Modeling and Multivariate Methods - SAS

Chapter 22 Scoring Tests Using Item Response Theory 527
Introduction to Item Response Theory
Changing the value of b merely shifts the curve from left to right, as shown in Figure 22.4. It corresponds to
the value of θ at the point where P(θ)=0.5. The parameter b can therefore be interpreted as item difficulty
where (graphically), the more difficult items have their inflection points farther to the right along their
x-coordinate.
Figure 22.4 Logistic curve for several values of b
Notice that
limP( θ)
= c
θ→ – ∞
and therefore c represents the lower asymptote, which can be non-zero. ICCs for several values of c are
shown graphically in Figure 22.5. The c parameter is theoretically pleasing, since a person with no ability of
the trait may have a non-zero chance of getting an item right. Therefore, c is sometimes called the
pseudo-guessing parameter.
Figure 22.5 Logistic model for several values of c
By varying these three parameters, a wide variety of probability curves are available for modeling. A sample
of three different ICCs is shown in Figure 22.6. Note that the lower asymptote varies, but the upper
asymptote does not. This is because of the assumption that there may be a lower guessing parameter, but as