When Do Item Response Function and Mantel-Haenszel Definitions of Differential Item Functioning Coincide?

A test item is typically considered free of differential item functioning (DIF) if its item response function is the same across demographic groups. A popular means of testing for DIF is the Mantel-Haenszel (MH) approach. Holland and Thayer (1988) showed that, under the Rasch model, identity of item response functions across demographic groups implies that the MH null hypothesis will be satisfied when the MH matching variable is test score, including the studied item. This result, however, cannot be generalized to the class of items for which item response functions are monotonic and local independence holds. Suppose that all item response functions are identical across groups, but the ability distributions for the two groups are stochastically ordered. In general, the population MH result will show DIF favoring the higher group on some items and the lower group on others. If the studied item is excluded from the matching criterion under these conditions, the population MH result will always show DIF favoring the higher group.