Modeling and Multivariate Methods - SAS

526 Scoring Tests Using Item Response Theory Chapter 22
Introduction to Item Response Theory
Figure 22.2 Example item response curve
The logistic model is the best choice to model this curve, since it has desirable asymptotic properties, yet is
easier to deal with computationally than other proposed models (such as the cumulative normal density
function). The model itself is
P( θ) = c + -------------------------------------
1 – c
1 + e a)
In this model, referred to as a 3PL (three-parameter logistic) model, the variable a represents the steepness of
the curve at its inflection point. Curves with varying values of a are shown in Figure 22.3. This parameter
can be interpreted as a measure of the discrimination of an item—that is, how much more difficult the item
is for people with high levels of the trait than for those with low levels of the trait. Very large values of a
make the model practically the step function shown in Figure 22.1. It is generally assumed that an examinee
will have a higher probability of getting an item correct as their level of the trait increases. Therefore, a is
assumed to be positive and the ICC is monotonically increasing. Some use this positive-increasing property
of the curve as a test of the appropriateness of the item. Items whose curves do not have this shape should be
considered as candidates to be dropped from the test.
Figure 22.3 Logistic model for several values of a