# Model-Robust Estimation of Multiple-Group Structural Equation Models
## 6 Empirical Example: ESS 2005 Data
### Results  
In the following, we present the estimates of RegML with the obtained optimal regularization parameter \(\kappa_{v}\) of 0.02. In Table 4, the country means for the six different estimators are presented. The maximum absolute difference between country means stemming from the different models ranged between 0.032 and 0.337, with an average of 0.116 (SD = 0.081). This showed considerable variability. Hence, the different robust and non-robust estimators differently weighted DIF effects in the computation of country means. We also computed the ranks of the 17 countries across the six different estimates. The maximum absolute country rank difference ranged between 0 (country C10, rank 1) and 8 (country C21, rank 2) with an average of 3.9 (SD = 2.1). A large range for country means was observed for country C21 (rank 2). The means for this country ranged between \(-0.01\) (RME) and 0.33 (ML). Note that robust approaches provided similar results, but they substantially differed from RME with \(p=1\), ULS, and ML. Similarly, for country C06 (rank 3), there was a range of country means with values between 0.06 and 0.26. However, low dependencies of country means from the model estimator choice were observed for countries C10 (rank1), C09 (rank 13), and C13 (rank 14). In Figure 2, average absolute differences in country means between different models are displayed. Large absolute differences are displayed in darker red-brown color, while small differences are colored in light yellow-orange. RME with \(p=0.25\) (i.e., \"RME0.25\") and RME with \(p=0.5\) (i.e., \"RME0.5\") provided similar country means with an average distance of 0.01. Notably, RegML also resulted in similar estimates to RME, with \(p=0.25\) or \(p=0.5\) (i.e., average distances of 0.02 or 0.03). In addition, ML and ULS resulted in similar estimates. Interestingly, RME with \(p=1\) performed slightly differently compared to the other approaches. As in the simulation studies, it did not show full robustness to outlying DIF effects in item intercepts. In Figure 3, the regularization paths of the estimated country means as a function of the regularization parameter \(\kappa_{\
u}\) are displayed. The RegML estimates of the country means are displayed on the left side of the figure, and the ML estimates are displayed on the right side of the figure. Interestingly, the paths are not monotone, and there is some variability in country mean estimates depending on the choice of the regularization parameter. In Table 5, the estimated country-specific item intercepts for regularized ML estimation are presented. Overall, three countries had three regularized item intercepts, 12 countries had two regularized item intercepts, and two countries had one regularized item intercept. In total, 35 out of 68 item intercepts were regularized (i.e., the country-specific item intercepts were estimated as zero). The regularization paths for country-specific item intercepts for four selected countries, C01, C09, C13, and C21, are displayed in Figure 4. One immediately recognizes that more and stronger deviations from invariance were observed with smaller values of the regularization parameter \(\kappa_{\
u}\). \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & & \multicolumn{6}{c}{**RME with \(p\) =**} \\ \cline{3-8}
**Rank** & **CNT** & **RegML** & **0.25** & **0.5** & **1** & **ULS** & **ML** \\ \hline
1 & C10 & 0.37 & 0.37 & 0.37 & 0.34 & 0.34 & 0.35 \\
2 & C21 & 0.01  
Figure 3: Empirical example: Regularization paths for estimated country means in regularized maximum likelihood estimation as a function of the regularization parameter \(\kappa_{\
u}\). Figure 2: Empirical example: Visualization of average absolute deviations in country means between different estimation methods.