Diagnostic classification models (DCMs) are important statistical tools in cognitive diagnosis. In this paper, we consider the issue of their identifiability. In particular, we focus on one basic and popular model, the DINA model. We propose sufficient and necessary conditions under which the model parameters are identifiable from the data. The consequences, in terms of the consistency of parameter estimates, of fulfilling or failing to fulfill these conditions are illustrated via simulation. The results can be easily extended to the DINO model through the duality of the DINA and DINO models. Moreover, the proposed theoretical framework could be applied to study the identifiability issue of other DCMs.

A new factor analysis (FA) procedure has recently been proposed which can be called matrix decomposition FA (MDFA). All FA model parameters (common and unique factors, loadings, and unique variances) are treated as fixed unknown matrices. Then, the MDFA model simply becomes a specific data matrix decomposition. The MDFA parameters are found by minimizing the discrepancy between the data and the MDFA model. Several algorithms have been developed and some properties have been discussed in the literature (notably by Stegeman in Comput Stat Data Anal 99:189–203, 2016), but, as a whole, MDFA has not been studied fully yet. A number of new properties are discovered in this paper, and some existing ones are derived more explicitly. The properties provided concern the uniqueness of results, covariances among common factors, unique factors, and residuals, and assessment of the degree of indeterminacy of common and unique factor scores. The properties are illustrated using a real data example.