# Implementation Aspects in Simulation Extrapolation-Based Stocking-Lord Linking
## 4 Simulation Study
### Results  
Table 1 presents bias of the estimated mean \(\widehat{\mu}\) and the estimated SD \(\widehat{\sigma}\) as a function of the DIF SD \(\tau\), the number of items \(I\), and the sample size \(N\). All linking methods were unbiased under the no DIF conditions. In the small DIF condition (i.e, \(\tau=0.25\)), the bias was negligible. In the large DIF condition (i.e., \(\tau=0.5\)), the mean \(\widehat{\mu}\) was slightly biased,  
and the SD \(\widehat{\sigma}\) was noticeably biased for the original SL linking method. Interestingly, the simulation-based SIMEX methods SE and SA had a bias in the case of a small number of items (i.e., \(I=10\)). This bias vanished for longer tests with a larger number of items. The replication-based methods RE and RA had the least bias.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline  & & & & \(\widehat{\mu}\) & & & & & \(\widehat{\sigma}\) & & \\ \cline{3-11}
**I** & **N** & **SL** & **RE** & **RA** & **SE** & **SA** & **SL** & **RE** & **RA** & **SE** & **SA** \\ \hline \multirow{6}{*}{10} & \multicolumn{6}{c}{_No DIF (DIF SD \(\tau=0\))_} \\  & 500 & 0.005 & 0.005 & 0.005 & 0.004 & 0.004 & 0.007 & 0.008 & 0.008 & 0.008 & 0.008 \\  & 1000 & 0.001 & 0.001 & 0.001 & 0.000 & 0.000 & 0.003 & 0.004 & 0.004 & 0.004 & 0.004 \\  & 2000 & \(-0.001\) & \(-0.001\) & \(-0.001\) & \(-0.001\) & \(-0.001\) & 0.002 & 0.002 & 0.002 & 0.002 \\  & 4000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & \(-0.001\) & \(-0.001\) & \(-0.001\) & \(-0.001\) \\  & 500 & 0.002 & 0.002 & 0.002 & 0.002 & 0.002 & 0.005 & 0.006 & 0.006 & 0.006 & 0.006 \\  & 1000 & 0.003 & 0.003 & 0.003 & 0.003 & 0.003 & 0.003 & 0.003 & 0.003 & 0.003 & 0.003 \\  & 2000 & 0.001 & 0.001 & 0.001 & 0.001 & 0.001 & 0.002 & 0.002 & 0.002 & 0.002 & 0.002 \\  & 4000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.001 & 0.001 & 0.001 & 0.001 & 0.001 \\  & 500 & \(-0.001\) & \(-0.001\) & \(-0.001\) & \(-0.002\) & \(-0.002\) & 0.007 & 0.008 & 0.008 & 0.008 & 0.008 \\  & 1000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.002 & 0.002 & 0.002 & 0.002 & 0.002 \\  & 2000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.001 & 0.002 & 0.002 & 0.002 & 0.002 \\  & 4000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.001 & 0.001 & 0.001 & 0.001 & 0.001 & 0.001 \\ \hline \multirow{6}{*}{10} & \multicolumn{6}{c}{_Small DIF (DIF SD \(\tau=0.25\))_} \\  & 500 & 0.001 & 0.003 & 0.003 & \(-0.004\) & \(-0.004\) & \(-0.002\) & 0.006 & 0.006 & 0.004 & 0.004 \\  & 1000 & 0.002 & 0.004 & 0.004 & \(-0.004\) & \(-0.004\) & \(-0.005\) & 0.003 & 0.003 & 0.001 & 0.001 \\  & 2000 & \(-0.001\) & 0.001 & 0.001 & \(-0.006\) & \(-0.007\) & \(-0.008\) & 0.000 & 0.000 & \(-0.001\) & \(-0.001\) \\  & 4000 & \(-0.001\) & 0.001 & 0.001 & \(-0.006\) & \(-0.007\) & \(-0.008\) & 0.000 & 0.000 & \(-0.001\) & \(-0.001\) \\  & 500 & \(-0.001\) & 0.001 & 0.001 & 0.003 & 0.003 & \(-0.004\) & 0.004 & 0.004 & 0.004 & 0.004 \\  & 1000 & \(-0.001\) & 0.001 & 0.001 & 0.002 & 0.003 & \(-0.006\) & 0.003 & 0.003 & 0.003 & 0.003 \\  & 2000 & \(-0.001\) & 0.001 & 0.001 & 0.003 & 0.003 & \(-0.009\) & 0.001 & 0.001 & 0.000 & 0.000 \\  &Table 2 presents the relative RMSE for the estimated parameters \(\widehat{\mu}\) and \(\widehat{\sigma}\). It is important to emphasize that all four SIMEX-based SL linking methods RE, RA, SE, and SA provided similar RMSE values. Hence, SIMEX-based linking with replication (methods RE and RA), as well as simulation-based linking with approximate estimation (method SA), serve as successful alternatives to the computationally demanding SIMEX-based linking with simulation and exact estimation (i.e., method SE).