Efficient Models for Cognitive Diagnosis With Continuous and Mixed-Type Latent Variables

The issue of latent trait granularity in diagnostic models is considered, comparing and contrasting latent trait and latent class models used for diagnosis. Relationships between conjunctive cognitive diagnosis models (CDMs) with binary attributes and noncompensatory multidimensional item response models are explored, leading to a continuous generalization of the Noisy Input, Deterministic ‘‘And’’ Gate (NIDA) model. A model that combines continuous and discrete latent variables is proposed that includes a noncompensatory item response theory (IRT) term and a term following the discrete attribute Deterministic Input, Noisy ‘‘And’’ Gate (DINA) model in cognitive diagnosis. The Tatsuoka fraction subtraction data are analyzed with the proposed models as well as with the DINA model, and classification results are compared. The applicability of the continuous latent trait model and the combined IRT and CDM is discussed, and arguments are given for development of simple models for complex cognitive structures.