Computerized Adaptive Testing — Item Response Theory Simulator (2D)

Simulated Examinee

True ability θ 0.0

Adaptive Engine

Item-selection rule

Last Response

No items administered yet — press "Next Item" to begin.

Live Estimate

Ability estimate θ̂

0.00

Standard error

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Items given

0 / 20

Percent correct

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Adaptive vs Fixed

Runs a full 20-item adaptive test and a full 20-item fixed-order test against the same hidden ability, on the same item bank, and plots both θ̂ trajectories below.

Adaptive: items to converge

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Fixed: items to converge

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How it works

Each item in the bank follows the 2-parameter logistic (2PL) IRT model: the probability a person of ability θ answers item i correctly is

P_i(θ) = 1 / (1 + exp(-a_i (θ - b_i)))
a_i = discrimination   b_i = difficulty

After every response the engine re-estimates ability by maximizing the log-likelihood of all answers so far, via a bounded Newton-Raphson step:

θ̂ ← θ̂ + Σ a_i(r_i - P_i) / Σ a_i² P_i(1-P_i)
SE(θ̂) = 1 / √(Σ a_i² P_i(1-P_i))    (Fisher information)

The maximum-information rule then picks, from the untested items, whichever one maximizes a_i² P_i(1-P̂_i) at the *current* θ̂ — the item that will shrink the standard error the most. The fixed-order test in the comparison administers the same bank in its original (unordered-by-difficulty) sequence, with no re-selection — the same paper-test approach adaptive testing replaced.

True ability θ — the hidden value the simulated examinee actually has; it drives whether each administered item is answered correctly, but the engine never sees it directly.
Run Comparison — instantly plays out a full adaptive test and a full fixed-order test on independent random draws from the same θ, and reports how many items each needed before its running estimate stayed within 0.3 of the true value.
Each test stops itself at 20 items or once SE(θ̂) drops below 0.30, whichever comes first — the standard stopping rule used by real computerized adaptive tests (e.g. GRE, NCLEX).

Item bank: 50 · tested 0

Untested item Answered correct Answered incorrect Adaptive θ̂ trajectory Fixed-order θ̂ trajectory