# Computational Aspects of \(L_{0}\) Linking in the Rasch Model
## 3 Numerical Illustration  
### Method  
In this Numerical Illustration, the properties of the estimated linking parameter \(\hat{\mu}\) were studied in a simplified setting in which no item responses were simulated. It was assumed that the difference \(\widehat{b}_{i2}-\widehat{b}_{i1}\) had a variance of \(V/N\), where \(N\) denotes the sample size. Hence, this illustration assumed equal sampling variances of item parameter differences, which might be violated in practice. However, this assumption eases the statistical treatment and aims at yielding clearly interpretable results. Furthermore, we assumed that we had \(I\) items, and a proportion \(\pi\) of the items had an unbalanced DIF effect \(\kappa\), while a proportion \(1-\pi\) of the items did not show DIF.  
From Section 2.4, we know that the linking parameter satisfies the estimating equation  
\[\frac{1}{I}\sum_{i=1}^{I}\rho_{\varepsilon}^{\prime}\Big{(}\widehat{b}_{i2}- \widehat{b}_{i1}+\hat{\mu}\Big{)}=\frac{1}{I}\sum_{i=1}^{I}\rho_{\varepsilon} ^{\prime}(\hat{\mu}-\mu+\kappa_{i}+u_{i})=0, \tag{18}\]  
where \(\mathsf{Var}(u_{i})=V/N\). Because all item parameter differences have equal sampling variances, (18) can be simplified to  
\[\sum_{i=1}^{I\pi}\rho_{\varepsilon}^{\prime}(\hat{\mu}-\mu+\kappa+u_{i})+\sum _{I\pi+1}^{I}\rho_{\varepsilon}^{\prime}(\hat{\mu}-\mu+u_{i})=0, \tag{19}\]  
assuming that \(I\pi\) is an integer. By using the bias Formula (15), we obtain  
\[\mathsf{Bias}(\hat{\mu})=-\frac{\pi\mathsf{E}[\rho_{\varepsilon}^{\prime}( \kappa+u_{i})]}{\pi\mathsf{E}[\rho_{\varepsilon}^{\prime\prime}(\kappa+u_{i} )]+(1-\pi)\mathsf{E}[\rho_{\varepsilon}^{\prime\prime}(u_{i})]}, \tag{20}\]  
where we use \(\mathsf{E}[\rho_{\varepsilon}^{\prime}(u_{i})]=0\) because \(u_{i}\) has a symmetric distribution. The variance can be computed as (see (16))  
\[\mathsf{Var}(\hat{\mu})=\frac{\pi\mathsf{Var}[\rho_{\varepsilon}^{\prime}( \kappa+u_{i})]+(1-\pi)\mathsf{Var}[\rho_{\varepsilon}^{\prime}(u_{i})]}{\big{\{} \pi\mathsf{E}[\rho_{\varepsilon}^{\prime\prime}(\kappa+u_{i})]+(1-\pi) \mathsf{E}[\rho_{\varepsilon}^{\prime\prime}(u_{i})]\big{\}}^{2}}, \tag{21}\]  
The root mean square error (RMSE) as the square root of the MSE can be obtained as  
\[\mathsf{RMSE}(\hat{\mu})=\sqrt{\mathsf{MSE}(\hat{\mu})}=\sqrt{\mathsf{ Bias}(\hat{\mu})^{2}+\mathsf{Var}(\hat{\mu})}. \tag{22}\]  
The expected values and variances in (20) and (21) can be numerically evaluated.  
The bias, SD, and RMSE are evaluated for \(\hat{\mu}\) for sample sizes \(N\) of 250, 500, and 1000, as well as the number of items \(I\) as 10, 20, and 40. We simulated fixed DIF effects \(\kappa=0.4\) and \(\kappa=0.8\), representing small and large DIF. The tuning parameter \(\varepsilon\) in the ratio loss function and Gaussian loss function \(\rho_{\varepsilon}\) was chosen as 1, 0.95, 0.9, 0.85, 0.8, 0.75, 0.7, 0.65, 0.6, 0.55, 0.45, 0.4, 0.35, 0.3, 0.25, 0.2, 0.15, 0.1, 0.095, 0.09, 0.085, 0.08, 0.075, 0.07, 0.065, 0.06, 0.055, 0.05, 0.045, 0.04, 0.035, 0.03, 0.025, 0.02, 0.015, 0.01, 0.009, 0.008, 0.007, 0.006, 0.005, 0.004, 0.003, 0.002, and 0.001. In total, 46 \(\varepsilon\) values were evaluated in this Numerical Illustration. We only report results for the ratio loss function because the findings for the Gaussian loss function were very similar. The statistical software R (Version 4.4.1; [50]) was used for the analysis in this study. Symbolic derivatives of the two loss functions were computed with the R package Deriv (Version: 4.1.6; [51]). Replication material for this illustration can be retrieved from [[https://osf.io/un6q4](https://osf.io/un6q4)]([https://osf.io/un6q4](https://osf.io/un6q4)) (accessed on 20 February 2025).