Modeling and Multivariate Methods - SAS

Chapter 22 Scoring Tests Using Item Response Theory 525
Introduction to Item Response Theory
Introduction to Item Response Theory
Psychological measurement is the process of assigning quantitative values as representations of characteristics
of individuals or objects, so-called psychological constructs. Measurement theories consist of the rules by which
those quantitative values are assigned. Item response theory (IRT) is a measurement theory.
IRT uses a mathematical function to relate an individual’s probability of correctly responding to an item to
a trait of that individual. Frequently, this trait is not directly measurable and is therefore called a latent trait.
To see how IRT relates traits to probabilities, first examine a test question that follows the Guttman “perfect
scale” as shown in Figure 22.1. The horizontal axis represents the amount of the theoretical trait that the
examinee has. The vertical axis represents the probability that the examinee will get the item correct. (A
missing value for a test question is treated as an incorrect response.) The curve in Figure 22.1 is called an
item characteristic curve (ICC).
Figure 22.1 Item characteristic curve of a perfect scale item
P(Correct Response)
1.0
0.5
b
Trait Level
This figure shows that a person who has ability less than the value b has a zero percent chance of getting the
item correct. A person with trait level higher than b has a 100% chance of getting the item correct.
Of course, this is an unrealistic item, but it is illustrative in showing how a trait and a question probability
relate to each other. More typical is a curve that allows probabilities that vary from zero to one. A typical
curve found empirically is the S-shaped logistic function with a lower asymptote at zero and upper
asymptote at one. It is markedly nonlinear. An example curve is shown in Figure 22.2.