# Model-Robust Estimation of Multiple-Group Structural Equation Models
## 7 Discussion
As pointed out in this article, the multiple-group SEM with unmodeled item intercepts corresponds to the estimation in the violation of measurement invariance (i.e., in the presence of differential item functioning). Linking approaches, such as invariance alignment [32; 71] or robust Haberman linking [30], also deal with the estimation of factor means in one- or multidimensional confirmatory factor models. They do so by first estimating model parameters of the factor model separately in each group in the first step. In thesecond step, group-specific estimated item intercepts and factor loadings are transformed with a robust or non-robust linking function to identify factor means and factor standard deviations. This approach is advantageous if there is a misfit in the mean and the covariance structure. If the covariance structure is correctly specified, a joint robust moment estimation approach of the multiple-group SEM could have higher efficiency. In practical applications, it has been shown that misspecification more frequently occurs in the mean structure than in the covariance structure [72]. Hence, the robust moment estimation approach proposed in this article might be a viable alternative to the frequently employed invariance alignment technique. Future research could thoroughly compare robust moment estimation with invariance alignment or robust Haberman linking. It has been argued that regularized estimation could also be applied with a fixed tuning parameter [73]. Recent research indicated that model parameter estimation could improve when using a fixed tuning parameter instead of an optimal tuning parameter based on the minimal information criterion [20]. Notably, regularized estimation in SEM is also referred to as penalized estimation [74]. In this framework, the penalty function \(\mathcal{P}(x)\propto|x|^{1/2}\) is interpreted as a fixed prior distribution that has a density \(f(x)\propto\exp(-|x|^{1/2})\)[74] using the so-called alignment loss function (see [30; 32]). An anonymous reviewer also suggested investigating the Huber loss function [10; 38] in addition to the \(L_{p}\) power loss function \(\rho(x)=|x|^{p}\) (\(p\leq 1\)). As a disadvantage, the Huber loss function requires an additional tuning parameter. We have some limited empirical experience in using the Huber loss for robust Haberman linking [30]. In this situation, the Huber loss function performed in an inferior way to the \(L_{p}\) loss function. This was particularly the case for asymmetrically distributed (i.e., asymmetrically structured) model deviations. Studying a wider range of robust loss functions in multiple-group SEM might be an interesting topic for future research, although we are somewhat skeptical that the Huber loss function will outperform the \(L_{p}\) loss function in the conditions of our simulation studies. Finally, it is always a substantive question of whether model errors are allowed to have outliers or not [75]. That is, should statistical models automatically remove particular means or covariances for estimating the parameter \(\mathbf{\theta}\)? We attacked such a mechanistic approach to model estimation and argued that model deviations, such as violations of measurement invariance, should not automatically result in model refinement [76]. Hence, we do not think that there is always an optimal loss function in every application [75]. If this were true, the statistical inference would always be based on maximum likelihood estimation. However, we believe that researchers intentionally choose a statistical parameter of interest and apply M-estimation theory [77] for the statistical inference that is not based on correctly specified models. In this sense, SEMs can be estimated with non-robust estimation approaches (such as maximum likelihood or unweighted least squares estimation) using an intentionally misspecified model [78; 79]. Appearing model errors can be included as additional uncertainty in standard errors of the estimated \(\mathbf{\theta}\) parameter. In SEM, ref.