# Implementation Aspects in Regularized Structural Equation Models
## 4 Simulation Study 1: Noninvariant Item Intercepts (DIF)
### Method  
The setup of the simulation study mimics the one presented in [43]. Datasets were simulated from a one-dimensional factor model involving five items and three groups. The factor variable \(\eta_{1}\) was normally distributed with group means \(\alpha_{1,1}=0\), \(\alpha_{2,1}=0.3\), and \(\alpha_{3,1}=0.8\). The group variances were set to \(\phi_{1,11}=1\), \(\phi_{2,11}=1.5\), and \(\phi_{3,11}=1.2\), respectively. All factor loadings were set to 1, and all measurement error variances were set to 1 in all groups and uncorrelated with each other. The factor variable, as well as the residual variables, were normally distributed. Some non-zero group-specific item intercepts were simulated that indicate measurement noninvariance. These differential item functioning (DIF [44]) effects in item intercepts were simulated in one and only one of the five items in each group. In the first group, the fourth item intercepts had a DIF effect \(\delta\). In the second group, the first item had a DIF effect \(-\delta\), while the second item had a DIF effect \(-\delta\) in the third group. The DIF effect \(\delta\) was chosen as either 0.3 or 0.6. The sample size per group was chosen as \(N=500\) or \(N=1000\). A regularized multiple-group one-dimensional SEM was specified as the analysis model. In this model, invariant factor loadings were assumed. For identification reasons, the mean of the factor variable in the first group was fixed at 0, and the standard deviation in the first group was fixed at 1. The SCAD penalty function was imposed on groupspecific item intercepts. In the penalty function, the powers \(p=1\) and \(p=0.5\) were investigated. The SEM was estimated on a grid of regularization parameters between 0.025 and 0.40, with increments of 0.025. The exact estimation approach was implemented by the coordinate descent (CD). In the differentiable approximation (DA) of the non-differentiable SCAD penalty function, we chose \(\varepsilon=10^{-4}\). The optimal regularization parameter \(\lambda\) was obtained by minimizing the BIC. Because no estimated item intercepts are exactly set to 0 in the estimation, the thresholds \(\tau=0.01\), 0.02, and 0.05 were chosen as the cutoff values for treating model parameters as a value of 0 in the BIC computation. Furthermore, the smoothed direct BIC minimization (DIR) approach of [PERSON] and [PERSON] was carried out using \(\varepsilon=0.01\). This relatively large value was found to be optimal in preliminary simulation studies in which the tuning parameter \(\varepsilon\) was varied as 0.1, 0.01, 0.001, and 0.0001. The lowest RMSE and a small bias for parameter estimates were obtained for \(\varepsilon=0.01\). For the direct BIC minimization method DIR and regularized ML estimation for a set of fixed regularization parameters \(\lambda\), standard errors were computed by means of the delta method described in Section 2.3. Confidence intervals were calculated based on the normal distribution assumption (i.e., the confidence interval of an estimate \(\hat{\theta}\) was computed as \([\hat{\theta}-1.96\cdot\mathsf{SE}(\hat{\theta}),\hat{\theta}+1.96\cdot\mathsf{ SE}(\hat{\theta})]\), where \(\mathsf{SE}(\hat{\theta})\) is the estimated standard error). In total, 1,000 replications were conducted for all 2 (DIF effect size \(\delta\)) \(\times\) 2 (sample size \(N\)) = 4 conditions of the simulation study. We investigated the estimation quality of factor means and factor variances. Bias and root mean square error (RMSE) were utilized to assess the performance of different estimators. Let \(\hat{\theta}_{r}\) be a model parameter estimate in replication \(r=1,\ldots,R\). The bias was estimated by  
\[\text{Bias}(\hat{\theta})=\frac{1}{R}\sum_{r=1}^{R}(\hat{\theta}_{r}-\theta)\, \tag{25}\]  
where \(\theta\) denotes the true parameter value.