed residual error variables were chosen as 0.115, 0.464, 0.572, 0.345, 0.411, 0.122, 0.536, 0.680, 0.383, and 0.627. All cross-loadings were simulated with the size \(\delta\). In the simulation, \(\delta\) was chosen as 0.2 or 0.4. Furthermore, the sample size \(N\) was chosen to be either 500 or 1000. The two-dimensional factor model with the SCAD penalty function on the cross-loadings was specified as the analysis model. For identification reasons, the variances of the  
Figure 3: Simulation Study 2: Data-generating model. \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline  & & & \multicolumn{4}{c}{**Bias**} & \multicolumn{4}{c}{**Coverage**} \\ \cline{3-11}  & & & & \multicolumn{2}{c}{\(p=1\)} & \multicolumn{2}{c}{\(p=0.5\)} & \multicolumn{2}{c}{\(p=1\)} & \multicolumn{2}{c}{\(p=0.5\)} \\ \cline{4-11}  & & & \multicolumn{2}{c}{**BIC**} & \multicolumn{2}{c}{**DA with \(\lambda=\)**} & \multicolumn{2}{c}{**DA with \(\lambda=\)**} & \multicolumn{2}{c}{**BIC**} & \multicolumn{2}{c}{**DA with \(\lambda=\)**} & \multicolumn{2}{c}{**DA with \(\lambda=\)**} \\ \cline{4-11}
**Par** & \(N\) & \(\delta\) & **DIR** & **0.05** & **0.10** & **0.10** & **0.15** & **DIR** & **0.05** & **0.10** & **0.10** & **0.15** \\ \hline \multirow{4}{*}{\(\
u_{3,2}\)} & \multirow{4}{*}{500} & 0.3 & 0.02 & 0.01 & **0.17** & 0.02 & **0.12** & 94.2 & 93.5 & **29.6** & 92.0 & **48.4** \\  & & 0.6 & 0.00 & 0.00 & 0.00 & 0.03 & 0.00 & 95.8 & 95.1 & 95.4 & **86.1** & 95.4 \\ \cline{1-1}  & & 0.3 & 0.00 & 0.00 & **0.19** & 0.00 & **0.11** & 95.6 & 95.2 & **23.7** & 94.5 & **52.1** \\ \cline{1-1}  & & 0.6 & 0.00 & 0.00 & 0.00 & 0.02 & 0.00 & 95.3 & 95.0 & 94.9 & **85.9** & 95.0 \\ \hline \end{tabular} Note. Par = parameter; \(\alpha_{5,1}\) = factor mean in group \(g=2,3\); \(\phi_{5,11}\) = factor variance in group \(g=2,3\); \(\upsilon_{g,i}\) = item intercept of item \(i\) in group \(g=2,3\); estimation using the optimal regularization parameter based on the BIC; \(p=\) power used in the penalty function; \(\lambda=\) fixed regularized parameter; DIR = direct BIC minimization using the differentiable approximation of O’Neill and Burke (2023) with \(\varepsilon=0.01\); DA = differentiable approximation using the threshold parameter \(\tau=0.02\); Absolute biases larger than 0.04 are printed in bold. Coverage rates smaller than 91 or larger than 98 are printed in bold.  
\end{table} 
Table 3: _Cont._factor variables were fixed to 1. The estimation method followed those used in Simulation Study 1. Again, \(\epsilon=10^{-4}\) was employed in the differentiable approximation method (DA) utilizing thresholds \(\tau=0.01\), \(0.02\), and \(0.04\) in the BIC minimization. The smoothed direct BIC minimization approach (DIR) was again conducted with \(\epsilon=0.01\). In total, \(1000\) replications were conducted for all 2 (size of cross-loadings \(\delta\)) \(\times\) 2 (sample size \(N\)) = 4 conditions of the simulation study. We analyzed the estimation quality of model parameter estimates through bias, RMSE, and coverage rates. The models were again estimated using the sirt::mgsem() function in the R [45] package sirt [46].