A Unified Theory of the Completeness of Q-Matrices for the DINA Model

Diagnostic classification models in educational measurement describe ability in a knowledge domain as a composite of specific binary skills called “cognitive attributes,” each of which an examinee may or may not have mastered. Attribute Hierarchy Models (AHMs) account for the possibility that attributes are dependent by imposing a hierarchical structure such that mastery of one or more attributes is a prerequisite of mastering one or more other attributes. Thus, the number of meaningfully defined attribute combinations is reduced, so that constructing a complete Q-matrix may be challenging. (The Q-matrix of a cognitively diagnostic test documents which attributes are required for solving which item; the Q-matrix is said to be complete if it guarantees the identifiability of all realizable proficiency classes among examinees.) For structured Q-matrices (i.e., the item attribute profiles are restricted to reflect the hierarchy postulated to underlie the attributes), the conditions of completeness have been established. However, sometimes, a structured Q-matrix cannot be assembled because the items of the test in question have attribute profiles that do not conform to the prerequisite structure imposed by the postulated attribute hierarchy. A Q-matrix composed of such items is called “unstructured.” In this article, the completeness conditions of unstructured Q-matrices for the DINA model are presented. Specifically, there exists an entire range of Q-matrices that are all complete for DINA-AHMs. Thus, the theoretical results presented here can be combined with extant insights about Q-completeness for models without attribute hierarchies into a unified framework on the completeness of Q-matrices for the DINA model.