6.3 Global Identifiability for General CDMs
It is important to note that identifiability is a property of both CDM and the Q-matrix. The following condition is related to more general models, such as R-RUM, A-CDM, and G-DINA model.
Identifiability condition #2(C2)
For any CDM, the following conditions are sufficient (but not necessary) for global model identificatoin(Xu, 2019).
- The Q-matrix has the following forms, where \(I_k\) is an identity matrix.
\[ Q=\left(\begin{array}{l} \mathcal{I}_K \\ \mathcal{I}_K \\ Q^{\prime} \end{array}\right) \] 2. There exist at least on item in \(Q'\) such that students with \(\alpha=e_k\) have different positive response probability from that of students with \(\alpha=0\).
The condition C2 indicates that each attribute is required at least two items, to ensure sufficient information to identify the model parameters for each attribute.
Hence, we can have follwoing sufficient conition.
Identifiability condition #3(C3)
For any CDM, the following condition is sufficient(but not necessary) for global model identification. The Q-matrix consists of three identity matrices.
\[ Q=\left(\begin{array}{l} I_K \\ I_K \\ I_K \\ Q^{\prime} \end{array}\right) \]
Xu (2019) recommended that we use a complete Q-matrix for the CDMs, and at least three items for each attribute to assess identifiability.