# Implementation Aspects in Regularized Structural Equation Models
## 4 Simulation Study 1: Noninvariant Item Intercepts (DIF)
### Results  
Figures 1 and 2 display the absolute bias and the RMSE of the factor mean \(\alpha_{2,1}\) in the second group as a function of the regularization parameter \(\lambda\) for the two sample sizes, \(N=500\) and \(N=1000\), and the two powers of the penalty function \(p=1\) and \(p=0.5\), respectively. It can be seen in the two figures that there is a range of values of the regularization parameter \(\lambda\), which results in unbiased and least variable (i.e., in terms of RMSE) estimates. The optimal fixed regularized parameter was larger for \(p=0.5\) than for \(p=1\). However, the minimal RMSE was similar for \(p=1\) and \(p=0.5\) in Simulation Study 1. Furthermore, the RMSE of the factor mean estimate based on the optimal regularization parameter selected by the minimal BIC did not generally outperform a well-chosen fixed regularization parameter \(\lambda\). Interestingly, Figure 2 illustrates in the condition of a larger DIF effect \(\delta=0.6\) that too small regularization parameters \(\lambda\) resulted in biased parameter estimates. The issue occurred both for \(p=1\) and \(p=0.5\). Moreover, by comparing Figures 1 and 2, it is evident that the optimal regularization parameter is a function of the size of DIF effects \(\delta\). That is, larger DIF effects \(\delta\) resulted in larger regularization parameters \(\lambda\). Figure 1: Simulation Study 1: Absolute bias and root mean square error (RMSE) of the factor mean \(\alpha_{2,1}\) in the second group as a function of the regularization parameter \(\lambda\) for a DIF effect of the item intercept of \(\delta=0.3\) for sample sizes \(N=500\) and \(N=1000\) and powers \(p=1\) and \(p=0.5\) of the penalty function. The RMSE of the estimate obtained by the optimal BIC and coordinate descent (BIC-CD) is displayed by the blue line. The location of the average optimal regularization parameter obtained by BIC-CD is displayed by the blue triangle. Table 1 presents the bias and the RMSE of the estimated group means of the second and the third group. In this table, the direct BIC minimization approach DIR is compared with the exact estimation approach (CD) and the differentiable approximation (DA) using the optimal regularization parameter \(\lambda\) based on the minimal BIC, as well as for fixed regularization parameters \(\lambda=0.05\) for the power \(p=1\) and \(\lambda=0.10\) for \(p=0.5\). The DA estimation approach is shown using the threshold \(\tau=0.02\). Values of the fixed  
Figure 2: Simulation Study 1: Absolute bias and root mean square error (RMSE) of the factor mean \(\alpha_{2,1}\) in the second group as a function of the regularization parameter \(\lambda\) for a DIF effect of the item intercept of \(\delta=0.6\) for sample sizes \(N=500\) and \(N=1000\) and powers \(p=1\) and \(p=0.5\) of the penalty function. The RMSE of the estimate obtained by the optimal BIC and coordinate descent (BIC-CD) is displayed by the blue line. The location of the average optimal regularization parameter obtained by BIC-CD is displayed by the blue triangle. regularization parameters were chosen based on the findings in Figures 1 and 2. Overall, there was only a negligible bias in factor mean estimates. Regarding RMSE, all different estimation methods resulted in relatively similar estimates. Furthermore, there were essentially no differences between the exact solution (CD) and the approximate solution (DA). If researchers seek toknow a close-to-optimal regularization parameter \(\lambda\), regularized ML estimation must not involve the choice of an optimal \(\lambda\) based on a minimal BIC. Finally, the computationally cheap direct BIC minimization approach (DIR) performed similarly to BIC estimation that requires fitting a regularized SEM at a sequence of regularization parameters \(\lambda\). A slight increase in RMSE of the DIR method was only observed for \(N=500\) and \(\delta=0.3\), which was the consequence of slightly biased parameter estimates.