# Implementation Aspects in Simulation Extrapolation-Based Stocking-Lord Linking
## 3 Proposed Computational Shortcuts in SIMEX-Based Linking
### Applying the SIMEX Approach to Linking Methods  
The SIMEX method can be used to reduce the bias in SL linking [23] if uniform DIF is present. The idea is to interpret DIF effects as measurement errors that can be handled with SIMEX. The parameter estimate \(\widehat{\mathcal{S}}=(\widehat{\mu},\widehat{\sigma})\) in SL linking as obtained by  
\[\widehat{\mathcal{S}}=(\mu^{*},\sigma^{*})=\operatorname*{arg\,min}_{ \delta}\;H(\boldsymbol{\delta},\widehat{\boldsymbol{a}}_{1},\widehat{ \boldsymbol{b}}_{1},\widehat{\boldsymbol{a}}_{2},\widehat{\boldsymbol{b}}_{2}). \tag{11}\]  
As shown in Refs. [20,30], the bias of \(\widehat{\mathcal{S}}\) is primarily a function of the DIF variance \(\tau^{2}=\text{Var}(e_{i})\) and can be expressed as  
\[\text{Bias}(\widehat{\mathcal{S}})=-\frac{1}{2}\tau^{2}\boldsymbol{H}_{ \delta\delta}(\delta_{0};\gamma,\boldsymbol{0})^{-1}\sum_{i=1}^{I}\boldsymbol {H}_{\delta e_{i}e_{i}}(\delta_{0};\gamma,\boldsymbol{0}), \tag{12}\]  
where \(\boldsymbol{H}_{\delta\delta}(\delta_{0};\gamma,\boldsymbol{0})\) denotes the Hessian matrix of the linking function \(H\) with respect to \(\delta\), evaluated at the true linking parameter \(\delta_{0}\), joint item parameters \(\gamma\) and no DIF effects (i.e., \(\boldsymbol{e}=(e_{1},\ldots,e_{I})=\boldsymbol{0}\)). Moreover, the random DIF variance \(\tau^{2}\) is item-wise weighted by \(\boldsymbol{H}_{\delta e_{i}e_{i}}(\delta_{0};\gamma,\boldsymbol{0})\), where \(\boldsymbol{H}_{\delta e_{i}e_{i}}\) involves second-order derivatives of the gradient \(\boldsymbol{H}_{\delta}\) with respect to DIF effects \(e_{i}\), evaluated at \(\boldsymbol{e}=\boldsymbol{0}\). The application of the SIMEX method to a linking method requires an estimate of the DIF variance \(\tau^{2}\). The true uniform DIF effects \(e_{i}\) are unknown, but can be estimated using the formula  
\[\widehat{e}_{i}=\widehat{\mu}+\widehat{\sigma}\widehat{b}_{i2}-\widehat{b}_{ i1}. \tag{13}\]  
Subsequently, the DIF variance \(\tau^{2}\) can be consistently estimated by  
\[\hat{\tau}^{2}=\frac{1}{I}\sum_{i=1}^{I}(\widehat{e}_{i}-\overline{e})^{2}- \frac{1}{I}\sum_{i=1}^{I}v_{\widehat{e}_{i}}, \tag{14}\]  
where \(v_{\widehat{e}_{i}}\) represents the variance estimate due to sampling error of the estimated DIF effect estimate \(\widehat{e}_{i}\) and \(\overline{e}=I^{-1}\sum_{i=1}^{I}\widehat{e}_{i}\). In the application of SIMEX to SL linking, adjusted DIF effects \(e_{i}^{*}(\lambda)=\widehat{e}_{i}+u_{i}(\lambda)\) are calculated for \(\lambda=0.5\), \(1.0\), \(1.5\), and \(2.0\), where the values \(u_{i}(\lambda)\) are random draws from a normal distribution with zero mean and a variance \(\lambda\hat{\tau}^{2}\). The newly created DIF effects \(e_{i}^{*}(\lambda)\) have a larger variance of \((1+\lambda)\hat{\tau}^{2}\). Next, adjusted item difficulties \(b_{i2}^{*}(\lambda)\) (i.e., pseudo-item parameters) are computed as  
\[b_{i2}^{*}(\lambda)=\frac{1}{\sigma^{*}}\left(\widehat{b}_{i1}+e_{i}^{*}( \lambda)-\mu^{*}\right)=\widehat{b}_{i2}+\frac{1}{\sigma^{*}}u_{i}(\lambda). \tag{15}\]  
All pseudo-item parameters \(b_{i2}^{*}(\lambda)\) are then gathered into the vector \(\boldsymbol{b}_{2}^{*}(\lambda)\). The SIMEX-based linking estimate of the mean \(\mu\) and the standard deviation \(\sigma\) are now a function of \(\lambda\) and are defined as  
\[\widehat{\mathcal{S}}(\lambda)=(\widehat{\mu}(\lambda),\widehat{\sigma}( \lambda))=\operatorname*{arg\,min}_{\delta}\;H(\boldsymbol{\delta},\widehat{ \boldsymbol{a}}_{1},\widehat{\boldsymbol{b}}_{1},\widehat{\boldsymbol{a}}_{2},\boldsymbol{b}_{2}^{*}(\lambda)). \tag{16}\]  
These estimates are derived from item parameters that exhibit a larger DIF variance than those present in the original data. In the second stage of SIMEX, the regression functions \(\widehat{\mu}(\lambda)\) and \(\widehat{\sigma}(\lambda)\) are specified as a quadratic function of \(\lambda\) (see (10)).