Item Response Theory

Definition

Item response theory (IRT) models the probability that a subject answers a test item correctly as a function of a latent trait ("ability") $\theta$ and item-specific parameters. The foundational logistic models (Birnbaum 1968) describe each item by an item characteristic curve (ICC) — an S-shaped function of $\theta$ — so that, conditional on $\theta$, item responses are independent (local independence). IRT underlies modern educational and psychological measurement and computerized adaptive testing.

Key Ideas

• The item characteristic curve. For a dichotomous item, the ICC gives $\Pr(\text{correct}\mid\theta)$ as a monotone S-curve; its shape is set by the item parameters.
• One-, two-, and three-parameter logistic models. 1PL / Rasch: only a difficulty $b_g$, all items equally discriminating. 2PL: adds discrimination $a_g$ (slope), $\Pr=\{1+\exp[-a_g(\theta-b_g)]\}^{-1}$. 3PL: adds a guessing lower asymptote $c_g$, $\Pr=c_g+(1-c_g)\{1+\exp[-a_g(\theta-b_g)]\}^{-1}$.
• Local independence. Conditional on $\theta$, the item responses are independent — the same assumption that makes IRT a latent-variable measurement model and links it to factor analysis.
• Information functions. The item information $I_g(\theta)$ measures how precisely an item pins down $\theta$ near its difficulty; the test information is the sum, and its reciprocal is the (asymptotic) variance of the ML ability estimate. Information peaks where an item is most discriminating and near its difficulty.
• Estimation. Ability $\theta$ and item parameters are estimated by maximum likelihood (joint, marginal, or Bayesian); marginal ML integrates $\theta$ over a population distribution (Bock-Aitkin).

Why It Matters

• The engine of standardized testing. The 2PL/3PL logistic models are the basis of large-scale assessments (SAT, GRE, licensing exams) and of computerized adaptive testing, which serves the item most informative at the examinee's current ability estimate.
• Item-free / person-free measurement. Because item and person parameters are on a common scale, IRT supports test equating and item banking — comparing examinees who took different item sets.
• A bridge to latent-variable statistics. IRT is the same object as factor analysis of discretized variables (normal-ogive twin of the logistic ICC) and a close relative of the latent class and multilevel ordinal models.

Open Questions

• Model choice and fit. Choosing among 1PL/2PL/3PL (and beyond — graded response, nominal, multidimensional) trades parsimony against fit; the 3PL guessing parameter is notoriously hard to estimate.
• Dimensionality. The standard models assume a single latent trait; multidimensional IRT relaxes this but complicates identification and interpretation.
• Logistic vs. normal ogive. The logistic ICC is convenient but the exact factor-analysis equivalence uses the normal-ogive link; the two differ slightly in the tails.

Related

• Rasch Model
• IRT–Factor Analysis Equivalence
• Latent Class Model
• Multilevel Ordinal Regression
• Birnbaum (1968) Some Latent Trait Models
• Allan Birnbaum