# Model-Robust Estimation of Multiple-Group Structural Equation Models
## 5 Simulation Study 2: Noninvariant Item Intercepts (DIF)
### Method
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c c c c c} \hline \hline  & & & \multicolumn{10}{c}{**RME with _p =_**} & \\ \cline{3-14}  & & & \multicolumn{3}{c}{**0.25**} & \multicolumn{3}{c}{**0.5**} & \multicolumn{3}{c}{**1**} & \multicolumn{3}{c}{**ULS**} & \multicolumn{3}{c}{**ML**} \\ \cline{3-14}
**Par** & \(\delta\) & **N** & **DF** & **JK** & **BS** & **DF** & **JK** & **BS** & **DF** & **JK** & **BS** & **DF** & **JK** & **BS** & **OI** & **DF** & **JK** & **BS** \\ \hline \multirow{6}{*}{\(\phi_{12}\)} & & 500 & 95.8 & 95.1 & 95.5 & 95.5 & 94.8 & 94.9 & 95.3 & 94.4 & 95.0 & 94.9 & 94.2 & 94.6 & 95.0 & 95.1 & 94.2 & 94.6 \\  & 0 & 1000 & 95.2 & 94.7 & 94.9 & 95.1 & 94.6 & 94.7 & 94.8 & 94.4 & 94.7 & 94.6 & 94.2 & 94.3 & 94.7 & 94.7 & 94.4 & 94.4 \\  & 2500 & 95.4 & 94.7 & 95.3 & 95.3 & 94.7 & 94.9 & 95.3 & 94.6 & 94.9 & 95.2 & 94.7 & 94.7 & 95.2 & 95.2 & 94.7 & 94.9 \\  & 500 & 94.8 & 94.5 & 94.6 & 94.5 & 94.1 & 94.5 & 93.7 & 93.0 & 93.3 & 94.7 & 94.0 & 94.3 & 93.5 & 94.4 & 93.7 & 94.4 \\  & 0.6 & 1000 & 94.3 & 94.0 & 94.7 & 94.3 & 93.7 & 94.1 & 93.7 & 93.3 & 93.1 & 94.7 & 94.0 & 94.4 & 93.6 & 94.6 & 94.0 & 94.1 \\  & 2500 & 95.1 & 94.5 & 95.0 & 95.0 & 94.3 & 94.7 & 95.0 & 94.2 & 94.1 & 95.2 & 94.5 & 94.8 & 94.3 & 95.2 & 94.3 & 94.6 \\ \hline \multirow{6}{*}{\(\lambda_{11}\)} & & 500 & 95.8 & 95.4 & 96.0 & 95.6 & 95.1 & 95.9 & 95.2 & 94.5 & 95.1 & 94.7 & 94.0 & 94.3 & 95.0 & 95.1 & 93.9 & 94.7 \\  & 0 & 1000 & 95.2 & 94.3 & 95.6 & 95.1 & 93.9 & 95.1 & 94.6 & 93.7 & 94.4 & 94.4 & 93.4 & 93.7 & 94.6 & 94.7 & 93.8 & 93.8 \\  & 2500 & 94.5 & 93.8 & 94.6 & 94.4 & 93.8 & 94.1 & 94.2 & 93.9 & 94.3 & 94.2 & 93.8 & 94.0 & 94.6 & 94.6 & 93.8 & 94.1 \\  & 500 & 95.5 & 94.9 & 96.0 & 95.6 & 94.7 & 96.2 & 95.6 & 94.5 & 95.3 & 94.9 & 93.8 & 94.5 & 88.5 & 96.2 & 95.7 & 96.4 \\  & 0.6 & 1000 & 95.1 & 94.5 & 95.5 & 94.9 & 94.5 & 95.0 & 94.7 & 94.4 & 94.8 & 94.6 & 93.8 & 94.4 & 88.5 & 95.6 & 94.7 & 95.9 \\  & 2500 & 95.1 & 94.7 & 95.2 & 95.0 & 94.7 & 94.9 & 95.3 & 94.5 & 94.9 & 95.2 & 94.3 & 94.9 & 89.0 & 95.4 & 94.5 & 95.4 \\ \hline \multirow{6}{*}{\(\psi_{11}\)} & & 500 & 96.6 & 96.2 & 97.6 & 96.4 & 95.8 & 96.9 & 95.8 & 94.8 & 95.4 & 95.1 & 94.3 & 94.4 & 95.0 & 95.0 & 94.4 & 94.9 \\  & 0 & 1000 & 96.1 & 95.3 & 96.7 & 95.9 & 95.1 & 95.9 & 95.5 & 94.8 & 95.3 & 95.3 & 94.6 & 94.9 & 95.1 & 95.1 & 94.7 & 94.9 \\  & 2500 & 95.0 & 94.7 & 95.3 & 95.1 & 94.6 & 95.1 & 95.1 & 94.6 & 94.9 & 95.0 & 94.5 & 94.7 & 95.0 & 95.0 & 94.3 & 94.4 \\ \cline{1-1}  & 500 & 96.1 & 95.7 & 97.1 & 96.1 & 95.6 & 96.9 & 95.7 & 95.2 & 95.8 & 94.9 & 94.5 & 94.6 & 88.7 & 96.7 & 96.5 & 96.8 \\ \cline{1-1}  & 0.6 & 1000 & 95.3 & 95.0 & 95.9 & 95.4 & 95.0 & 95.8 & 95.3 & 94.7 & 95.3 & 94.9 & 94.2 & 94.7 & 88.6 & 95.8 & 95.1 & 96.4 \\ \cline{1-1}  & 2500 & 95.1 & 94.7 & 95.0 & 95.1 & 94.6 & 95.0 & 94.9 & 94.5 & 94.8 & 95.1 & 94.5 & 94.7 & 88.8 & 95.2 & 94.8 & 95.6 \\ \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; DF = standard error estimation based on delta formula; JK = standard error estimation based on jackknife; BS = standard error estimation based on bootstrap; OT = standard error estimation based on observed information.  
\end{table} 
Table 2: Focused Simulation Study 1A: Coverage rate at confidence level 95% of model parameter estimates as a function of sample size \(N\) and size of residual error correlations \(\delta\) for different standard error estimation methods. misspecified analysis models. ML, ULS, RegML with the SCAD penalty on group-specific item intercepts, and RME with powers \(p=0.25\), 0.5, and 1 were utilized. In RegML, the optimal regularization parameter \(\kappa_{\
u}\) was chosen based on the BIC.