# Computational Aspects of \(L_{0}\) Linking in the Rasch Model
## 4 Simulation Study
### Results  
Table 2 presents the absolute bias as a function of the DIF effect size \(\kappa\), the number of items \(I\), and the sample size \(N\). Overall, the absolute bias generally decreased as \(\kappa\) increased from 0.4 to 0.8. The bias in the estimated \(\hat{\mu}\) was highest for the smallest number of items (\(I=10\)) and slightly decreased as the number of items increased. This reduction in bias was more pronounced for smaller sample sizes. As the sample size \(N\) increased, the bias systematically decreased, with the lowest bias observed for the largest sample size (\(N=1000\)). For the data-driven \(\varepsilon\) choices \(\varepsilon_{0.2}\) and \(\varepsilon_{0.4}\), the bias also decreased as the sample size increased. The biases for the ratio and Gaussian loss functions were relatively similar, although the bias was slightly smaller under conditions with a larger DIF effect size (\(\kappa=0.8\)) compared to a smaller DIF effect size (\(\kappa=0.4\)). Additionally, the bias decreased as \(\varepsilon\) values in the \(\rho_{\varepsilon}\) function were reduced. Specifically, the bias was slightly higher for \(\varepsilon=0.2\) compared to \(\varepsilon=0.4\), particularly at smaller \(N\). These findings suggest that while the choice of the loss function influenced estimation accuracy, its effect diminished as \(I\) and \(N\) increased. Table 3 presents the relative RMSE of the estimated mean \(\hat{\mu}\) as a function of the DIF effect size \(\kappa\), the number of items \(I\), and the sample size \(N\). In general, larger \(\varepsilon\) values performed better in smaller samples and with fewer items. Thus, the optimal \(\varepsilon\) for minimizing the RMSE depends on \(\kappa\), \(I\), and \(N\). The data-driven estimates \(\varepsilon_{0.2}\) and \(\varepsilon_{0.4}\) generally outperformed the fixed choice \(\varepsilon=0.01\) based on the loss function \(\rho_{\varepsilon}\). Overall, it can be concluded that \(\varepsilon_{0.4}\) was superior to \(\varepsilon_{0.2}\). Although \(\varepsilon_{0.2}\) produced a lower RMSE than \(\varepsilon_{0.4}\) in many simulation conditions, the differences were typically small. However, in scenarios where \(\varepsilon_{0.2}\) performed worse than \(\varepsilon_{0.4}\), its RMSE exceeded the reference value of 100 and was substantially higher than that of \(\varepsilon_{0.4}\). This supports the use of \(\varepsilon_{0.4}\) as a conservative default choice. In most conditions, the Gaussian loss function slightly outperformed the ratio loss function. In large samples (\(N=1000\)), smaller \(\varepsilon\) values, such as \(\varepsilon=0.05\) or \(\varepsilon=0.01\), proved effective. To provide further insight into the absolute bias and relative RMSE, two additional figures are presented below. Figure 3 presents the absolute bias, SD, and RMSE of the estimated mean \(\hat{\mu}\) using the ratio loss function for \(I=20\) items as a function of the DIF effect size \(\kappa\) and sample size \(N\). The absolute bias increased with larger \(\varepsilon\), while the SD decreased, illustrating a bias-variance trade-off in the RMSE. This trade-off results in an optimal \(\varepsilon\) value that minimizes the RMSE. As shown in Figure 3, the optimal \(\varepsilon\) was smaller for a larger DIF effect size (i.e., \(\kappa=0.8\)) than for \(\kappa=0.4\). Additionally, the optimal \(\varepsilon\) decreased with increasing sample size \(N\). As expected, the RMSE was lower for larger sample sizes. Figure 4 presents the relative RMSE of the estimated mean based on data-driven \(\varepsilon_{y}\) values for the ratio loss function \(\rho_{\varepsilon}\) with \(\varepsilon=\varepsilon_{y}\). The results indicate that an optimal RMSE was achieved for a specific \(y\) value, depending on the DIF effect size \(\kappa\), the number of items \(I\), and the sample size \(N\). Overall, the findings in Figure 4 support the selection of \(\varepsilon_{0.2}\) and \(\varepsilon_{0.4}\), as presented in Tables 2 and 3.