Item Response Theory

3Unidimensional IRT Models

There we measure shadows, and we search among ghostly errors of measurement for landmarks that are scarcely more substantial.

(Source: Edward Powell Hubble)

Item response theory (IRT) deals with the statistical analysis of data in which responses of each of a number of respondents to each of a number of items or trials are assigned to defined mutually exclusive categories. Although its potential applications are much broader, IRT was developed mainly in connection with educational measurement, where the main objective is to measure individual student achievement. Prior to the introduction of IRT, the statistical treatment of achievement data was based entirely on what is now referred to as “classical” test theory. That theory is predicated on the test score (usually the student's number of correct responses to the items presented) as the observation. It assumes that the number of items is sufficiently large to justify treating the test score as if it were a continuous measurement with specified origin and unit. On the further assumption that the items are randomly sampled from a larger item domain, a classical method of estimating measurement error due to item sampling is to score random halves of the test items separately, compute the product‐moment correlation of the two scores, and apply the so‐called Spearman–Brown formula to extend the correlation to that of the full test to obtain its “split‐half” reliability. The complement of the reliability ...