# On the Identifiability of Diagnostic Classification Models 

Guanhua Fang, Jingchen Liu, and Zhiliang Ying 

June 6, 2017 

## Abstract 

This paper establishes fundamental results for statistical inference of diagnostic classification models (DCM). The results are developed at a high level of generality, applicable to essentially all diagnostic classification models. In particular, we establish identifiability results of various modeling parameters, notably item response probabilities, attribute distribution, and Q -matrix- induced partial information structure. Consistent estimators are constructed. Simulation results show that these estimators perform well under various modeling settings. We also use a real example to illustrate the new method. The results are stated under the setting of general latent class models. For DCM with a specific parameterization, the conditions may be adapted accordingly. 

## 1 Introduction 

Cognitive diagnosis is an important area which has become increasingly popular in educational assessment, psychiatric evaluation, and many other disciplines. An incomplete list of recent devel- opments of cognitive diagnosis models include the rule space method [27, 28], the reparameterized unified/fusion model (RUM) [8, 11, 29], the conjunctive (noncompensatory) DINA and NIDA models [14, 26, 7, 30], the compensatory DINO and NIDO models [31, 30], the attribute hierarchy method [17], clustering methods [5], and the G-DINA model [6]; see also [13, 33, 22] for more approaches to cognitive diagnosis. A comprehensive review of diagnostic classification models can be found in [21]. Statistical inference methods have also been developed for these models, such as item parameter estimation/calibration [20, 24], Q -matrix estimation [3, 4, 18, 19]. 

