# Implementation Aspects in Simulation Extrapolation-Based Stocking-Lord Linking
## 4 Simulation Study  
### Method  
This simulation study employed the 2 PL model to generate item responses of subjects from two groups. In the first group, the ability variable \(\theta\) in the IRT model defined in (1) was normally distributed with a mean of zero and a standard deviation of one. For the second group, the ability variable \(\theta\) had a mean of \(\mu=0.3\) and a standard deviation of \(\sigma=1.2\). The number of items \(I\) in the simulation was set to 10, 20, or 30. The group-specific item parameters \(a_{ig}\) and \(b_{ig}\) in the 2 PL model for \(i=1,\ldots,I\) and \(g=1,2\) were based on common item parameters \(a_{i}\) and \(b_{i}\), which remained fixed across replications in the simulation, along with randomly generated uniform and normally distributed DIF effects. Note that uniform DIF effects \(e_{i}\) were simulated in each replication of the simulation study. For the case of \(I=10\) items, the common item discriminations \(a_{i}\) were selected as follows: 1.14, 1.05, 1.13, 0.79, 0.70, 1.13, 1.28, 0.87, 0.80, and 1.11. The mean of the \(a_{i}\) parameters was \(M=1.000\) and the standard deviation was \(SD=0.194\), representing a test with moderately discriminating items and moderate variability in item discrimination. The common item difficulties \(b_{i}\) were chosen as 0.86, \(-0.60\), 1.45, \(-1.18\), \(-0.71\), \(-0.94\)\(-0.19\), \(0.17\), \(1.55\), and \(-0.41\). The \(b_{i}\) parameters had a mean \(M=0.000\) and a standard deviation \(SD=0.979\), resulting in a test with items of average difficulty. For item counts of \(I=20\) and \(I=30\), the item parameters of the 10 items were duplicated in multiples of 10. Uniform DIF effects were simulated by adding a normally distributed random DIF effect with zero mean and DIF variance \(\tau^{2}\) to item difficulties in the second group. More formally, we defined \(b_{i1}=b_{i}\) and \(b_{i2}=b_{i}+e_{i}\). The group-specific item discriminations \(a_{ig}\) were chosen to be equal (i.e., \(a_{i1}=a_{i2}=a_{i}\)). The DIF SD \(\tau\) was chosen as 0, 0.25, and 0.50, referring to situations of no DIF, small DIF, and large DIF. Furthermore, item responses from the 2 PL model were simulated for finite sample sizes \(N=500\), 1000, 2000, and 4000. In each of the 4 (sample size \(N\)) \(\times\) 3 (number of items \(I\))\(\times\) 3 (DIF SD \(\tau\)) = 36 cells of the simulation, 3000 replications were conducted. In total, five different variants of SL linking were applied to each of the simulated datasets. First, method \"SL\" denotes the original SL linking that is prone to bias in the presence of random DIF. The SL linking methods \"RE\" and \"RA\" denote the SIMEX-based linking approach utilizing replication (see Section 3.3) and the exact and approximate (see Section 3.4) SL estimate, respectively. The replication methods in this simulation study used a replication factor of \(M=41\) (see Section 3.3). For example, the pseudo-datasets in SIMEX included \(MI=820\) pseudo-items in the condition with \(I=20\) items. The SL methods \"SE\" and \"SA\" denote the simulation-based linking methods utilizing the exact and approximate SL estimates, respectively. The simulation approach for SIMEX in this study used 100 Monte Carlo replications for the computation of the parameter curves \(\widehat{\delta}(\lambda)\). For all five SL methods, the empirical bias was computed for the estimated group mean \(\widehat{\mu}\) and the estimated SD \(\widehat{\sigma}\). Let \(\psi\) denote the parameter \(\mu\) or \(\sigma\) and let \(\widehat{\psi}\) represent the corresponding linking parameter estimate. Additionally, let \(\widehat{\psi}_{r}\) denote the parameter estimate in the \(r\)th replicated dataset for \(r=1,\ldots,R\).