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Adaptive Testing

Design and implement adaptive testing systems using Item Response Theory (IRT). Use when working with computerized adaptive tests (CAT), psychometric assessm...

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Design and implement adaptive testing systems using Item Response Theory (IRT). Use when working with computerized adaptive tests (CAT), psychometric assessm...

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Version: 1.0.3

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Adaptive Testing with IRT

Design computerized adaptive tests that measure ability efficiently and accurately using Item Response Theory.

Core Concept

Adaptive tests adjust difficulty in real-time based on student responses. A correct answer → harder question. Incorrect → easier question. The result: accurate ability estimates in ~50% fewer questions than fixed-length tests. Key advantage: Traditional tests waste time on too-easy or too-hard questions. Adaptive tests spend time where measurement matters most — near the student's ability level.

Quick Decision Tree

You need to...SeeUnderstand IRT models and parametersIRT FundamentalsDesign a new adaptive testTest Design WorkflowChoose item selection algorithmItem SelectionDecide when to stop the testStopping RulesCalibrate new questionsreferences/calibration.mdImplement CAT algorithmreferences/implementation.md

The 3-Parameter Logistic (3PL) Model

Most adaptive tests use the 3PL model. Each question has three parameters: a (discrimination) — How well the question differentiates ability levels. Higher = steeper curve. Typical range: 0.5 to 2.5 b (difficulty) — The ability level where P(correct) = 0.5. Range: -3 to +3 (standardized scale) c (guessing) — Probability of guessing correctly. Usually 0.2 to 0.25 for multiple choice Probability of correct response: P(correct | ability, a, b, c) = c + (1 - c) / (1 + e^(-a(ability - b))) Simpler models: 2PL: Set c = 0 (no guessing parameter) 1PL (Rasch): Set c = 0 and a = 1 for all items (only difficulty varies) Use 3PL for high-stakes tests. Use 2PL/1PL when sample size is small (<500 responses per item).

Information and Standard Error

Information measures how precisely an item estimates ability at a given level. Peak information occurs when ability ≈ difficulty (b parameter). Standard Error (SE) is the inverse of information: SE = 1 / sqrt(Information) Goal of CAT: Maximize information (minimize SE) at the student's true ability level.

1. Define Test Specifications

Purpose: Placement, diagnostic, certification, progress monitoring? Content domain: Single skill or multidimensional? Target population: What ability range (-3 to +3)? Constraints: Time limit, minimum/maximum length, content balance

2. Build Item Bank

Minimum bank size: 10× the average test length. For a 20-item CAT, you need ≥200 calibrated items. Distribution targets: Difficulty (b): Spread across expected ability range Discrimination (a): Target 1.0 to 2.0 (high discrimination) Exposure: No item used >20% of the time Content balancing: If testing math, ensure geometry/algebra/etc. are proportionally represented.

3. Choose Algorithms

Pick one from each category: Item selection: (see below) Maximum Information Randomesque (MFI + exposure control) Content balancing Ability estimation: Maximum Likelihood Estimation (MLE) Expected A Posteriori (EAP) — better for extreme scores Weighted Likelihood (WLE) Stopping rule: (see below) Fixed length Standard error threshold Information threshold

4. Simulate Performance

Before going live, simulate 1000+ test sessions with known abilities. Check: Average test length SE at different ability levels Item exposure rates Content balance adherence Adjust if needed.

Maximum Fisher Information (MFI)

Rule: Select the item with highest information at current ability estimate. Pros: Optimal precision, shortest tests Cons: Overuses "best" items, poor security Use when: Pilot testing, low-stakes practice

Randomesque (MFI + Exposure Control)

Rule: Select from top N items by information (e.g., top 5), choose randomly from that set. Pros: Balances precision and security Cons: Slightly longer tests than pure MFI Use when: Operational tests, default choice

a-Stratified

Rule: Start with high-discrimination items (high a), use mid-discrimination later. Pros: Fast initial ability estimate Cons: Complex to implement Use when: Very large item banks, research settings

Content Balancing

Rule: Track content area usage, prioritize underrepresented areas when selecting next item. Implementation: Weight information by content constraint satisfaction. Use when: Blueprint requirements, multidimensional tests

Fixed Length

Stop after N items (e.g., 20 questions). Pros: Predictable time, simple Cons: May over/under-test some students Use when: Time limits matter, simple implementation needed

Standard Error Threshold

Stop when SE < target (e.g., SE < 0.3). Pros: Consistent precision across ability levels Cons: Variable test length (harder to schedule) Typical targets: Low-stakes: SE < 0.4 Medium-stakes: SE < 0.3 High-stakes: SE < 0.25 Use when: Precision matters more than time

Combined Rule

Stop when (SE < target) OR (length ≥ max) OR (length ≥ min AND ability estimate stable). Use when: Production systems (safest approach)

Starting Ability Estimate

Options: Population mean (θ = 0) Prior information (e.g., grade level, previous test) First question is medium difficulty, estimate from there Never start at extremes (-3 or +3).

Handling Extreme Response Patterns

All correct or all incorrect: MLE fails. Use EAP or Bayesian prior to regularize. Rapid changes: If ability estimate jumps >1.0, consider response anomaly (cheating, guessing).

Exposure Control

Track how often each item is used. Flag items used >20% of the time. Consider: Randomesque selection (above) Sympson-Hetter method (advanced) Periodic item bank refresh

Multidimensional IRT (MIRT)

If testing multiple skills (e.g., algebra + geometry), use separate ability estimates per dimension. Select items to balance information across dimensions. Warning: MIRT requires larger item banks and more complex calibration.

Common Mistakes

❌ Too few items in bank → High exposure, security risk ✅ Aim for 10× average test length ❌ Poorly distributed difficulties → Accurate only in narrow ability range ✅ Spread items across -2 to +2 difficulty ❌ Ignoring content balance → May skip important topics ✅ Build content constraints into item selection ❌ Using MLE for all incorrect → Returns -∞ ✅ Use EAP or cap estimates at -3/+3 ❌ No exposure control → Same items every test ✅ Use randomesque or Sympson-Hetter

When to Load References

NeedFileCalibrate new items (collect data, estimate parameters)references/calibration.mdImplement CAT algorithm (code patterns, libraries)references/implementation.md

Real-World Example: K-12 Math Placement

Setup: Item bank: 300 questions, b from -2 (basic) to +2 (advanced) Target: SE < 0.35 or max 25 questions Content: 40% algebra, 30% geometry, 30% statistics Algorithm: Randomesque (top 5), EAP estimation Flow: Start at θ = 0 (grade-level average) Select item: b ≈ 0, content area needed Student answers → update ability estimate (EAP) Select next: maximize information at new θ, respect content balance, randomesque from top 5 Stop when SE < 0.35 or 25 questions reached Report: ability estimate + placement recommendation Result: Average 18 questions, 95% of students placed within ±0.5 grade levels of true ability.

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3 Docs

SKILL.md Primary doc
references/calibration.md Docs
references/implementation.md Docs