# Implementation Aspects in Simulation Extrapolation-Based Stocking-Lord Linking
## 2 Stocking-Lord Linking  
In this section, we briefly describe the SL linking approach. Assume that the 2 PL model holds in both groups. In the first group, the item discriminations and item intercepts are defined as \(a_{i1}=a_{i}\) and \(b_{i1}=b_{i}\), respectively. For identification purposes, we further assume \(\theta\sim\mathsf{N}(0,1)\) in this group. In contrast, for the second group, the ability distribution follows \(\theta\sim\mathsf{N}(\mu,\sigma^{2})\), where \(\mu\) and \(\sigma\) are the mean and the SD, respectively. The item discriminations are assumed to be invariant across groups, implying that \(a_{i2}=a_{i}\). Additionally, a random uniform DIF [16] effect \(e_{i}\) is added to the common item difficulty \(b_{i}\) to obtain the item difficulty \(b_{i2}\) in the second group. That is, we define  
\[b_{i2}=b_{i}+e_{i}\,,\,\text{where}\,\,\mathsf{E}(e_{i})=0\,\,\text{and}\,\, \text{Var}(e_{i})=\tau^{2}. \tag{3}\]  
The variance \(\tau^{2}\) is referred to as the DIF variance [24], and \(\tau\) is designated as the DIF SD. We now describe the first step of the linking approach. The 2 PL model is separately fitted for each group, assuming a standard normal distribution \(\mathsf{N}(0,1)\) for the ability variable \(\theta\). The estimated item parameters \(\widehat{a}_{i1}\) and \(\widehat{b}_{i1}\) will deviate from the true data-generating parameters \(a_{i1}=a_{i}\) and \(b_{i1}=b_{i}\) due to sampling error (of subjects). In the second group, the original item parameters \(a_{i2}\) and \(b_{i2}\) cannot be directly recovered, as the 2 PL model is fit with \(\theta\sim\mathsf{N}(0,1)\), while the true model specifies \(\theta\sim\mathsf{N}(\mu,\sigma^{2})\). Straightforward algebra yields  
\[a_{i2}(\theta-b_{i2})=a_{i2}(\sigma\theta^{*}+\mu-b_{i2})\,\,\text{for}\,\, \theta^{*}\sim\mathsf{N}(0,1). \tag{4}\]Therefore, the identified item parameters in the second group can be expressed as  
\[a_{i2}^{*}=a_{i}\sigma\text{ and }b_{i2}^{*}=\sigma^{-1}(b_{i}+e_{i}-\mu). \tag{5}\]  
Due to sampling errors, the estimated item parameters \(\widehat{a}_{i1}\) and \(\widehat{b}_{i1}\) (or \(\widehat{a}_{i2}\) and \(\widehat{b}_{i2}\)) will also deviate from \(a_{i1}^{*}\) and \(b_{i1}^{*}\) (or \(a_{i2}^{*}\) and \(b_{i2}^{*}\)). We now collect all estimated item parameters in vectors \(\widehat{a}_{g}\) and \(\widehat{b}_{g}\) for the two groups \(g=1,2\). The Stocking-Lord linking method, as introduced in [21], employs the linking function \(H\) that is defined as  
\[H(\mu,\sigma,\widehat{\mathbf{a}}_{1},\widehat{\mathbf{b}}_{1},\widehat{\mathbf{a}}_{2}, \widehat{\mathbf{b}}_{2})=\sum_{t=1}^{T}\omega_{t}\left[\sum_{i=1}^{I}\Psi( \widehat{a}_{i1}(\sigma\theta_{t}+\mu-\widehat{b}_{i1}))-\sum_{i=1}^{I}\Psi( \widehat{a}_{i2}(\theta_{t}-\widehat{b}_{i2}))\right]^{2}. \tag{6}\]  
The weights \(\omega_{t}\) are known, and a grid for the ability variable \(\theta\) is chosen, ranging between \(\theta_{1}\) and \(\theta_{T}\) for \(\theta_{t}\) with \(t=1,\ldots,T\). In applications, the grid could be chosen to have equidistant values, and the weights \(\omega_{t}\) could either be equal to 1 or are proportional to a discretized density of a normal distribution with an SD \(\sigma_{0}\) larger than 1 (e.g., an SD of 2), that is, one chooses \(\omega_{t}\propto\exp(-\theta_{t}^{2}/(2\sigma_{0}^{2}))\) with \(\sum_{t=1}^{T}\omega_{t}=1\). We would like to point out that the linking function (6) is referred to as asymmetric Stocking-Lord linking because it aligns the test characteristic function (TCF) from the first group with that of the second group.