Attribute Hierarchy Models in Cognitive Diagnosis: Identifiability of the Latent Attribute Space and Conditions for Completeness of the Q-Matrix

Complete Q-Matrices in General Attribute Structure Models

“…Characterizations of the identifiability of AHMs as well as general attribute structure models are provided in Section 6. This generalizes a recently published result on the identifiability of AHMs by Cai et al (2018), and corrects a false theorem by Köhn and Chiu (2018). While proofs are formulated within the KST framework, results are immediately recasted in CDM terms.…”

“…In other words, a Q-matrix is complete if and only if it contains the (k × k) identity matrix as a submatrix, with k the number of attributes. In recently published papers Cai et al (2018) and Köhn and Chiu (2018) independently claim that they have generalized this result to attribute hierarchies (as suggested by Leighton et al (2004)). The following formulation restates the claim within the terminology introduced above.…”

“…This is a clear limitation, and one would be interested in a characterization of the general case, which refrains from imposing such kind of constraints, and is sometimes referred to as independent Q-matrix design (de la Torre, Hong, & Deng, 2010;Liu, Huggins-Manley, & Bradshaw, 2017;Templin & Bradshaw, 2014). This is exactly the situation considered in Theorem 1 of Köhn and Chiu (2018). However, we argue that the "only if" part of this theorem is invalid, and the "proof" provided by the authors is flawed.…”

“…The following formulation restates the claim within the terminology introduced above. Notice that Köhn and Chiu (2018) consider the DINA model and use a different, but in this context equivalent notion of completeness (see Appendix for details).…”

“…Characterizations of interesting properties of CDMs (like, e.g., identifiability) were obtained at first under the assumption of independent attributes. How these results change when attributes are interdependent is currently a highly active area (e.g., Cai, Tu, & Ding, 2018;Köhn & Chiu, 2018). This research strand is known under the label Attribute Hierarchy Models (AHMs).…”