Nothing lasts forever – only item administration: An Item Response Theory algorithm to shorten tests
4-PL - Item Response Function
\[P(x_{pi}= 1| \theta_p, b_i, a_i, c_i, d_i) = c_i + (d_i -c_i) \dfrac{\exp[a_i(\theta_p - b_i)]}{1 + \exp[a_i(\theta_p - b_i)]}\]
4-PL - Item Information Function
\[ \text{IIF}_{i}(\theta) = \dfrac{a_i^2[P(\theta)-c_i]^2[d_i - P(\theta)]^2}{(d_{i}-c_i)^2 P(\theta)Q(\theta)}\]
4-PL - Test Information Function
\[TIF = \sum_{i = 1}^{||B||} IIF_i\] (\(B\): Set of items in a test (\(||X||\) cardinality of set \(X\)))
Not a property of the item!
\(d\) depends on the \(r\) rank of the item presentation during the administration, \(d_r\):
Competing algorithms
\[\text{TIF}^*\]
\[k = 0\]
\[k = 0\]
\[k = 0\]
\[k = 0\]
\[k = 0\]
\[k = 0\]
\[k = 1\]
\[k = 1\]
\[k = 1\]
\[k = 1\]
Frank
At \(k = 0\): \(\text{TIF}^0(\theta) = 0 \, \forall \theta\), \(Q^0 = \emptyset\).
For \(k \geq 0\),
- \(A^k = B \setminus Q^k\)
- \(\forall i \in A^k\), \(p\text{TIF}_{i}^k = \frac{\text{TIF}^k + \text{IIF}_{i}}{||Q^k||+1}\), with \(d_i = 1\), \(\forall i\)
\(i^* = \arg \min_{i \in A^k} (|\text{TIF}^* - \text{pTIF}_i^k|)\)
Termination criterion: \(|\text{TIF}^* - \text{pTIF}_{i^*}^k| \geq |\text{TIF}_B - \text{TIF}^{k}|\):
FALSE: \(Q^{k+1} = Q^{k} \cup \{i^*\}\), \(\text{TIF}^{k+1} = p\text{TIF}_{i^*}\), iterates 1-4
TRUE: Stop, \(Q_{\text{Frank}} = Q^k\)
Léon
At \(k = 0\): \(\text{TIF}^0(\theta) = 0 \, \forall \theta\), \(Q^0 = \emptyset\).
For \(k \geq 0\),
- \(A^k = B \setminus Q^k\)
- \(\forall i \in A^k\), \(p\text{TIF}_{i}^k = \frac{\text{TIF}^k + \text{IIF}_{i}}{||Q^k||+1}\), with \(r = \{0, 1, \ldots, ||Q^k||-1\}\)
\(i^* = \arg \min_{i \in A^k} (|\text{TIF}^* - \text{pTIF}_i^k|)\)
Termination criterion: \(|\text{TIF}^* - \text{pTIF}_{i^*}^k| \geq |\text{TIF}_B - \text{TIF}^{k}|\):
FALSE: \(Q^{k+1} = Q^{k} \cup \{i^*\}\), \(\text{TIF}^{k+1} = p\text{TIF}_{i^*}\), iterates 1-4
TRUE: Stop, \(Q_{\text{Léon}} = Q^k\)
