# Implementation Aspects in Regularized Structural Equation Models
## 4 Simulation Study 1: Noninvariant Item Intercepts (DIF)
### Results
Table 2 compares the average number of regularized parameters of the exact approach (CD) and the differentiable estimation approach (DA) using the thresholds \(\tau\) of 0.01, 0.02, and 0.04. It turned out that the number of regularized item intercepts in the selected models was very similar. Only for \(N=500\) and \(\delta=0.3\) was the number of regularized parameters slightly too low. Notably, lower values of thresholds such as \(\tau=0.005\) or \(\tau=0.001\) would result in a substantially lower average number of regularized parameters. \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline  & & & & \multicolumn{4}{c}{\(p\) = 1} & \multicolumn{4}{c}{\(p\) = 0.5} \\ \cline{4-13}  & & & **BIC** & \multicolumn{2}{c}{**BIC**} & \multicolumn{2}{c}{\(\lambda\) = 0.05} & \multicolumn{2}{c}{**BIC**} & \multicolumn{2}{c}{\(\lambda\) = 0.10} \\ \cline{4-13}
**Par** & \(N\) & \(\delta\) & **DIR** & **CD** & **DA** & **CD** & **DA** & **CD** & **DA** & **CD** & **DA** \\ \hline \multirow{6}{*}{\(\alpha_{2,1}\)} & \multirow{6}{*}{500} & \multicolumn{4}{c}{_Bias_} & & & & & & & & & & \\  & & 0.3 & \(-0.012\) & \(-0.005\) & \(-0.005\) & \(-0.004\) & \(-0.005\) & \(-0.005\) & \(-0.006\) & \(-0.007\) & \(-0.007\) \\  & & 0.6 & \(0.000\) & \(0.000\) & \(0.001\) & \(-0.001\) & \(-0.001\) & \(0.000\) & \(0.000\) & \(-0.019\) & \(-0.019\) \\  & & 0.3 & \(-0.005\) & \(-0.001\) & \(0.000\) & \(0.000\) & \(0.000\) & \(-0.001\) & \(-0.001\) & \(-0.001\) & \(-0.001\) \\  & & 0.6 & \(0.001\) & \(0.002\) & \(0.002\) & \(0.002\) & \(0.002\) & \(0.002\) & \(0.002\) & \(-0.010\) & \(-0.010\) \\  & & 0.3 & \(-0.013\) & \(-0.007\) & \(-0.006\) & \(-0.006\) & \(-0.006\) & \(-0.007\) & \(-0.007\) & \(-0.008\) & \(-0.008\) \\  & & 0.6 & \(0.001\) & \(0.001\) & \(0.002\) & \(0.000\) & \(0.000\) & \(0.001\) & \(0.001\) & \(-0.017\) & \(-0.017\) \\  & & 0.3 & \(-0.004\) & \(0.000\) & \(0.001\) & \(0.001\) & \(0.001\) & \(0.000\) & \(0.000\) & \(0.000\) & \(0.000\) \\  & & 0.6 & \(0.004\) & \(0.004\) & \(0.004\) & \(0.004\) & \(0.004\) & \(0.004\) & \(0.004\) & \(-0.008\) & \(-0.008\) \\  & & \multicolumn{4}{c}{_RMSE_} & & & & & & & & & \\  & & 0.3 & \(0.090\) & \(0.086\) & \(0.086\) & \(0.086\) & \(0.086\) & \(0.083\) & \(0.083\) & \(0.082\) & \(0.082\) \\  & & 0.6 & \(0.081\) & \(0.081\) & \(0.081\) & \(0.081\) & \(0.081\) & \(0.080\) & \(0.080\) & \(0.084\) & \(0.084\) \\ \(\alpha_{2,1}\) & & \multirow{2}{*}{1000} & \multirow{2}{*}{0.3} & \multirow{2}{*}{0.057} & \multirow{2}{*}{0.056} & \multirow{2}{*}{0.056} & \multirow{2}{*}{0.056} & \multirow{2}{*}{0.056} & \multirow{2}{*}{0.056} & \multirow{2}{*}{0.056} & \multirow{2}{*}{0.056} & \multirow{2}{*}{0.056} \\  & & 0.6 & \(0.057\) & \(0.057\) & \(0.057\) & \(0.057\) & \(0.057\) & \(0.057\) & \(0.057\) & \(0.060\) & \(0.061\) \\  
Table 3 focuses on the coverage rates of selected model parameters. It can be seen that coverage rates for a model parameter were acceptable for both powers \(p=1\) and \(p=0.5\) if the respective parameter estimate was approximately unbiased. Interestingly, the coverage rates were also satisfactory for the direct BIC minimization approach (DIR). \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{3-10}  & & & \multicolumn{2}{c}{**DA with \(\tau\) =**} & \multicolumn{4}{c}{**DA with \(\tau\) =**} \\ \cline{3-10} \(N\) & \(\delta\) & **CD** & **0.01** & **0.02** & **0.04** & **CD** & **0.01** & **0.02** & **0.04** \\ \hline \multirow{3}{*}{500} & 0.3 & 11.86 & 11.45 & 11.93 & 12.08 & 11.95 & 11.96 & 11.96 & 11.96 \\  & 0.6 & 11.94 & 11.96 & 11.97 & 11.99 & 11.97 & 11.97 & 11.97 & 11.97 \\ \multirow{3}{*}{1000} & 0.3 & 11.95 & 11.87 & 11.98 & 12.00 & 11.95 & 11.96 & 11.96 & 11.96 \\  & 0.6 & 11.99 & 12.00 & 12.00 & 12.00 & 11.98 & 11.99 & 11.99 & 11.99 \\ \hline \hline \end{tabular}  
* _Note._\(p\) = power used in the penalty function; CD = coordinate descent; DA = differentiable approximation using a threshold parameter \(\tau\).  
\end{table}