Item response theory

Item response theory is a family of statistical models linking responses to test items with latent traits and item properties such as difficulty and discrimination.

Rasch model: A one-parameter logistic item response model in which items differ in difficulty while discrimination is fixed. It isolates item difficulty as the sole varying item parameter.
Latent variable: An unobserved quantity inferred from observable measurements or responses. Ability is modeled as a latent quantity rather than directly observed.
Computerized adaptive testing: A testing method that selects each next item based on a test taker's previous responses. Item information guides selection of items suited to the current ability estimate.
Classical test theory: A framework that models an observed test score as a true score plus measurement error. Unlike IRT, it centers analysis on total scores rather than item response probabilities.
Two-parameter logistic model: A logistic item response model with item-specific difficulty and discrimination parameters. It shows how item slopes alter the probability curve beyond difficulty alone.
Logistic function: An S-shaped function that maps real-valued inputs to probabilities between zero and one. Logistic models use this function to turn ability-item differences into response probabilities.
Differential item functioning: A difference in an item's response behavior across groups after matching people on the measured trait. IRT parameters can reveal items that function differently for comparable groups.
Factor analysis: A statistical method for explaining correlations among observed variables through latent factors. Both model latent traits, but IRT specifies probabilities for individual item responses.
Three-parameter logistic model: A logistic item response model that adds a lower asymptote for guessing to difficulty and discrimination. It represents the chance of a correct response by guessing on multiple-choice items.
Local independence: An assumption that observed responses are independent after conditioning on the relevant latent variables. It lets the model factor a response pattern into item-level probabilities.

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