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.
- A complete Q-matrix is designed to captures all necessary attributes associated with each item.
Identifiability condition #5 (C5)
For the DINA or DINO model, the Q-matrix is complete if and only if it consists of a diagnoal matrix with the following form
\[ Q=\begin{pmatrix} I_K \\ Q^* \end{pmatrix} \] 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,
\[ \eta_{j c}= \begin{cases}0 & \text { if } \alpha_c^T \mathbf{q}_j<\mathbf{q}_j^T \mathbf{q}_j \\ 1 & \text { otherwise }\end{cases} \]
and for the DINO model,
\[ \eta_{j c}= \begin{cases}0 & \text { if } \alpha_c^T \mathbf{q}_j=0 \\ 1 & \text { otherwise }\end{cases} \] Take the Q-matrix below as an example.
Code
| 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, ] %*% Q1[j, ]) >=
sum(Q[j, ]))
}
}
colnames(Gamma.matrix)<- c("00","10","01", "11")
rownames(Gamma.matrix)<- c("item 1", "item 2", "item 3", "item 4", "item 5")
knitr::kable(Gamma.matrix)| 00 | 10 | 01 | 11 | |
|---|---|---|---|---|
| item 1 | 0 | 1 | 0 | 1 |
| item 2 | 0 | 0 | 1 | 1 |
| item 3 | 0 | 1 | 1 | 1 |
| item 4 | 0 | 0 | 0 | 1 |
| item 5 | 0 | 0 | 0 | 1 |
At least two columns of \(\Gamma\) matrix are unique, so th model is partially identifiable.
Identifiability Conditions #6(C6)
When DINA or DINO model is used and the guessing and slip paramters are known, the following condition is sufficient and necessary for identification of population proportion parameters(Xu, 2019).
Any two columns in \(\Gamma\) are distinct.
Identifiability conditions #7 (C7)
For other CDMs, a diagonal matrix in the Q-matrix is sufficient but not necessary condition for Q-matrix completeness.