Comparing different specifications of mean-geometric mean linking

Mean–geometric mean (MGM) linking compares group differences on a latent variable 𝜃 within the two-parameter logistic (2PL) item response theory model. This article investigates three specifications of MGM linking that differ in the weighting of item difficulty differences: unweighted (UW), discrimination-weighted (DW), and precision-weighted (PW). These methods are evaluated under conditions where random DIF effects are present in either item difficulties or item intercepts. The three estimators are analyzed both analytically and through a simulation study. The PW method outperforms the other two only in the absence of random DIF or in small samples when DIF is present. In larger samples, the UW method performs best when random DIF with homogeneous variances affects item difficulties, while the DW method achieves superior performance when such DIF is present in item intercepts. The analytical results and simulation findings consistently show that the PW method introduces bias in the estimated group mean when random DIF is present. Given that the effectiveness of MGM methods depends on the type of random DIF, the distribution of DIF effects was further examined using PISA 2006 reading data. The model comparisons indicate that random DIF with homogeneous variances in item intercepts provides a better fit than random DIF in item difficulties in the PISA 2006 reading dataset.