# Model-Robust Estimation of Multiple-Group Structural Equation Models
## 2 Model-Robust Estimation in Structural Equation Modeling
### Robust Moment Estimation Using Robust Loss Functions
#### 2.2.1 Bias Derivation in the Presence of Model Errors  
In this section, we formally study the bias in the estimated parameter \(\mathbf{\theta}\) in the presence of model errors if the robust fitting function \(F_{\text{rob}}\) in (8) is utilized. General asymptotic results for misspecified models were presented in [6; 37]. In the following, we consider the slightly more specific minimization problem  
\[F_{\text{rob}}(\mathbf{\theta};\zeta)=\sum_{h=1}^{H}\rho(\zeta_{h}-\zeta_{h}(\mathbf{ \theta}))\, \tag{9}\]  
where \(\zeta~{}=~{}(\zeta_{1},\dots,\zeta_{H})\) is a vector that can contain all group-wise mean vectors \(\mathbf{m}_{g}\) and covariance matrices \(\mathbf{S}_{g}\) for \(g=1,\dots,G\). Let \(\text{vech}(\mathbf{S})\) denote the nonduplicated elements in a covariance matrix \(\mathbf{S}\). Then, the vector \(\zeta\) contains all sufficient statistics \(\zeta=(\mathbf{m}_{1},\dots,\mathbf{m}_{G},\mathbf{s}_{1},\dots,\mathbf{s}_{G})\), where \(\mathbf{s}_{g}~{}=~{}\text{vech}(\mathbf{S}_{g})\) for \(g=1,\dots,G\). We study the estimation problem using the optimization function in (9) at the population level. That is, we study the model differences between \(\zeta\) and \(\zeta(\mathbf{\theta})\). According to [26], we can frame the estimation problem (9) as a nonlinear regression  
\[\zeta~{}=~{}\zeta(\mathbf{\theta})+\mathbf{r} \tag{10}\]  
for some model error \(\mathbf{r}=(r_{1},\dots,r_{H})\). We now study the impact of sufficiently small model errors \(\mathbf{r}\). It is assumed that there exists a perfectly fitting SEM. That is, there exists a parameter vector \(\mathbf{\theta}_{0}\) such that \(\zeta_{0}~{}=~{}\zeta(\mathbf{\theta}_{0})~{}=~{}(\zeta_{01},\dots,\zeta_{0H})\). This situation corresponds to the absence of model error. We now replace the minimization problem (9) using a robust loss function with a weighted least squares estimation problem. This approach is ordinarily employed in robust statistics when estimating a robust regression model with iteratively weighted least squares estimation [38; 39]. Define \(\Delta\mathbf{\theta}~{}=~{}\mathbf{\theta}-\mathbf{\theta}_{0}\) and \(\mathbf{r}~{}=~{}\mathbf{\zeta}-\mathbf{\zeta}(\mathbf{\theta}_{0})\). Then, we can rewrite the minimization problem (9) as  
\[F_{\text{rob}}(\Delta\mathbf{\theta};\mathbf{\zeta})=\sum_{h=1}^{H}\rho(\zeta_{0h}+r_{h} -\zeta_{h}(\mathbf{\theta})). \tag{11}\]  
The nonlinear function \(\zeta_{h}(\mathbf{\theta})\) can be linearized around \(\mathbf{\theta}~{}=~{}\mathbf{\theta}_{0}\). Hence, we can write  
\[\zeta_{h}(\mathbf{\theta})\simeq\zeta_{0h}+\mathbf{a}_{h}^{\top}\Delta\mathbf{\theta}\, \tag{12}\]  
where \(\mathbf{a}_{h}~{}=~{}(\partial\zeta_{h})/(\partial\mathbf{\theta})\) evaluated at \(\mathbf{\theta}_{0}\). Moreover, we approximate the robust estimation problem in (11) by a weighted least squares problem [38; 40]  
\[F_{\text{rob}}(\Delta\mathbf{\theta};\mathbf{\zeta})=\sum_{h=1}^{H}\omega_{h}^{2}(r_{ h}-\mathbf{a}_{h}^{\top}\Delta\mathbf{\theta})^{2} \tag{13}\]  
using \(\omega_{h}~{}=~{}\rho(e_{h})/e_{h}\) for \(e_{h}~{}=~{}r_{h}-\mathbf{a}_{h}^{\top}\Delta\mathbf{\theta}\). Then, the change in the \(\mathbf{\theta}\) parameter estimate can be directly expressed as a function of model deviations \(\mathbf{r}\) using the weighted least squares regression formula [38]. Let \(\mathbf{\Omega}\) be the diagonal matrix containing weights \(\omega_{h}\) in the diagonal. Let \(A\) be the design matrix that contains rows \(\mathbf{a}_{h}^{\top}\) for \(h=1,\ldots,H\). Then, the asymptotic bias in the estimated parameter vector is given by  
\[\Delta\mathbf{\theta}~{}=~{}(\mathbf{A}^{\top}\mathbf{\Omega}\mathbf{A})^{-1}\mathbf{A}^{\top} \mathbf{\Omega}\mathbf{r}. \tag{14}\]  
If a robust loss function \(\rho\) is chosen, gross model errors \(r_{h}\) result in small weights \(\omega_{h}\). Hence, these model errors should only have a minor impact with respect to the parameter estimate \(\mathbf{\theta}\).