# Implementation Aspects in Simulation Extrapolation-Based Stocking-Lord Linking
## 5 Discussion  
The literature highlights that SL linking produces biased estimates when random DIF is present in item difficulties under the 2 PL model. To address this issue, a bias reduction method for SL linking using the SIMEX technique has been proposed. The core idea of applying SIMEX to SL linking is to treat random DIF as covariates affected by measurement  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**I**} & \multirow{2}{*}{**N**} & \multicolumn{4}{c}{\(\widehat{\mu}\)} & \multicolumn{4}{c}{\(\widehat{\sigma}\)} \\ \cline{3-12}  & & \multicolumn{1}{c}{**SL**} & \multicolumn{1}{c}{**RE**} & \multicolumn{1}{c}{**RA**} & \multicolumn{1}{c}{**SE**} & \multicolumn{1}{c}{**SA**} & \multicolumn{1}{c}{**SL**} & \multicolumn{1}{c}{**RE**} & \multicolumn{1}{c}{**RA**} & \multicolumn{1}{c}{**SE**} & \multicolumn{1}{c}{**SA**} \\ \hline \multirow{9}{*}{10} & \multirow{9}{*}{10} & \multicolumn{1}{c}{\(N\)_DIF (DIF SD_\(\tau=0\))_} & & & & & & & & & & \\  & 500 & 100.0 & 100.0 & 100.0 & 100.0 & 100 & 99.7 & 100.0 & 100.0 & 100.0 & 100 \\  & 1000 & 100.0 & 100.0 & 100.0 & 100.0 & 100 & 99.9 & 100.0 & 100.0 & 100.0 & 100 \\  & 2000 & 100.0 & 100.0 & 100.0 & 100.0 & 100 & 99.9 & 100.0 & 100.0 & 100.0 & 100 \\  & 4000 & 100.0 & 100.0 & 100.0 & 100.0 & 100 & 100.0 & 100.0 & 100.0 & 100 \\  & 500 & 99.9 & 100.0 & 100.0 & 100.0 & 100 & 99.7 & 100.0 & 100.0 & 100.0 & 100 \\
20 & 1000 & 99.9 & 100.0 & 100.0 & 100.0 & 100 & 99.9 & 100.0 & 100.0 & 100.0 & 100 \\  & 2000 & 100.0 & 100.0 & 100.0 & 100.0 & 100 & 100.0 & 100.0 & 100.0 & 100 \\  & 4000 & 100.0 & 100.0 & 100.0 & 100.0 & 100 & 100.0 & 100.0 & 100.0 & 100 \\  & 500 & 100.0 & 100.0 & 100.0 & 100.0 & 100 & 99.8 & 100.0 & 100.0 & 100.0 & 100 \\
30 & 1000 & 100.0 & 100.0 & 100.0 & 100.0 & 100 & 99.9 & 100.0 & 100.0 & 100.0 & 100 \\  & 2000 & 100.0 & 100.0 & 100.0 & 100.0 & 100 & 100.0 & 100.0 & 100.0 & 100.0 & 100 \\  & 4000 & 100.0 & 100.0 & 100.0 & 100.0 & 100 & 100.0 & 100.0 & 100.0 & 100.0 & 100 \\ \hline \multirow{9}{*}{10} & \multicolumn{4}{c}{_Small DIF DIF (DIF SD_\(\tau=0.25\))_} & & & & & & & & & \\  & 500 & 99.4 & 99.9 & 99.9 & 100.0 & 100 & 99.3 & 100.3 & 100.2 & 100.0 & 100 \\  & 1000 & 99.4 & 100.1 & 100.1 & 100.0 & 100 & 99.4 & 100.3 & 100.3 & 100.0 & 100 \\  & 2000 & 99.2 & 99.8 & 99.8 & 100.0 & 100 & 100.4 & 100.3 & 100.2 & 100.1 & 100 \\  & 4000 & 99.1 & 99.8 & 99.8 & 99.9 & 100 & 100.8 & 100.3 & 100.3 & 100.1 & 100 \\  & 500 & 99.4 & 100.0 & 100.0 & 100.0 & 100 & 99.4 & 100.1 & 100.1 & 100.0 & 100 \\
20 & 1000 & 99.3 & 100.0 & 100.0 & 100.0 & 100 & 99.5 & 100.1 & 100.1 & 100.0 & 100 \\  & 2000 & 99.2 & 100.0 & 100.0 & 100.0 & 100 & 101.1 & 10 errors. However, SIMEX-based linking is computationally intensive. This is because the nonlinear optimization of the SL linking function requires an iterative estimation, as no closed-form solution exists. Additionally, the SIMEX technique relies on simulation by introducing artificially generated random DIF into the item parameters, necessitating repeated estimation across numerous resimulated datasets. To mitigate the computational burden, this article explored two shortcuts for the SIMEX-based SL linking method, which depends heavily on Monte Carlo simulation. The first shortcut, referred to as the replication approach, substituted the assumed normal distribution of random DIF with a deterministic replication of items. This replication generated DIF items that approximated the exact normal distribution. The second shortcut replaced the iterative estimation in SL linking with a one-step minimization based on a Taylor expansion of the SL linking function. Since the resimulated datasets in SIMEX only slightly perturb the original item parameters, the one-step approach may provide a reasonable approximation. The two shortcuts were compared to the original SIMEX-based SL linking procedure through a simulation study.