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... | See |

|----------------|-----|

| Understand IRT models and parameters | IRT Fundamentals |

| Design a new adaptive test | Test Design Workflow |

| Choose item selection algorithm | Item Selection |

| Decide when to stop the test | Stopping Rules |

| Calibrate new questions | references/calibration.md |

| Implement CAT algorithm | references/implementation.md |

IRT Fundamentals

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.

Test Design Workflow

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.

Item Selection Strategies

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

Stopping Rules

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)

Practical Considerations

Starting Ability Estimate

Options:

1. Population mean (θ = 0)
2. Prior information (e.g., grade level, previous test)
3. 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

| Need | File |

|------|------|

| Calibrate new items (collect data, estimate parameters) | references/calibration.md |

| Implement 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:

1. Start at θ = 0 (grade-level average)
2. Select item: b ≈ 0, content area needed
3. Student answers → update ability estimate (EAP)
4. Select next: maximize information at new θ, respect content balance, randomesque from top 5
5. Stop when SE < 0.35 or 25 questions reached
6. Report: ability estimate + placement recommendation

Result: Average 18 questions, 95% of students placed within ±0.5 grade levels of true ability.

Further Reading

• Lord, F. M. (1980). *Applications of Item Response Theory to Practical Testing Problems*
• Wainer, H. (2000). *Computerized Adaptive Testing: A Primer* (2nd ed.)
• van der Linden, W. J., & Glas, C. A. W. (2010). *Elements of Adaptive Testing*

IRT packages:

• Python: mirt, girth, catsim
• R: mirt, TAM, catR
• Production: Custom implementation or AdaptiveTest.io