# Implementation Aspects in Simulation Extrapolation-Based Stocking-Lord Linking
###### Abstract  
Stocking-Lord (SL) linking is commonly used for group comparisons based on dichotomous item responses. Recently, the simulation extrapolation (SIMEX) method for SL linking has been introduced within the two-parameter logistic (2 PL) model to reduce the bias due to the presence of random differential item functioning (DIF). However, the original SIMEX approach can be computationally complex. This article evaluates two computational shortcuts to SIMEX-based SL linking through a simulation study: a quasi-Monte Carlo method that replicates items and a proposal using an approximate noniterative estimation instead of the iterative one in the SIMEX application of SL linking. The results demonstrate that these shortcuts yield linking parameter estimates with comparable statistical properties.  
Stocking-Lord linking; SIMEX; item response model; linking; 2 PL model; approximation 1220122  
IRT models are commonly employed to compare the distribution of \(\mathbf{X}\) across two groups, focusing on the parameters associated with the distribution of the ability variable \(\theta\) in the IRT model (1). This paper examines linking methods [9] for conducting group comparisons. Linking methods involve a two-step process. First, the IRT model is estimated separately for each group. In the second step, differences in the estimated item parameters are utilized to compute distributional differences of the \(\theta\) variable across the two groups using a linking method [9; 10; 11; 12; 13; 14].  
In this article, we apply linking methods in the presence of differential item functioning (DIF; [15; 16; 17]). DIF implies a heterogeneous functioning of items across groups, that is, item discrimination or item difficulty in the 2 PL model could differ across the two groups. It has been shown that the presence of random DIF [18; 19] can result in biased distribution parameter estimates in nonlinear linking methods [20].  
The Stocking-Lord (SL) linking method [21], as a nonlinear linking method [9], also yields biased distribution parameter estimates in the presence of random DIF effects [20]. To this end, the simulation extrapolation (SIMEX; [22]) measurement error correction method has been applied in SL linking [23]. It turns out that the SIMEX method is effective in reducing the bias in SL linking [23]. However, a purely simulation-based SIMEX method can be computationally demanding requiring repeated application of the iterative SL linking procedure many times. Therefore, two computational shortcuts are introduced and compared with the original SIMEX procedure in this article through a simulation study. It is hypothesized that these shortcuts will yield results comparable to those of the original SIMEX-based SL linking procedure.  
The article proceeds with the following structure. Section 2 briefly describes the SL linking method. Section 3 reviews the general concept of the SIMEX method and its application to SL linking. Furthermore, two computational shortcuts to SIMEX-based SL linking are proposed in this section. Section 4 presents the findings of a simulation study comparing the performance of estimators from different SIMEX implementations of SL linking. The article concludes with a discussion in Section 5.