# Computational Aspects of \(L_{0}\) Linking in the Rasch Model
## 4 Simulation Study  
### Method  
The Rasch model was used as the IRF in the data-generating model for two groups. For identification purposes, the mean of the latent variable \(\theta\) in the first group was fixed at 0, with an SD \(\sigma\) of 1. In the second group, the mean \(\mu\) was set to 0.3 to represent the difference in \(\theta\) between the groups, while the SD \(\sigma\) was set to 1.2. The simulation study was conducted for \(I=10\), 20, and 40 items. In the \(I=10\) condition, base item difficulty values \(b_{i}\) were set to \(-0.314\), \(0.411\), \(-1.097\), \(-0.542\), \(-1.854\), \(-0.403\), \(-0.895\), \(0.715\), \(0.841\), and \(0.139\), resulting in a mean of \(M=-0.300\) and an SD of 0.850. These item difficulties were used in the data generation for the first group. In the second group, the same values were applied, except for the first three items, where item difficulties were adjusted to \(b_{i2}=b_{i}+\kappa\). This introduced fixed DIF in 30% of the items. The DIF effect size \(\kappa\) was set to 0.4 and 0.8, representing small and large DIF conditions, respectively. For item sets larger than 10, the same 10-item parameter set was repeated accordingly. The item parameters are also available at [[https://osf.io/un6q4](https://osf.io/un6q4)]([https://osf.io/un6q4](https://osf.io/un6q4)) (accessed on 20 February 2025). Note that item parameters remained fixed across all replications within each condition. Sample sizes of \(N=125\), 250, 500, and 1000 per group were selected to reflect small- to large-scale applications of the Rasch model. A total of 5000 replications were conducted for each of the 24 conditions, corresponding to the combinations of 4 sample sizes (\(N\)) \(\times\) 3 item numbers (\(I\)) \(\times\) 2 DIF effect sizes (\(\kappa\)). For each simulated dataset, \(L_{0}\) linking was performed using various specifications. Both the ratio and Gaussian loss functions \(\rho_{\varepsilon}\) (see Section 2.2) were applied. The \(\rho_{\varepsilon}\) loss functions were evaluated for a sequence of 11 \(\varepsilon\) values: 1, 0.75, 0.50, 0.25, 0.10, 0.075, 0.05, 0.025, 0.01, 0.005, and 0.001. Additionally, as demonstrated in Section 3, the optimal \(\varepsilon\) value corresponding to the minimal RMSE is dependent on the sample size \(N\). To account for this, a data-driven \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{\(\mathbf{I}\)} & \multicolumn{2}{c}{\(\mathbf{\kappa=0.4}\), \(\mathbf{N=}\)} & \multicolumn{2}{c}{\(\mathbf{\kappa=0.8}\), \(\mathbf{N=}\)} \\ \cline{2-7}  & **250** & **500** & **1000** & **250** & **500** & **1000** \\ \hline
10 & 0.900 & 0.065 & 0.030 & 0.150 & 0.100 & 0.090 \\
20 & 0.200 & 0.050 & 0.030 & 0.100 & 0.100 & 0.090 \\
40 & 0.100 & 0.035 & 0.025 & 0.100 & 0.095 & 0.085 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Numerical Illustration: optimal \(\varepsilon\) value yielding the minimum root mean square error (RMSE) of the estimated mean \(\hat{\mu}\) using the ratio loss function as a function of the DIF effect size \(\kappa\), number of items \(I\), and sample size \(N\). approach for selecting the \(\varepsilon\) value was explored. Let \(\overline{V}\) denote the average variance of the estimated item difficulty differences \(\widehat{b}_{\bar{\eta}_{2}}-\widehat{b}_{\bar{\eta}_{1}}\). For a fixed \(y\in(0,1)\), \(\varepsilon_{y}\) is chosen such that  
\[\rho_{\varepsilon_{y}}\Big{(}\sqrt{\overline{V}}\Big{)}=y. \tag{23}\]  
In the simulation, the following \(y\) values were chosen: 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, and 0.9. The bias, standard deviation (SD), and root mean square error (RMSE) of the estimated mean, \(\hat{\mu}\), were calculated for all specifications of the \(L_{0}\) linking approach. The relative RMSE of \(\hat{\mu}\) was defined as the ratio of the RMSE for a given specification to the RMSE of a reference method, multiplied by 100. Across all conditions, the reference method was the ratio loss function with a fixed \(\varepsilon\) value of 0.01.