Disentangling Direct and Indirect Interactions in Polytomous Item Response Theory Models

Measurement is at the core of scientific discovery. However, some quantities, such as economic behavior or intelligence, do not allow for direct measurement. They represent latent constructs that require surrogate measurements. In other scenarios, non-observed quantities can influence the variables of interest. In either case, models with latent variables are needed. Here, we investigate fused latent and graphical models that exhibit continuous latent variables and discrete observed variables. These models are characterized by a decomposition of the pairwise interaction parameter matrix into a group-sparse component of direct interactions and a low-rank component of indirect interactions due to the latent variables. We first investigate when such a decomposition is identifiable. Then, we show that fused latent and graphical models can be recovered consistently from data in the high-dimensional setting. We support our theoretical findings with experiments on synthetic and real-world data from polytomous item response theory studies.