# Implementation Aspects in Simulation Extrapolation-Based Stocking-Lord Linking
## 3 Proposed Computational Shortcuts in SIMEX-Based Linking
### Approximate Estimation in SIMEX-Based Linking  
In this section, we describe a computational shortcut to SIMEX-based linking. The idea is to avoid the computationally demanding iterative estimation of SL linking and replace the estimation with a one-step updating approach that is based on a Taylor approximation of the linking function. The technique has also been used to derive the bias and variance of nonlinear linking functions [20; 30].  
Let \(\widehat{\delta}=(\widehat{\mu},\widehat{\sigma})\) represent the vector of parameter estimates that is the root of  
\[H_{\delta}(\widehat{\delta},\mathbf{a}_{1},\mathbf{b}_{1},\mathbf{a}_{2},\mathbf{b}_{2})=\mathbf{ 0}, \tag{18}\]  
where \(H_{\delta}\) is the vector of partial derivatives of the linking function \(H\) with respect to the parameter vector \(\delta=(\mu,\sigma)\). It is vital to highlight that the notation in (18) refers to original item parameters \(\mathbf{a}_{\delta}\) and \(b_{\delta}\) rather than the identified item parameters \(\widehat{\mathbf{a}}_{\delta}\) and \(\widehat{b}_{\delta}\) for \(g=1,2\). In the SIMEX-based approach to linking, we substitute the item parameters \(\mathbf{b}_{2}=(b_{12},\ldots,b_{l2})\) with pseudo-item parameters \(\mathbf{b}_{2}^{*}=(b_{12}^{*},\ldots,b_{l2}^{*})\) that include an additional error variance, depending on the parameter \(\lambda\). The parameter vector \(\delta^{*}\) based on pseudo-item parameters in SIMEX is obtained as the root of  
\[H_{\delta}(\delta^{*},\mathbf{a}_{1},\mathbf{b}_{1},\mathbf{a}_{2},\mathbf{b}_{2}^{*})=\mathbf{0}. \tag{19}\]  
In the following, we apply a Taylor expansion of \(H_{\delta}\) in (19) around \(\widehat{\delta}\) in the first order and \(\mathbf{b}_{2}\) up to the second order. We obtain  
\[\begin{array}{rcl}\mathbf{0}&=&H_{\delta}(\delta^{*},\mathbf{a},\mathbf{b}_{1},\mathbf{a}, \mathbf{b}_{2}^{*})\\ &=&H_{\delta}(\widehat{\delta},\mathbf{a}_{1},\mathbf{b}_{1},\mathbf{a}_{2},\mathbf{b}_{2})+H_ {\delta\delta}(\widehat{\delta},\mathbf{a}_{1},\mathbf{b}_{1},\mathbf{a}_{2},\mathbf{b}_{2}) \Big{(}\delta^{*}-\widehat{\delta}\Big{)}\\ &&\qquad+\sum_{i=1}^{I}H_{\delta b_{i2}}(\widehat{\delta},\mathbf{a}_{1},\mathbf{b}_{1 },\mathbf{a}_{2},\mathbf{b}_{2})(b_{i2}^{*}-b_{i2})\\ &&\qquad+\frac{1}{2}\sum_{i=1}^{I}H_{\delta b_{i2}b_{i2}}(\widehat{\delta}, \mathbf{a}_{1},\mathbf{b}_{1},\mathbf{a}_{2},\mathbf{b}_{2})(b_{i2}^{*}-b_{i2})^{2}.\end{array} \tag{20}\]  
Due to (18), we obtain an approximate estimate in a one-step iteration as  
\[\delta^{*}-\widehat{\delta}=-H_{\delta\delta}^{-1}\Bigg{[}\sum_{i=1}^{I}H_{ \delta b_{i2}}(b_{i2}^{*}-b_{i2})+\frac{1}{2}\sum_{i=1}^{I}H_{\delta b_{i2}b_ {i2}}(b_{i2}^{*}-b_{i2})^{2}\Bigg{]}. \tag{21}\]  
Arguments in the derivatives are omitted in (21). For instance, we use the short notation \(H_{\delta\delta}=H_{\delta\delta}(\widehat{\delta},\mathbf{a}_{1},\mathbf{b}_{1},\mathbf{ a}_{2},\mathbf{b}_{2})\). As a consequence, the approximation (21) to the linking estimation problem is a linear function in \(\delta\). Moreover, the dependence with respect to \(\mathbf{b}_{2}\) is allowed to be quadratically related, and the approximation error will typically decrease for smaller differences \(\mathbf{b}_{2}^{*}-\mathbf{b}_{2}\). The approximate estimation of the linking method can be utilized in both cases, for pseudo-item parameters \(\mathbf{b}_{2}^{*}\) that emerge from simulation or replication. However, the reduction in computational demand is more important for SIMEX-based linking that relies on simulation.  
In an implementation of an approximate estimation in SIMEX-based SL linking, the required partial derivatives can be analytically computed in a straightforward way or can be computed via numerical differentiation.