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Ensuring fairness in instruments like survey questionnaires or educationaltests is crucial. One way to address this is by a Differential Item Functioning(DIF) analysis, which examines if different subgroups respond differently to aparticular item, controlling for their overall latent construct level. DIFanalysis is typically conducted to assess measurement invariance at the itemlevel. Traditional DIF analysis methods require knowing the comparison groups(reference and focal groups) and anchor items (a subset of DIF-free items).Such prior knowledge may not always be available, and psychometric methods havebeen proposed for DIF analysis when one piece of information is unknown. Morespecifically, when the comparison groups are unknown while anchor items areknown, latent DIF analysis methods have been proposed that estimate the unknowngroups by latent classes. When anchor items are unknown while comparison groupsare known, methods have also been proposed, typically under a sparsityassumption -- the number of DIF items is not too large. However, DIF analysiswhen both pieces of information are unknown has not received much attention.This paper proposes a general statistical framework under this setting. In theproposed framework, we model the unknown groups by latent classes and introduceitem-specific DIF parameters to capture the DIF effects. Assuming the number ofDIF items is relatively small, an $L_1$-regularised estimator is proposed tosimultaneously identify the latent classes and the DIF items. A computationallyefficient Expectation-Maximisation (EM) algorithm is developed to solve thenon-smooth optimisation problem for the regularised estimator. The performanceof the proposed method is evaluated by simulation studies and an application toitem response data from a real-world educational test.

DIF Analysis with Unknown Groups and Anchor Items