# Model-Robust Estimation of Multiple-Group Structural Equation Models
## 5 Simulation Study 2: Noninvariant Item Intercepts (DIF)
### Results
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c c} \hline \hline  & & & \multicolumn{6}{c}{**Bias**} & \multicolumn{6}{c}{**Relative RMSE**} \\ \cline{3-13}  & & & & \multicolumn{6}{c}{**RME with \(p\) =**} & \multicolumn{6}{c}{**RME with \(p\) =**} & \\ \cline{3-13}
**Par** & \(\delta\) & \(N\) & **RegML** & **0.25** & **0.5** & **1** & **ULS** & **ML** & **RegML** & **0.25** & **0.5** & **1** & **ULS** & **ML** \\ \hline \multirow{8}{*}{\(\alpha_{2}\)} & \multirow{8}{*}{0} & 500 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 100 & 103 & 102 & 100 & 100 & 100 \\  & & 1000 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 100 & 102 & 101 & 100 & 100 & 100 \\  & & 2500 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 100 & 100 & 100 & 100 & 100 & 100 \\  & & 500 & \(-0.02\) & \(\mathbf{-0.03}\) & \(\mathbf{-0.03}\) & \(\mathbf{-0.07}\) & \(\mathbf{-0.12}\) & \(\mathbf{-0.12}\) & 101 & 100 & 100 & **113** & **152** & **152** \\  & & 1000 & 0.00 & \(-0.01\) & \(-0.01\) & \(-0.02\) & \(\mathbf{-0.06}\) & \(\mathbf{-0.12}\) & \(\mathbf{-0.12}\) & 100 & 103 & 106 & **140** & **231** & **230** \\  & & 2500 & 0.00 & \(-0.01\) & \(-0.01\) & \(-0.01\) & \(\mathbf{-0.05}\) & \(\mathbf{-0.12}\) & \(\mathbf{-0.12}\) & 100 & 103 & 107 & **169** & **356** & **355** \\  & & 500 & 0.00 & \(-0.01\) & \(-0.02\) & \(\mathbf{-0.07}\) & \(\mathbf{-0.24}\) & \(\mathbf{-0.24}\) & 100 & 104 & 104 & **129** & **303** & **302** \\  & & 0.6 & 1000 & 0.00 & \(-0.01\) & \(-0.01\) & \(\mathbf{-0.06}\) & \(\mathbf{-0.24}\) & \(\mathbf{-0.24}\) & 100 & 102 & 103 & **144** & **435** & **433** \\  & & 2500 & 0.00 & 0.00 & \(-0.01\) & \(\mathbf{-0.05}\) & \(\mathbf{-0.24}\) & \(\mathbf{-0.24}\) & 100 & 101 & 104 & **171** & **698** & **692** \\ \hline \multirow{8}{*}{\(\alpha_{3}\)} & \multirow{8}{*}{0.3} & 500 & 0.01 & 0.01 & 0.01 & 0.01 & 0.01 & 0.01 & 0.01 & 100 & 103 & 102 & 101 & 100 & 100 \\  & & 1000 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 100 & 102 & 101 & 100 & 100 & 100 \\  & & 2500 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 100 & 101 & 101 & 100 & 100 & 100 \\  & & 500 & \(-0.02\) & \(\mathbf{-0.03}\) & \(\mathbf{-0.03}\) & \(\mathbf{-0.07}\) & \(\mathbf{-0.12}\) & \(\mathbf{-0.11}\) & 100 & 103 & 102 & **115** & **153** & **149** \\ \cline{1-1}  & & 1000 & 0.00 & \(-0.01\) & \(-0.02\) & \(\mathbf{-0.06}\) & \(\mathbf{-0.12}\) & \(\mathbf{-0.11}\) & 100 & 104 & 107 & **139** & **223** & **216** \\ \cline{1-1}  & & 2500 & 0.00 & \(-0.01\) & \(-0.01\) & \(\mathbf{-0.01}\) & \(\mathbf{-0.05}\) & \(\mathbf{-0.12}\) & \(\mathbf{-0.12}\) & 100 & 103 & 108 & **167** & **344** & **332** \\ \cline{1-1}  & & 500 & 0.00 & \(-0.01\) & \(-0.01\) & \(\mathbf{-0.06}\) & \(\mathbf{-0.23}\) & \(\mathbf{-0.21}\) & 100 & 103 & 102 & **124** & **286** & **267** \\ \cline{1-1}  & & 0.6 & 1000 & 0.00 & \(-0.00\) & \(-0.01\) & \(\mathbf{-0.05}\) & \(\mathbf{-0.23}\) & \(\mathbf{-0.22}\) & 100 & 103 & 103 & **139** & **421** & **390** \\ \cline{1-1}  & & 2500 & 0.00 & \(-0.01\) & \(-0.01\) & \(\mathbf{-0.05}\) & \(\mathbf{-0.24}\) & \(\mathbf{-0.22}\) & 100 & 101 & 104 & **174** & **693** & **641** \\ \hline \hline \end{tabular} _Note._ Par = model parameter; ML = maximum likelihood estimate; RegML = regularized maximum likelihood estimate using the Bayesian information criterion for regularization parameter selection; RME = robust moment estimation with power \(p\) in the loss function \(\rho(x)=|x|^{p}\); ULS = unweighted least squares estimate; biases with absolute values of at least 0.03 are printed in bold. RMSE values of at least 110 are printed in bold.  
\end{table} 
Table 3: Simulation Study 2: Bias and relative root mean square error (RMSE) of model parameter estimates as a function of sample size \(N\) and size of DIF effects \(\delta\).