IRTmodel

IRTmodel refers to the family of statistical models within item response theory that relate individuals' latent traits to their observed item responses. In IRT, each person is assumed to have an unobserved ability, often denoted theta, and each item is described by parameters that determine how the probability of a particular response changes with theta. The core ideas are local independence (responses to different items are conditionally independent given theta) and monotonicity (the probability of a higher or more favorable response increases with theta). IRT models are used to calibrate items and estimate person abilities on a common scale, enabling meaningful comparisons across tests.

Common IRT models include the 1-parameter logistic (1PL) model, also known as the Rasch model, which uses a single difficulty parameter b for each item. The 2-parameter logistic (2PL) model adds a discrimination parameter a, allowing items to vary in how sharply they differentiate between ability levels. The 3-parameter logistic (3PL) model further includes a guessing parameter c to account for incidental correct responses, particularly in multiple-choice formats.

For items with more than two response categories, polytomous models are used. The graded response model (GRM) handles ordered categories, while the partial credit model (PCM) and generalized partial credit model (GPCM) address various partial credit schemes.

Estimating IRT models typically involves marginal maximum likelihood or Bayesian methods, often using the EM algorithm. Applications include test development, scale linking, and computerized adaptive testing (CAT). Assumptions such as unidimensionality and local independence are important for valid parameter estimation, and model fit is assessed with item fit statistics and overall fit measures.