Item Response Theory

Analyze → Scale → Item Response Theory. Reliability analysis tells you how well a set of items hangs together as one score; IRT opens the items themselves. Each binary item (right/wrong, yes/no, symptom present/absent) is modeled as a function of a latent trait θ — ability, severity, attitude — with its own difficulty (where on the trait the item starts being passed) and, in the 2PL model, its own discrimination (how sharply it separates people just below from just above that point). The payoff is a question no reliability coefficient can answer: where on the trait does this test measure well, and which items do the work?

The data layout

One row per person, one column per item — each item binary (exactly two observed values; numeric or string). The higher observed value is treated as the keyed — correct or endorsed — response, and the output names it per item, so a 0/1 coding needs no preparation. Cases incomplete on any item are dropped; anything non-binary is refused by name.

The dialog

Tick the Items, choose the Model, and keep Plot information curves on for the figures:

• 2PL (the default) — each item gets its own discrimination and difficulty. The general-purpose choice.
• Rasch — one shared discrimination, per-item difficulties. Fewer parameters, item difficulties on one common yardstick, and the model of choice when items are meant to differ only in difficulty. (Fit both and compare AIC/BIC: if Rasch fits comparably, its simplicity wins.)

Estimation is marginal maximum likelihood (Bock–Aitkin EM) with the latent trait fixed to N(0,1) — deterministic, so the same run always reproduces the same tables.

Reading the output

• IRT Model Summary — N, items, parameters, the marginal log-likelihood, AIC/BIC (for comparing Rasch vs 2PL on the same items), and the EM convergence.
• IRT Item Parameters — per item, with standard errors:
  • Difficulty (b) — the trait level at which a person has a 50% chance of the keyed response. Items spread across a range of b cover the trait; a test whose difficulties bunch together measures only one narrow band.
  • Discrimination (a) — how steeply the response probability rises at b. Higher discriminates better near its difficulty; a very low a (say below ~0.5) marks an item barely related to the trait — a candidate for revision. The parameterization is the classical P(keyed) = logistic(a·(θ − b)).
• Item Information Curves — each item's Fisher information over θ: a peak at its difficulty, higher and narrower the more discriminating the item. Read it as a map of which items work where.
• Test Information Curve — the sum, with SE(θ) = 1/√I(θ): the measurement precision profile of the whole test. A tall curve over the range you care about (e.g. around a cut score) is the goal; information collapsing at the extremes is why extreme scores are always less precise.

Relation to the other scale tools

• Reliability (alpha) — one number for the whole scale, assuming items are interchangeable. IRT replaces it with item-level parameters and a precision curve.
• Confirmatory Factor Analysis — tests a hypothesized factor structure for continuous-ish indicators; a one-factor CFA of binary items is the rough cousin of the 2PL. Use CFA for structure questions across many constructs, IRT for measurement questions within one.
• Latent Class Analysis — also models categorical items, but with a categorical latent variable (subgroups) instead of a continuous trait.

Scope (v1). Rasch and 2PL for binary items only: no 3PL (guessing parameter — its estimation is notoriously fragile), no polytomous models (graded response / partial credit), no saved ability (EAP) scores, no item-fit statistics, and no item characteristic (trace-line) plots — the information curves are the v1 figures. A frequency weight (Weight Cases) is honored as case replication; a declared survey design is not covered and the output says so.