Development of Short Test Forms within Item Response Theory: The shortIRT Package
2026-05-22
Item Response Function
According to the 4-Parameter Logistic Model:
\[P(x_{pi}=1|\theta_p, b_i, a_i, c_i, e_i) = P(\theta) = c_i + (e_i-c_i)\dfrac{\exp[a_i(\theta_p - b_i)]}{1 + \exp[a_i(\theta_p - b_i)]}\]
Item Characteristics Curves (ICCs)
Information Functions
\[ \text{IIF}_{i}(\theta) = \dfrac{a_i^2[P(\theta)-c_i]^2[e_i - P(\theta)]^2}{(e_{i}-c_i)^2 P(\theta)[1-P(\theta)]}\]
\[TIF(\theta) = \sum_{i = 1}^{|B|} IIF_i(\theta)\] (\(B\): Set of items in a test (\(|X|\) cardinality of set \(X\)))
Warning!
Tip
bench()
bench(item_par, iifs = NULL, theta = NULL, num_item = NULL)set.seed(1312)
n = 10
item_par = data.frame(b = runif(n, -3,3),
a = runif(n, .7, 1.5),
c = runif(n, 0, .10),
e = 1)
theta = rnorm(1000)
stf = bench(item_par, theta = theta, num_item = 3)
summary(stf)
plot(stf)- 1
- Define the item parameters in the item bank
- 2
- Random values for the latent trait
- 3
- Generate the STF with the benchmark procedure
- 4
- Summary of the obtained STF
- 5
- Plot the resulting TIF (as compared to the TIF obtained from \(B\))
The item selection is based on the benchmark procedure.
The procedure selected the following 3 dichotomous items:
10 2 9
with parameters:
b a c e
10 2.243866 1.390367 0.02130947 1
2 -1.246596 1.283032 0.02087452 1
9 -1.171959 1.344295 0.07825710 1
These items maximize the information for thetas equal to:
2.251 -1.215 -1.077
define_targets()
define_targets(theta, num_targets = NULL, method = c("equal", "clusters"))
theta_target()
theta_target(targets, item_par)The item selection is based on the theta-target procedure with cluster-defined targets.
The procedure selected the following 3 dichotomous items:
2 6 8
with parameters:
b a c e
2 -1.2465959 1.283032 0.02087452 1
6 0.8283301 1.198013 0.01028904 1
8 0.3331977 1.208366 0.09370018 1
These items maximize the information for thetas equal to:
-1.207019 1.213158 0.01562251
The item selection is based on the theta-target procedure with equally-spaced targets.
The procedure selected the following 3 dichotomous items:
10 8 2
with parameters:
b a c e
10 2.2438664 1.390367 0.02130947 1
8 0.3331977 1.208366 0.09370018 1
2 -1.2465959 1.283032 0.02087452 1
These items maximize the information for thetas equal to:
2.483388 0.04465118 -2.394086
Cut-off based STFs
The item selection is based on the theta-target procedure with user-defined targets.
The procedure selected the following 3 dichotomous items:
10 6 4
with parameters:
b a c e
10 2.2438664 1.3903672 0.02130947 1
6 0.8283301 1.1980125 0.01028904 1
4 0.7957328 0.9924159 0.08716286 1
These items maximize the information for thetas equal to:
2 2 2
ISA
\(\mathbf{TIF^*}\): TIF target describing the desired characteristic of a test
\(B\): Set of items of the item bank , items \(i = \{1,2, \ldots, ||B||\}\)
\(Q_{\text{ILA}}^k\subset B\): Set of items selected by ILA up to iteration \(k\)
\(k = 0\), \(Q^0 = \emptyset\), \ \(\theta_{target} := \arg\max |\mathbf{TIF}^* - \mathbf{TIF}^{0}|\), where \(\mathbf{TIF^0} = (0, 0, \ldots, 0)\) \
\(k = 1\), \(Q^1 = Q^{0} \cup \arg\min_{i \in B\setminus Q^0} |\theta_{target} - b_i|\), \(||Q^1|| = 1\) \(|\mathbf{TIF}^* - \mathbf{TIF}^1| \geq |\mathbf{TIF}^* - \mathbf{TIF}^{0}|\) \(\rightarrow\) , \(k = 2\)\ \(\theta_{target} := \arg\max |\mathbf{TIF}^* - \mathbf{TIF}^{1}|\)\
\(Q^2 = Q^{1} \cup \arg\min_{i \in B\setminus Q^1} |\theta_{target} - b_i|\), \(||Q^2|| = 2\) \(|\mathbf{TIF}^* - \mathbf{TIF}^2| \geq |\mathbf{TIF}^* - \mathbf{TIF}^{1}|\) \(\rightarrow\) , \(k = 3\)\ \(\theta_{target} := \arg\max |\mathbf{TIF}^* - \mathbf{TIF}^{2}|\)\
\(Q^3 = Q^{2} \cup \arg\min_{i \in B\setminus Q^2} |\theta_{target} - b_i|\), \(||Q^3|| = 3\) \ \(|\mathbf{TIF}^* - \mathbf{TIF}^3| \geq |\mathbf{TIF}^* - \mathbf{TIF}^{2}|\) \(\rightarrow\) \(\rightarrow\) , \(Q_{\textbf{ILA}} = Q^2\)
