6.6 Partial Identifiability: Completeness of the Q-matrix
A Q-matrix is said to be complete if it guarantees the identifiability of all possible proficiency classes.
Identifiability Conditions #5
For the DINA or DINO model, the Q-matrix is complete if and only if it consists of a diagonal matrix with the following form (Köhn & Chiu, 2018)
\[ Q=\begin{pmatrix} I_K \\ Q^{\star} \end{pmatrix} \]
To better understand the partial identifiability of different CDMs, we can consider an indicator matrix \(\Gamma\).
- The rows in \(\Gamma\) represent items
- The columns represent different attribute profiles
- The element \(\Gamma_{jc}=\eta_{jc}\), where for the DINA model, \[\begin{equation} \eta_{jc}= \begin{cases} 0 & \quad \text{if } \mathbf{\alpha}_{c}^T\mathbf{q}_j < \mathbf{q}_{j}^T\mathbf{q}_j\\ 1 & \quad \text{otherwise} \end{cases} \end{equation}\] and for the DINO model, \[\begin{equation} \eta_{jc}= \begin{cases} 0 & \quad \text{if } \mathbf{\alpha}_{c}^T\mathbf{q}_j =0\\ 1 & \quad \text{otherwise} \end{cases} \end{equation}\]
| Attribute 1 | Attribute 2 | |
|---|---|---|
| Item 1 | 1 | 0 |
| Item 2 | 0 | 1 |
| Item 3 | 1 | 1 |
| Item 4 | 1 | 1 |
| Item 5 | 1 | 1 |
Under the DINA model, the \(\Gamma\) matrix is given below:
Code
all.patterns <- GDINA::attributepattern(K = 2)
C <- nrow(all.patterns)
J <- nrow(Q)
Gamma.matrix <- matrix(NA, J, C)
for (l in 1:C) {
for (j in 1:J) {
Gamma.matrix[j, l] <- 1 * (drop(all.patterns[l, ] %*% Q[j, ]) >=
sum(Q[j, ]))
}
}| 00 | 10 | 01 | 11 | |
|---|---|---|---|---|
| Item 1 | 0 | 1 | 0 | 1 |
| Item 2 | 0 | 0 | 1 | 1 |
| Item 3 | 0 | 0 | 0 | 1 |
| Item 4 | 0 | 0 | 0 | 1 |
| Item 5 | 0 | 0 | 0 | 1 |
Identifiability Conditions #6
When DINA or DINO model is used and the guessing and slip parameters are known, the following condition is sufficient and necessary for identification of population proportion parameters (G. Xu, 2019).
Any two columns in \(\Gamma\) are distinct.
Identifiability Conditions #7
For other CDMs, a diagonal matrix in the Q-matrix is sufficient but not necessary condition for Q-matrix completeness. (Köhn & Chiu, 2018)