The DINA model as a constrained general diagnostic model: Two variants of a model equivalency

The 'deterministic-input noisy-AND' (DINA) model is one of the more frequently applied diagnostic classification models for binary observed responses and binary latent variables. The purpose of this paper is to show that the model is equivalent to a special case of a more general compensatory family of diagnostic models. Two equivalencies are presented. Both project the original DINA skill space and design Q-matrix using mappings into a transformed skill space as well as a transformed Q-matrix space. Both variants of the equivalency produce a compensatory model that is mathematically equivalent to the (conjunctive) DINA model. This equivalency holds for all DINA models with any type of Q-matrix, not only for trivial (simple-structure) cases. The two versions of the equivalency presented in this paper are not implied by the recently suggested log-linear cognitive diagnosis model or the generalized DINA approach. The equivalencies presented here exist independent of these recently derived models since they solely require a linear - compensatory - general diagnostic model without any skill interaction terms. Whenever it can be shown that one model can be viewed as a special case of another more general one, conclusions derived from any particular model-based estimates are drawn into question. It is widely known that multidimensional models can often be specified in multiple ways while the model-based probabilities of observed variables stay the same. This paper goes beyond this type of equivalency by showing that a conjunctive diagnostic classification model can be expressed as a constrained special case of a general compensatory diagnostic modelling framework.