Computerized Adaptive Testing: Item Response Engine

Item Bank

Bank size 25 items

Guessing floor c 0.15

Simulated Examinee

True ability θ 0.5

Stop when SE ≤ 0.30

Bank generated — ready to administer.

Live Estimate

θ estimate (EAP)

0.00

Standard error

0.99

Items administered

0 / 25

Correct responses

Next item selected (max Fisher info) Administered — correct Administered — incorrect Waiting — brighter = more informative now

How it works

Each bar is a test item with its own 3-parameter logistic (3PL) item response curve:

P_i(θ) = c_i + (1−c_i) / (1 + e^(−a_i(θ−b_i)))
a = discrimination, b = difficulty, c = guessing floor

Bar height = P_i at the current ability estimate; footprint size scales with a_i; x-position is the item's difficulty b_i.

Item selection — the adaptive engine always administers the un-used item with maximum Fisher information at the current θ estimate:

I_i(θ) = a_i² (P_i−c_i)² (1−P_i) / [(1−c_i)² P_i]

The amber bar is that item. The ribbon along the back row is the bank's total information curve ΣI_i(θ) — its peak shows where the bank measures ability most precisely.

Ability update — after each simulated response u_i ∈ {0,1}, the ability estimate is the Bayesian expected-a-posteriori (EAP) mean over a standard-normal N(0,1) prior, evaluated by quadrature over θ ∈ [−4,4]:

θ̂ = ∫ θ · L(θ) g(θ) dθ / ∫ L(θ) g(θ) dθ,  L(θ) = Π P_i(θ)^u_i (1−P_i(θ))^(1−u_i)
SE(θ̂) = √[ ∫ (θ−θ̂)² L(θ) g(θ) dθ / ∫ L(θ) g(θ) dθ ]

The test stops once SE drops to the chosen threshold — fewer, better-targeted items than a fixed-length test needs to reach the same precision. This is the same core loop (item bank + Fisher-information selection + Bayesian ability update) used inside real computerized adaptive tests such as the GRE and NWEA MAP.

x-axis: item difficulty (b) · white marker = true θ · blue marker = θ estimate

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