# Item Characteristic Curve Models

## Abstract

In the first chapter the properties of the item characteristic curve were defined in terms of verbal descriptors. While this is useful to obtain an intuitive understanding of item characteristic curves, it lacks the precision and rigor needed by a theory. Consequently, in this chapter the reader will be introduced to three mathematical models for the item characteristic curve. These models provide mathematical equations for the relation of the probability of correct response to ability. Each model employs one or more item parameters whose numerical values define a particular item characteristic curve. Such mathematical models are needed if one is to develop a measurement theory that can be rigorously defined and is amenable to further growth. In addition, these models and their parameters provide a vehicle for communicating information about an item’s technical properties. For each of the three models, the mathematical equation will be used to compute the probability of correct response at several ability levels. Then the graph of the corresponding item characteristic curve will be shown. The goal of the chapter is to have you develop a sense of how the numerical values of the item parameters for a given model relate to the shape of the item characteristic curve.

## References

- Birnbaum, A. (1968). Some latent trait models and their use in inferring an examinee’s ability. Part 5. In F. M. Lord & M. R. Novick (Eds.),
*Statistical theories of mental test scores.*Reading, MA: Addison-Wesley.Google Scholar