Item Response Theory

Dan Mungas, Ph.D.Department of Neurology

University of California, Davis

What is it?

Why should anyone care?

IRT Basics

Item Response Theory - What Is It

• Modern approach to psychometric test development– Mathematical measurement theory– Associated numeric and computational methods

• Widely used in large scale educational, achievement, and aptitude testing

• More than 50 years of conceptual and methodological development

Item Response Theory - Methods

• Dataset consists of rectangular table– rows correspond to examinees– columns correspond to items

• IRT applications simultaneously estimate examinee ability and item parameters– iterative, maximum likelihood estimation algorithms– processor intensive, no longer a problem

Basic Data Structure

Subject Item1 Item2 Item3 Item4

S1 X11 X12 X13 X14

S2 X21 X22 X23 X24

S3 X31 X32 X33 X34

S4 X41 X42 X43 X44

Item Types

• Dichotomous• Multiple Choice• Polytomous

– Information is greater for polytomous item than for the same item dichotomized at a cutpoint

What is the item level response

• Smallest discrete unit (e.g. Object Naming)• Sum of correct responses (trials in word list

learning test)• For practical reasons, continuous measures might

have to be recoded into ordinal scales with reduced response categories (10, 15)

Item Response Theory - Basic Results

• Item parameters– difficulty– discrimination– correction for guessing

• most applicable for multiple choice items

• Subject Ability (in the psychometric sense)– Capacity to successfully respond to test items (or propensity to

respond in a certain direction)– Net result of all genetic and environmental influences– Measured by scales composed of homogenous items

• Item difficulty and subject ability are on the same scale

Item Characteristic Curves

0.0

0.2

0.4

0.6

0.8

1.0

-3 -2 -1 0 1 2 3Ability

Proportion Correct

Item 1 Item 2 Item 3

Item Response Theory - Outcomes

• Item-Level Results– Item Characteristic Curve (ICC)

• non-linear function relating ability to probability of correct response to item

– Item Information Curve (IIC)• non-linear function showing precision of measurement

(reliability) at different ability points

– Both curves are defined by the item parameters

Item Characteristic Curves

0.0

0.2

0.4

0.6

0.8

1.0

-3 -2 -1 0 1 2 3Ability

Proportion Correct

Item 1 Item 2 Item 3

Information Curves

0.00.51.01.52.02.53.03.54.0

-3 -2 -1 0 1 2 3Ability

Information

Item 1 Item 2 Item 3 Total

0.0

0.2

0.4

0.6

0.8

1.0

-3 -2 -1 0 1 2 3Ability

Proportion Correct

Item 1 Item 2 Item 3

0.00.51.01.52.02.53.03.54.0

-3 -2 -1 0 1 2 3Ability

Information

Item 1 Item 2 Item 3 Total

Item Response Theory - Outcomes

• Test-Level Results– Test Characteristic Curve (TCC)

• non-linear function relating ability to expected total test score

– Test Information Curve (TIC)• non-linear function showing precision of measurement

(reliability) at different ability points

– Both sum of item level functions of included items

Test Characteristic CurveMini-Mental State Examination

0

5

10

15

20

25

30

-3.0 -2.0 -1.0 0.0 1.0 2.0 3.0Ability Metric

Total Score

Information Curves

Item Response Theory - Fundamental Assumptions

• Unidimensionality - items measure a homogenous, single domain

• Local independence - covariance among items is determined only by the latent dimension measured by the item set

IRT Models

• 1PL (Rasch)– Only Difficulty and Ability are estimated– Discrimination is assumed to be equal across items

• 2PL– Discrimination, Difficulty and Ability are estimated– Guessing is assumed to not have an effect

• 3PL – Discrimination, Difficulty, Guessing, and Ability are

estimated (multiple choice items)

Item Response Theory - Invariance Properties

• Invariance requires that basic assumptions are met• Item parameters are invariant across different

samples– Within the range of overlap of distributions– Distributions of samples can differ

• Ability estimates are invariant across different item sets– Assumes that ability range of items spans ability range

of subjects that is of interest

Why Do We Care -Applications of IRT in Health Care Settings

• Refined scoring of tests• Characterization of psychometric properties of

existing tests• Construction of new tests

Test Scoring

• IRT permits refined scoring of items that allows for differential weighting of items based on their item parameters

Physical Function Scale Hays, Morales & Reise (2000)

Item LIMITED LIMITED NOT LIMITEDA LOT A LITTLE AT ALL

Vigorous activities, running,Lifting heavy objects,Strenuous sports 1 2 3

Climbing one flight 1 2 3

Walking more than 1 mile 1 2 3

Walking one block 1 2 3

Bathing / dressing self 1 2 3

Preparing meals / doing laundry 1 2 3

Shopping 1 2 3

Getting around inside home 1 2 3

Feeding self 1 2 3

How to Score Test

• Simple approach: there are numbers that will be circled; total these up, and we have a score.

• But: should “limited a lot” for walking a mile receive the same weight as “limited a lot” in getting around inside the home?

• Should “limited a lot” for walking one block be twice as bad as “limited a little” for walking one block?

