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Item response theory (IRT) models are stochastic models for responses of persons to items, where the influences of items and persons on the responses are modelled by disjunctive sets of parameters. In the framework of educational and psychological measurement, the person parameter can usually be labelled ability or proficiency; in the sequel, the term ability will be used. The definition of separate parameters for persons and items supports a comprehensive framework for many important issues in educational and psychological measurement, such as test scoring, validity, local reliability, test equating, calibration of item banks using incomplete designs, differential item functioning, optimal test construction and computerized adaptive testing (see, for instance, Lord, 1980, or Hambleton & Swaminathan, 1985).

In many instances, it suffices to assume that ability is ...

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