# Implementation Aspects in Regularized Structural Equation Models
## 6 Summary of Simulation Findings  
In this section, the main findings of the two simulation studies are discussed regarding the research questions posed in Section 3.  
RQ1: Using a Fixed Regularization Parameter \(\lambda\) Can Be Advantageous Regarding Bias and RMSE  
First, the findings of Simulation Study 2 demonstrated that using a fixed regularized parameter \(\lambda\) instead of an optimally chosen \(\lambda\) by means of minimizing BIC can result in more efficient estimates of structural parameters (research question RQ1). This finding undermines the fact that obtaining efficient parameter estimates is not necessarily related to the search for a parsimonious model in terms of minimal information criteria.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \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{3-11}  & & & \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{3-11}
**Par** & \(N\) & \(\delta\) & **DIR** & **0.025** & **0.05** & **0.025** & **0.05** & **DIR** & **0.025** & **0.05** & **0.025** & **0.05** \\ \hline \multirow{4}{*}{\(\phi_{12}\)} & 500 & 0.2 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 94.2 & 96.2 & 93.8 & 97.3 & 97.9 \\  & 0.4 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 94.3 & 96.0 & 94.7 & 98.0 & 98.3 \\  & 0.2 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 95.1 & 95.9 & 94.7 & 97.0 & 97.0 \\  & 1000 & 0.4 & 0.00 & 0.00 & 0.00 & 0.00 & 95.1 & 95.8 & 95.1 & 97.7 & 96.9 \\  & 500 & 0.2 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 94.4 & 95.4 & 94.6 & **99.6** & 96.7 \\  & 0.4 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 94.9 & 95.8 & 95.0 & **99.7** & 96.9 \\  & 1000 & 0.2 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 94.7 & 94.9 & 94.5 & **99.8** & 95.9 \\ \hline \hline \end{tabular} Note.: Par = parameter; \(\phi_{12}\) = factor correlation; \(\lambda_{il}\) = factor loading of \(i\)th item on the \(d\)th factor; 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 \(\epsilon=0.01\); 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 6: Simulation Study 2: Bias and coverage rates as a function of sample size \(N\) and the size of cross-loadings \(\delta\).  
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline  & & & \multicolumn{4}{c}{\(p=1\)} & \multicolumn{4}{c}{\(p=0.5\)} \\ \cline{3-11}  & & & \multicolumn{4}{c}{**DA with \(\tau=\)**} & \multicolumn{4}{c}{**DA with \(\tau=\)**} \\ \cline{3-11} \multirow{2}{*}{\(N\)} & \(\delta\) & **CD** & **0.01** & **0.02** & **0.04** & **CD** & **0.01** & **0.02** & **0.04** \\ \hline \multirow{2}{*}{500} & 0.3 & 6.50 & 5.85 & 6.64 & 7.03 & 6.98 & 7.01 & 7.01 & 6.99 \\  & 0.6 & 6.92 & 6.72 & 6.94 & 6.98 & 6.94 & 6.95 & 6.95 & 6.94 \\  & 0.3 & 6.72 & 6.35 & 6.86 & 6.99 & 6.95 & 6.96 & 6.96 & 6.91 \\  & 0.6 & 6.97 &