Development of Short Test Forms within Item Response Theory: The shortIRT Package

Livio Finos\(^{1,2}\), Ottavia M. Epifania\(^{2, 3}\), Pasquale Anselmi\(^{1}\), Egidio Robusto\(^{1}\)

\(^1\) University of Padova, Padova \(^2\) Psicostat, Padova \(^3\) University of Trento, Rovereto

Item Response Theory

Item Response Function

According to the 4-Parameter Logistic Model:

\[P(x_{pi}=1|\theta_p, b_i, a_i, c_i, e_i) = c_i + (e_i-c_i)\dfrac{\exp[a_i(\theta_p - b_i)]}{1 + \exp[a_i(\theta_p - b_i)]}\]

Information Functions

\[ \text{IIF}_{i}(\theta) = \dfrac{a_i^2[P(\theta)-c_i]^2[d_i - P(\theta)]^2}{(d_{i}-c_i)^2 P(\theta)Q(\theta)}\]

\[TIF = \sum_{i = 1}^{|B|} IIF_i\] (\(B\): Set of items in a test (\(|X|\) cardinality of set \(X\)))

Development of short test forms (STFs)

Brevissimi accenni ad ata (?)

Some proposed alternatives

Benchmark Procedure

Create a short test form composed of \(N\) items from an item bank \(B\) \(\rightarrow\) Select the \(N\) items with the highest IIFs:

The IIFs of the items of item bank are sorted in decreasing order:

\[\mathit{iif} = (\displaystyle \max_{1 < i < B} IIF_i, \ldots \displaystyle, \min_{1 < i < B} IIF_i) \]

Items with IIFs from 1 to \(N\), \(N < ||B||\), are selected to be included in the short test form

Aim: STF with \(N = 3\) items from a full-length test of 10 items:

item \( b_i \) \( a_i \) \( \text{IIF}_i \)
1 -0.67 0.71 0.08
2 0.50 1.19 0.15
3 -2.43 0.25 0.01
4 2.12 1.98 0.24
5 1.72 0.39 0.03
6 -2.28 1.62 0.19
7 0.64 0.50 0.05
8 -2.51 1.68 0.19
9 -0.66 0.44 0.04
10 0.72 0.33 0.02

Aim: STF with \(N = 3\) items from a full-length test of 10 items:

item \( b_i \) \( a_i \) \( \text{IIF}_i \)
4 2.12 1.98 0.24
6 -2.28 1.62 0.19
8 -2.51 1.68 0.19
2 0.50 1.19 0.15
1 -0.67 0.71 0.08
7 0.64 0.50 0.05
9 -0.66 0.44 0.04
5 1.72 0.39 0.03
10 0.72 0.33 0.02
3 -2.43 0.25 0.01

Aim: STF with \(N = 3\) items from a full-length test of 10 items:

item \( b_i \) \( a_i \) \( \text{IIF}_i \)
4 2.12 1.98 0.24
6 -2.28 1.62 0.19
8 -2.51 1.68 0.19
2 0.50 1.19 0.15
1 -0.67 0.71 0.08
7 0.64 0.50 0.05
9 -0.66 0.44 0.04
5 1.72 0.39 0.03
10 0.72 0.33 0.02
3 -2.43 0.25 0.01

\(\theta\)-target procedure

The shortIRT package

General package structure, available on CRAN

Methods for S3 classes

bench()

theta_target()