  Ensuring fairness in instruments like survey questionnaires or educational
tests is crucial. One way to address this is by a Differential Item Functioning
(DIF) analysis, which examines if different subgroups respond differently to a
particular item, controlling for their overall latent construct level. DIF
analysis is typically conducted to assess measurement invariance at the item
level. 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 have
been proposed for DIF analysis when one piece of information is unknown. More
specifically, when the comparison groups are unknown while anchor items are
known, latent DIF analysis methods have been proposed that estimate the unknown
groups by latent classes. When anchor items are unknown while comparison groups
are known, methods have also been proposed, typically under a sparsity
assumption -- the number of DIF items is not too large. However, DIF analysis
when both pieces of information are unknown has not received much attention.
This paper proposes a general statistical framework under this setting. In the
proposed framework, we model the unknown groups by latent classes and introduce
item-specific DIF parameters to capture the DIF effects. Assuming the number of
DIF items is relatively small, an $L_1$-regularised estimator is proposed to
simultaneously identify the latent classes and the DIF items. A computationally
efficient Expectation-Maximisation (EM) algorithm is developed to solve the
non-smooth optimisation problem for the regularised estimator. The performance
of the proposed method is evaluated by simulation studies and an application to
item response data from a real-world educational test.
