Abstract: Hierarchical latent attribute models (HLAMs) are a family of discrete latent variable models that are attracting increasing attention in the educational, psychological, and behavioral sciences. An HLAM includes a binary structural matrix and a directed acyclic graph specifying hierarchical constraints on the configurations of the latent attributes. These components encode practitioners’ design information and carry important scientific meaning. However, despite the popularity of HLAMs, the fundamental issue of identifiability remains unaddressed. The existence of the attribute hierarchy graph leads to a degenerate parameter space, and the potentially unknown structural matrix further complicates the identifiability problem. Here, we identify the latent structure and model parameters underlying an HLAM, and develop sufficient and necessary identifiability conditions. These results directly and sharply characterize the effects on identifiability of different attribute types in the graph. The proposed conditions provide insights into diagnostic test designs under the attribute hierarchy, and serve as tools that we can use to assess the validity of an estimated HLAM.

Key words and phrases: Attribute hierarchy graph, cognitive diagnosis, identifiability, Q-matrix.