# Implementation Aspects in Regularized Structural Equation Models
## 7 Discussion and Conclusions  
In this article, implementation aspects of regularized maximum likelihood estimation of SEMs were investigated. We obtained some insights into how regularized SEMs could be efficiently implemented in practice. In contrast to statements in the literature, differentiable approximations of the non-differentiable penalty functions in regularized SEM perform comparably well to specialized estimation methods if tuning parameters in these approximations are thoughtfully chosen.  
Our preliminary conclusion for regularized SEM estimation from our simulation studies is that the direct BIC minimization approach or the fixed regularization parameter approach should deserve more attention in future research. By focusing on these approaches, the computational burden of regularized SEM is noticeably reduced. Future research might investigate whether the findings obtained for SEMs transfer to other models involving latent variables such as item response models [47; 48; 49; 50; 51; 52], latent class models [53; 54; 28; 55], or mixture models [56; 57].  
In this article, we focused on a differentiable approximation of the BIC. However, the same approximation technique could be applied to estimating regularized SEMs that minimize the AIC. However, we have preliminary simulation evidence that convergenceissues appeared more frequently when minimizing the differentiable approximation of the AIC compared to BIC.  
Hopefully, the availability of the direct BIC minimization approach could lead to more widespread use of regularized estimation. Nevertheless, the regularization approaches discussed in this paper still hinge on the assumption that there is sparsity with respect to regularized model parameters. Such sparse models or parameter deviations might not always be appropriate for modeling real-world datasets.  
The simulation studies showed that the optimal regularization parameter \(\lambda\) regarding the bias and RMSE of the model parameters of interest does not necessarily coincide with the optimal \(\lambda\) obtained by minimizing the BIC. Determining the optimal regularization parameter \(\lambda\) for particular regularized SEMs is, therefore, difficult for researchers. Maybe only simulation studies that involve a similarly complex model and a similar sample size could help to determine an appropriate \(\lambda\). If the researcher's interest lies in the interpretation of model parameters in a regularized SEM, it is uncertain as to why model fitting is aimed at minimizing a prediction error, as in BIC, because such a criterion can only be weakly related to estimating optimal model parameters (see [58]).  
This research received no external funding.  
The authors declare no conflicts of interest.  
The following abbreviations are used in this manuscript:  
\begin{tabular}{l l} AIC & Akaike information criterion \\ BIC & Bayesian information criterion \\ CD & coordinate descent \\ CFA & confirmatory factor analysis \\ DA & differentiable approximation \\ DGM & data-generating model \\ DIF & differential item functioning \\ LASSO & least absolute shrinkage and selection operator \\ ML & maximum likelihood \\ RMSE & root mean square error \\ SCAD & smoothly clipped absolute deviation \\ SEM & structural equation model \\ \end{tabular}