# Model-Robust Estimation of Multiple-Group Structural Equation Models
## 2 Model-Robust Estimation in Structural Equation Modeling
### Regularized Maximum Likelihood Estimation  
Regularized estimation of SEMs might be a more direct approach for modeling misspecifications in means and covariances [43; 44; 45; 46; 47; 48; 49; 50]. In this case, model errors are represented as outliers in item intercepts (i.e., in the vector \(\mathbf{v}\)) and nondiagonal entries in the residual error covariance (i.e., in the matrix \(\mathbf{\Upsilon}\)). In regularized estimation, a nonidentified SEM is typically utilized, and only outlying entries in \(\mathbf{v}\) and \(\mathbf{\Psi}\) are estimated differently from zero. To ensure the identifiability of model parameters, regularized estimation relies on sparsity assumptions [51; 52; 53] in item intercepts and residual correlations. The main idea in regularized ML (RegML) estimation is that a penalty function is subtracted from the likelihood. The penalty function \(\mathcal{P}\) controls the sparsity in a subset of the estimated parameter vector \(\mathbf{\theta}\). For example, the subset in \(\mathbf{\theta}\) might refer to nondiagonal residual error correlations or group-specific item intercepts. We now discuss the choice of the penalty function. For a scalar parameter \(x\), the lasso penalty is a popular penalty function used in regularization [52], and it is defined as  
\[\mathcal{P}_{\text{Lasso}}(x,\kappa)~{}=~{}\kappa|x|~{}, \tag{22}\]  
where \(\kappa\) is a nonnegative regularization parameter that controls the extent of sparsity in the obtained parameter estimate. It is known that the lasso penalty introduces bias in estimatedparameters. To circumvent this issue, the smoothly clipped absolute deviation (SCAD; [54]) penalty has been proposed. The SCAD penalty is defined as  
\[\mathcal{P}_{\text{SCAD}}(x,\kappa)\ =\ \left\{ \begin{array}{ll}\kappa|x|&\text{if }|x|< \kappa\\ -(x^{2}-2a\kappa|x|^{2}+\kappa^{2})(2(a-1))^{-1}&\text{if }\kappa\leq|x|\leq a \kappa\\ (a+1)\kappa^{2}&\text{if }|x|>a\kappa\end{array} \right. \tag{23}\]  
with \(a>2\). In many studies, the recommended value of \(a=3.7\) (see [54]) has been adopted (e.g., [51; 55; 56]). The SCAD penalty retains the penalization rate and the induced bias of the lasso for model parameters close to zero, but continuously relaxes the rate of penalization as the absolute value of the model parameters increases. Note that \(\mathcal{P}_{\text{SCAD}}\) has the property of the lasso penalty around zero, but has zero derivatives for \(x\) values strongly differing from zero. We now describe RegML estimation in more detail. In the general multiple-group SEM, the parameter vector \(\theta\) is extended to include group-specific residual error correlations \(\mathbf{\psi}_{g}\) (which is the vectorized version of \(\mathbf{\
abla}_{g}\)) and group-specific item intercepts \(\mathbf{\
u}_{g}\) (\(g=1,\ldots,G\)). By including all effects in the SEM, the parameter \(\theta\) would not be identified. Therefore, a penalty function is subtracted from the log-likelihood function, and the following function is minimized  
\[F_{\text{ML}}(\theta;\{\overline{x}_{g}\},\{S_{g}\})-N^{*}\sum_{g=1}^{G}\sum_{ h}\mathcal{P}_{\text{SCAD}}(\
u_{gh},\kappa_{\mathbf{\
u}})-N^{*}\sum_{g=1}^{G}\sum_{ h}\mathcal{P}_{\text{SCAD}}(\psi_{gh},\kappa_{\mathbf{\psi}}) \tag{24}\]  
using a sample size \(N^{*}\) that can be equal to \(\sum_{g=1}^{G}N_{g}\). Note that, for brevity, we suppress to indicate the dimensionality of the regularized parameters \(\mathbf{\psi}_{g}\) and \(\mathbf{\
u}_{g}\). In addition, note that there are two regularization parameters \(\kappa_{\mathbf{\
u}}\) and \(\kappa_{\mathbf{\psi}}\). In practice, minimization of (24) for fixed values of the two \(\kappa\) parameters results in a subset of \(\mathbf{\
u}_{g}\) and \(\mathbf{\psi}_{g}\) parameters that are different from zero, where the remaining parameters have been set to zero. Typically, the two \(\kappa\) regularization parameters are unknown nuisance parameters in (24) that must also be estimated.