# Model-Robust Estimation of Multiple-Group Structural Equation Models
## 2 Model-Robust Estimation in Structural Equation Modeling
### Multiple-Group Structural Equation Modeling  
We now describe different estimation approaches for multiple-group SEMs. Note that some identification constraints must be imposed to estimate the covariance structure model (4) (see [2]). For modeling multivariate normally distributed data without any missing values, the empirical mean vector \(\overline{x}\) and the empirical covariance matrix \(S\) are sufficient statistics for estimating \(\mu\) and \(\Sigma\). Hence, they are also sufficient statistics for the parameter vector \(\theta\). Now, assume that there are \(G\) groups with sample sizes \(N_{g}\) and empirical means \(\overline{x}_{g}\) and covariance matrices \(\mathbf{S}_{g}\) (\(g=1,\ldots,G\)). The population mean vectors are denoted by \(\mu_{g}\), and the population covariance matrices are denoted by \(\mathbf{\Sigma}_{g}\). The model-implied mean vector shall be denoted by \(\mu_{g}(\theta)\) and the model-implied covariance matrix by \(\mathbf{\Sigma}_{g}(\theta)\). Note that the parameter vector \(\theta\) does not have an index \(g\) to indicate that there can be common and unique parameters across groups. As a typical application of a CFA, equal factor loadings and item intercepts across groups are imposed (i.e., measurement invariance holds) by assuming the same loading matrix \(\Lambda\) and the same intercept vector \(\boldsymbol{v}\) across groups. The maximum likelihood (ML) function for the parameter \(\theta\) is given by (see [2;4])  
\[F_{\text{ML}}(\boldsymbol{\theta};\{\overline{x}_{g}\},\{\mathbf{S}_{g}\}) = -\sum_{g=1}^{G}\frac{N_{g}}{2}\Big{(}-I\log(2\pi)+\log|\mathbf{ \Sigma}_{g}(\boldsymbol{\theta})|+\text{tr}(\mathbf{S}_{g}\mathbf{\Sigma}_{g} (\boldsymbol{\theta})^{-1})+(\overline{x}_{g}-\boldsymbol{\mu}_{g}( \boldsymbol{\theta}))^{\top}\mathbf{\Sigma}_{g}(\boldsymbol{\theta})^{-1}( \overline{x}_{g}-\boldsymbol{\mu}_{g}(\boldsymbol{\theta}))\Big{)}\, \tag{5}\]  
where \(\{\overline{x}_{g}\}\) and \(\{\mathbf{S}_{g}\}\) are the sets of the empirical mean vectors and empirical covariance matrices of all groups, respectively. In practice, the model-implied covariance matrix will be misspecified [22; 23; 24], and \(\theta\) is a pseudo-true parameter defined as the maximizer of the fitting function \(F_{\text{ML}}\) in (5). A more general class of fitting functions is weighted least squares (WLS) estimation [3; 4; 25]. The parameter vector \(\theta\) is determined as the minimizer of  
\[F_{\text{WLS}}(\boldsymbol{\theta};\{\overline{x}_{g}\},\{\mathbf{S}_{g}\}) = \sum_{g=1}^{G}(\overline{x}_{g}-\boldsymbol{\mu}_{g}(\boldsymbol{ \theta}))^{\top}\boldsymbol{W}_{1g}(\overline{x}_{g}-\boldsymbol{\mu}_{g}( \boldsymbol{\theta}))+(\mathbf{s}_{g}-\mathbf{\sigma}_{g}(\boldsymbol{\theta} ))^{\top}\boldsymbol{W}_{2g}(\mathbf{s}_{g}-\mathbf{\sigma}_{g}(\boldsymbol{ \theta}))\, \tag{6}\]  
where matrices \(\mathbf{\Sigma}\) and \(\mathbf{S}\) have been replaced by vectors \(\boldsymbol{\sigma}\) and \(\mathbf{s}\) that collect all nonduplicated elements of the matrices in vectors. The weight matrices \(\boldsymbol{W}_{1g}\) and \(\boldsymbol{W}_{2g}\) (\(g=1,\ldots,G\)) can also depend on parameters that must be estimated. Diagonally weighted least squares (DWLS) estimation involves choosing diagonal weight matrices \(\boldsymbol{W}_{g1}\) and \(\boldsymbol{W}_{g2}\). The unweighted least squares (ULS) estimation is obtained when these matrices are identity matrices. Interestingly, the minimization in (6) can be interpreted as a nonlinear least squares estimation problem with sufficient statistics \(\{\overline{x}_{g}\}\) and \(\{\mathbf{S}_{g}\}\) as input data [26]. It has been shown that ML estimation can be approximately written as DWLS estimation [27] with particular weight matrices.