Equivalency of the DINA Model and a Constrained General Diagnostic

This report shows that the deterministic-input noisy-AND (DINA) model is a special case of more general compensatory diagnostic models by means of a reparameterization of the skill space and the design (Q-) matrix of item by skills associations. This reparameterization produces a compensatory model that is equivalent to the (conjunctive) DINA model, and is valid for all types of complex structure Q-matrices, not only for trivial cases. This equivalency uses the GDM as the basis, is not based on recent developments of diagnosis models such as G-DINA or LCDM. Model equivalency is a topic of some relevance as soon as researchers want to draw conclusions derived from any particular model-based estimates. It can be shown that for multidimensional models, there are often multiple ways to specify different sets of latent variables and their relationships to observed variables. This report goes beyond showing that multiple versions of a design matrix lead to the same model-based conditional probability space; it shows that a conjunctive diagnostic classification model can be expressed as a constrained special case of a compensatory diagnostic modeling framework.