# Model-Robust Estimation of Multiple-Group Structural Equation Models
###### Abstract  
Structural equation models (SEM) are widely used in the social sciences. They model the relationships between latent variables in structural models, while defining the latent variables by observed variables in measurement models. Frequently, it is of interest to compare particular parameters in an SEM as a function of a discrete grouping variable. Multiple-group SEM is employed to compare structural relationships between groups. In this article, estimation approaches for the multiple-group are reviewed. We focus on comparing different estimation strategies in the presence of local model misspecifications (i.e., model errors). In detail, maximum likelihood and weighted least-squares estimation approaches are compared with a newly proposed robust \(L_{p}\) loss function and regularized maximum likelihood estimation. The latter methods are referred to as model-robust estimators because they show some resistance to model errors. In particular, we focus on the performance of the different estimators in the presence of unmodelled residual error correlations and measurement noninvariance (i.e., group-specific item intercepts). The performance of the different estimators is compared in two simulation studies and an empirical example. It turned out that the robust loss function approach is computationally much less demanding than regularized maximum likelihood estimation but resulted in similar statistical performance. structural equation modeling;  
In SEM, more structured relationships among the latent variables can be imposed. A matrix \(\mathbf{B}\) of regression coefficients is specified such that  
\[\mathbf{\eta}\ =\ \mathbf{B}\mathbf{\eta}+\mathbf{\xi}\ \text{with}\ \text{E}(\mathbf{\xi})\ =\ \mathbf{\alpha}\ \text{and}\ \text{Var}(\mathbf{\xi})\ =\ \mathbf{\Phi}. \tag{3}\]  
Hence, the mean vector and the covariance matrix are represented in SEM as  
\[\mathbf{\mu}(\mathbf{\theta})\ =\ \mathbf{\
u}+\mathbf{\Lambda}(\mathbf{I}-\mathbf{B})^{-1}\mathbf{\alpha}\ \text{and}\ \mathbf{\Sigma}(\mathbf{\theta})\ =\ \mathbf{\Lambda}(\mathbf{I}-\mathbf{B})^{-1}\mathbf{\Phi}[(\mathbf{I}-\mathbf{B})^{-1}]^{ \top}\mathbf{\Lambda}^{\top}+\mathbf{\
abla}\, \tag{4}\]  
where \(\mathbf{I}\) denotes the identity matrix. In practice, SEM parsimoniously parametrizes the mean vector and the covariance matrix using a parameter \(\mathbf{\theta}\) as a summary. As is true with all statistical models, these restrictions are unlikely to hold in practice, and model assumptions in SEM are only an approximation of a true data-generating model. In SEM, model deviations (i.e., model errors) in covariances emerge as a difference between a population covariance matrix \(\mathbf{\Sigma}\) and a model-implied covariance matrix \(\mathbf{\Sigma}(\mathbf{\theta})\) (see [8; 9]). Similarly, there can be model errors in the mean vector leading to a difference between the population mean vector \(\mathbf{\mu}\) and the model-implied mean vector \(\mathbf{\mu}(\mathbf{\theta})\). As a consequence, the SEM is misspecified at the population level. Note that the model errors are defined at the population level in infinite sample sizes. With finite samples in real-data applications, the empirical covariance matrix \(\mathbf{S}\) estimates the population covariance matrix \(\mathbf{\Sigma}\) and the mean vector \(\mathbf{\overline{x}}\) estimates the population mean vector \(\mathbf{\mu}\). In this article, we investigate estimators that possess some kind of resistance against model deviations. That is, the presence of some amount of model errors does not impact the parameter estimate \(\mathbf{\theta}\). This kind of robustness is referred to as model robustness and follows the principles of robust statistics [10; 11; 12]. While in classical robust statistics, observations (i.e., cases or subjects) do not follow an imposed statistical model and should be treated as outliers, model errors in SEM occur as residuals in the modeled mean vector and the modeled covariance matrix.