Identifiability of Partial-Mastery Cognitive Diagnostic Models

Abstract

Partial-mastery (PM) cognitive diagnostic models (CDMs) extend traditional CDMs by replacing binary latent attribute mastery indicators with continuous mastery scores for multiple latent attributes. In PM-CDMs, each subject is characterized by a fixed continuous latent mastery vector, from which item-specific binary attribute profiles are independently generated. This formulation provides a bridge between classical CDMs and continuous latent variable models. Despite growing interest in PM-CDMs, their identifiability properties remain unexplored. In this work, we establish the first identifiability results for PM-CDMs. We derive sufficient conditions for identifiability that are direct analogues of established conditions for traditional CDMs. To develop the main argument, we use symbolic computation on a minimal example with five items and two latent attributes to show that the Jacobian of the model parameterization is generically nonzero. Combining tools from real analysis and algebraic statistics, we prove that this local property implies generic finite-to-one identifiability of the item parameters and the marginal distributions of the relevant latent attributes. We further show that if the $Q$-matrix contains such identifiable local structures for all attribute pairs, identifiability extends to the full PM-CDM. These findings provide a rigorous theoretical foundation for estimation and inference in partial-mastery cognitive diagnostic models.