iRT

Item Response Theory (IRT) is a family of mathematical models used to analyze the relationship between individuals’ latent traits, such as ability or personality characteristics, and their discrete item responses on assessments. In IRT, the probability of a given response to an item is a function of the person’s latent trait level, θ, and item parameters that describe the item’s characteristics. The core idea is that item characteristics remain stable across different samples, allowing more precise measurement than traditional score-based approaches.

Common models include the one-parameter logistic model (1PL or Rasch), which uses a single difficulty parameter; the two-parameter logistic model (2PL), which adds a discrimination parameter; and the three-parameter logistic model (3PL), which also includes a guessing parameter. For items with nonbinary responses, polytomous models such as the graded response model or the partial credit model are used. Key concepts include the item characteristic curve (ICC), which plots response probability against θ, item difficulty (b), discrimination (a), and guessing (c) where applicable. Assumptions underlying IRT include unidimensionality (a single dominant trait), local independence (responses are independent once θ is accounted for), and monotonicity (higher θ increases the likelihood of a higher response).

Estimation methods, often via maximum likelihood or Bayesian approaches, yield item and ability parameters. IRT provides information functions that quantify measurement precision at different θ levels and underpins applications such as computerized adaptive testing, test development, and cross-test comparability. Compared with classical test theory, IRT offers item-level invariance and greater scalability, but it requires larger samples and careful model checking to mitigate issues such as model misspecification and local dependence.