Does random differential item functioning occur in one or two groups? Implications for bias and variance in asymmetric and symmetric Haebara and Stocking-Lord linking

Linking methods are frequently applied to analyze the performance of two groups on a set of items. This article shows analytically and by simulation that the occurrence of differential item functioning (DIF) can induce bias and additional variance in parameter estimates of the linking method. Interestingly, the bias of the parameter estimates of a linking method depends on whether random DIF occurs in one or two groups (i.e., the type of DIF effects). The findings are shown by means of asymmetric and symmetric Haebara and Stocking-Lord linking. Moreover, the latter linking methods were compared with recently proposed corresponding SIMEX-based variants of linking in a simulation study. It turned out that SIMEX-based linking provided unbiased estimates of whether the correct assumption of the type of DIF effects has been implemented.