Robust and Nonrobust Linking of Two Groups for the Rasch Model with Balanced and Unbalanced Random DIF: A Comparative Simulation Study and the Simultaneous Assessment of Standard Errors and Linking Errors with Resampling Techniques

In this article, the Rasch model is used for assessing a mean difference between two groups for a test of dichotomous items. It is assumed that random differential item functioning (DIF) exists that can bias group differences. The case of balanced DIF is distinguished from the case of unbalanced DIF. In balanced DIF, DIF effects on average cancel out. In contrast, in unbalanced DIF, the expected value of DIF effects can differ from zero and on average favor a particular group. Robust linking methods (e.g., invariance alignment) aim at determining group mean differences that are robust to the presence of DIF. In contrast, group differences obtained from nonrobust linking methods (e.g., Haebara linking) can be affected by the presence of a few DIF effects. Alternative robust and nonrobust linking methods are compared in a simulation study under various simulation conditions. It turned out that robust linking methods are preferred over nonrobust alternatives in the case of unbalanced DIF effects. Moreover, the theory of M-estimation, as an important approach to robust statistical estimation suitable for data with asymmetric errors, is used to study the asymptotic behavior of linking estimators if the number of items tends to infinity. These results give insights into the asymptotic bias and the estimation of linking errors that represent the variability in estimates due to selecting items in a test. Moreover, M-estimation is also used in an analytical treatment to assess standard errors and linking errors simultaneously. Finally, double jackknife and double half sampling methods are introduced and evaluated in a simulation study to assess standard errors and linking errors simultaneously. Half sampling outperformed jackknife estimators for the assessment of variability of estimates from robust linking methods.