How IRT Can Help

• IRT provides us with a data-driven means of rational scoring for such measures

• Items that are more discriminating are given greater weight

• In practice, the simple sum score is often very good; improvement is at the margins

Description of Psychometric Properties

• The Test Information Curve (TIC) shows reliability that continuously varies by ability– Depicts ability levels associated with high and low

reliability• The standard error of measurement is directly

related to information value (I())– SEM = 1 / sqrt(I())

• SEM and I() also have a direct correspondence to traditional r– r = 1 - 1/ I()

I(), SEM, r

I() SEM (s.d. units) r1 1.00 0.002 0.71 0.504 0.50 0.759 0.33 0.89

12 0.29 0.9216 0.25 0.9425 0.20 0.9636 0.17 0.97

TICs for English and Spanish language Versions of Two Scales

0

4

8

12

16

-3 -2 -1 0 1 2 3Ability

Information

Object Naming English Object Naming Spanish3MSE English 3MSE Spanish

Mungas et al., 2004

Construction of New Scales

• Items can be selected to create scales with desired measurement properties

• Can be used for prospective test development• Can be used to create new scales from existing

tests/item pools

• IRT will not overcome inadequate items

TICs from an Existing Global Cognition Scale and Re-Calibrated Existing Cognitive Tests

05

101520253035

40 55 70 85 100 115 130Ability (standard score metric)

Information

Global Memory Executive Mattis DRS

Mungas et al., 2003

Principles of Scale Construction

• Information corresponds to assessment goals– Broad and flat TIC for longitudinal change measure in

population with heterogenous ability– For selection or diagnostic test, peak at point of ability

continuum where discrimination is most important

– But normal cognition spans a 4.0 s.d. range, and is even greater in demographically diverse populations

Other Issues In IRT

• Polytomous IRT models are available– Useful for ordinal (Likert) rating scales

• Each possible score of the item (minus 1) is treated like a separate item with a different difficulty parameter

• Information is greater for polytomous item than for the same item dichotomized at a cutpoint

Other Issues in IRT

• Applicable to broad range of content domains• IRT certainly applies to cognitive abilities• Also applies to other health outcomes

– Quality of life– Physical function– Fatigue– Depression– Pain

Other Issues in IRT

• Differential Item Function - Test Bias• IRT provides explicit methods to evaluate and

quantify the extent to which items and tests have different measurement properties in different groups– e.g. racial and ethnic groups, linguistic groups, gender

English and Spanish Item Characteristic Curves for “Lamb/Cordero” Item

0.00

0.20

0.40

0.60

0.80

1.00

-3 -2 -1 0 1 2 3Ability Metric

Probability of Correct Response

EnglishSpanish

English and Spanish Item Characteristic Curves for “Stone/Piedra” Item

0.00

0.20

0.40

0.60

0.80

1.00

-3 -2 -1 0 1 2 3Ability Metric

Probability of Correct Response

EnglishSpanish

Differential Item Function (DIF)

• DIF refers to systematic bias in measuring “true” ability - doesn’t address group differences in ability

Challenges/ Limitations of IRT

• Large samples required for stable estimation– 150-200 for 1PL– 400-500 for 2PL– 600-1000 for 3PL

• Analytic methods are labor intensive– There are a number of (expensive *) applications

readily available for IRT analyses– Evaluation of basic assumptions, identification of

appropriate model, and systematic IRT analysis require considerable expertise and labor

* but, R!!

Computerized Adaptive Testing (CAT)

• IRT based computer driven method• Selects items that most closely match examinee’s

ability• Administers only items needed to achieve a pre-

specified level of precision in measurement (information, s.e.m., reliability)

Why CAT

• Efficiency– Administration -

• Standardization• Time efficiency• Data collection

– Scoring• Computer can implement complex scoring algorithms

CAT Example 1

CAT Example 2

Practical Considerations for CAT

What You Need for CAT

• Computer technology– Item Selection– Item Administration– Scale Scoring

• Item bank with IRT parameters– Range of item difficulty relevant to measurement needs

What is Straightforward/Easy?

• Dichotomous items• Multiple choice items• Ordered polytomous response scales

– Up to 10-15 response options

Technical Challenges

• Continuous response scales (memory, timed tasks)– Can be recoded into smaller number of ordered

response ranges• Lose information

Methodological Challenges

• Sample size requirements– Minimally 300-600 cases for stable estimation of item

parameters• Differential Item Function and Measurement Bias

– Essentially involves item calibration within groups of interest

• e.g., age, education, language, gender, race

– Available literature provides minimal guidance

References

• Hays, R. D., Morales, L. S., & Reise, S. P. (2000). Item response theory and health outcomes measurement in the 21st century. Med Care, 38(9 Suppl), II28-42.

• Mungas, D., Reed, B. R., & Kramer, J. H. (2003). Psychometrically matched measures of global cognition, memory, and executive function for assessment of cognitive decline in older persons. Neuropsychology, 17(3), 380-392.

• Mungas, D., Reed, B. R., Crane, P. K., Haan, M. N., & González, H. (2004). Spanish and English Neuropsychological Assessment Scales (SENAS): Further development and psychometric characteristics. Psychological Assessment, 16(4), 347-359.