# Model-Robust Estimation of Multiple-Group Structural Equation Models
## 7 Discussion  
In this article, we have demonstrated that, using model-robust SEM estimation methods, such as regularized maximum likelihood estimation and robust moment estimation, provides some resistance to model errors for modeled means and covariances. Importantly, this robustness property is only guaranteed if model errors are sparsely distributed. That is, most of the model errors are zero (or approximately zero), while only a few model errors are allowed to differ from zero. The plausibility of this assumption should be evaluated for each empirical application that potentially involves model error. Moreover it is noted that robust moment SEM estimation approaches did not result in practically relevant efficiency losses compared to non-robust approaches (i.e., maximum likelihood or unweighted least squares estimation) in the absence of model errors. Hence, robust estimation approaches can be recommended in empirical applications with moderate to large sample sizes (i.e., \(N\geq 500\)). Importantly, robust moment estimation is computationally much less demanding than regularized estimation. Moreover, it also provides valid standard errors that are much more difficult to obtain with regularized ML estimation. We applied robust moment estimation and regularized ML estimation for unmodeled item intercepts in Simulation Study 2 in Section 5. In this case, there is essentially no need to utilize the more computationally expensive regularization technique. The crucial assumption is that there are only a few model deviations in the modeled mean vectors and covariance matrices. This property is also referred to as the sparsity assumption [51], which means that the majority of the entries in mean vectors and covariance matrices are correctly specified. The methods discussed in this article will typically fail if the majority of or all the elements in the mean vectors and covariance matrices are misspecified [65; 66]. This will likely be the case if the item loadings in the confirmatory factor model vary across groups. This situation is referred to as nonuniform DIF. Model deviations in item loadings lead to model deviations in the covariance matrix that is not as sparse as deviations in the mean structure. Hence, this is a situation in which regularized ML estimation might be preferred over robust moment estimation. The treatment in this article was based on the multivariate normal distribution utilizing mean vectors and covariance matrices as sufficient statistics. Sometimes, researchers tend to model ordinal items by relying on underlying latent normally distributed variables [67; 68]. In this case, thresholds and polychoric correlations replace mean vectors and covariance matrices in the weighted least squares fitting function. The approach based on model-robust fitting function can also be simply transferred to the case of modeling ordinal items. Future research might compare the performance of robust moment estimation with limited information methods relying on thresholds and polychoric correlations with regularized maximum likelihood estimation. Clearly, the computational advantages of model-robust limited information methods would be even more pronounced compared to the case of continuous items. It should be noted that multivariate normality is not a requirement for obtaining consistent estimates in correctly specified structural equation models [37]. This article investigates the robustness of parameter estimates regarding model misspecification in the mean and the covariance structure. There is a distinct literature that focuses on robust estimation of SEM in the violation of multivariate normality [69; 70]. These approaches might result in more efficient estimates in the case of heavy-tailed or contaminated distributions. However, outliers in this article are defined in the sense that some entries of the mean vector and the covariance matrix are misspecified infinite sample sizes (i.e., at the population level).