ItemResponseTheorie

Item Response Theory (IRT), in German often called Item-Response-Theorie, is a family of probabilistic models for describing the relationship between an individual's latent trait and their responses to test items. The core idea is that each person has a latent ability or trait theta, and each item has parameters that determine how likely a particular response is as theta changes.

For dichotomous items (correct/incorrect), common models are the one-parameter logistic model (1PL, Rasch model), which uses a single difficulty parameter for each item; the two-parameter logistic model (2PL) adds a discrimination parameter; and the three-parameter logistic model (3PL) adds a guessing parameter. Item characteristic curves show p_i(theta) as an S-shaped function. For polytomous items, models such as the graded response or partial credit extend the approach. The information function measures how much information an item or test provides about theta at each level of ability.

Estimation typically relies on marginal maximum likelihood or Bayesian methods, producing item parameters and person abilities. The models assume unidimensionality (a single latent trait), local independence of item responses given theta, and monotone relationships between theta and response probability. In practice, extensions to multidimensional IRT, differential item functioning analyses, and computerized adaptive testing are common.

IRT is widely used in educational testing and psychology for test construction, score interpretation, and item analysis. It supports test equating, linking scores across forms, and computerized adaptive testing (CAT). The development of IRT traces to the work of Georg Rasch in the 1960s and later expansions by Birnbaum, Lord, and others.