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Item Response Theory in Health
Measurement
Outline
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Contrast IRT with classical
test theory
Introduce basic concepts in
IRT
Illustrate IRT methods with
ADL and IADL scales
Discuss empirical comparisons
of IRT and CTT
Advantages and disadvantages
of IRT
When would it be appropriate
to use IRT?
Test Theory
 Any
item in any health measure has two
parameters:
 The
level of ability required to answer the question
correctly. (In health this translates into the level of
health at which the person doesn’t report this problem)
 The
level of discrimination of the item: how accurately
it distinguishes well from sick
Classical Test Theory
Most common paradigm for scale development and
validation in health
 Few theoretical assumptions, so broadly applicable
 Partitions observed score into True Score + Error
 Probability of a given item response is a function of
person to whom item is administered and nature of item
 Item difficulty: proportion of examinees who answer item
correctly (in health: item severity…)
 Item discrimination: biserial correlation between item and
total test score.

Classical test theory
Probability of ‘no’ answer depends on type of item
(difficulty) and the level of physical functioning (e.g. SF36 bathing vs. vigorous activities)
 Some limitations
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Item difficulty, discrimination, and ability are confounded
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Sample dependent; item difficulty estimates will be different in different
samples. Estimate of ability is item dependent
Difficult to compare scores across two different tests because not
on same scale
 Often, ordinal scale of measurement for test
 Assumes equal errors of measurement at all levels of ability
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Item Response Theory
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Complete theory of measurement and item selection
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Theoretically, item characteristics are not sample
dependent; estimates of ability are not item dependent
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Item scores on same scale as ability
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Puts all individual scores on standardized, interval level
scale; easy to compare between tests and individuals
Item Response Theory
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Assumes that a normally distributed latent trait underlies
performance on a measure
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Assumes unidimensionality
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Assumes local independence
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All items measuring the same construct
Items are uncorrelated with each other when ability is held
constant
Given unidimensionality, any reponse to an item is a
monotonically increasing function of the latent trait (item
characteristic curve)
Example of item characteristic curves
(Note the a parameter: 2.82 for the steep curve, 0.98
for the shallow curve)
Differential Item Functioning
Assuming that the measured ability is unidimensional and
that the items measure the same ability, the item curve
should be unique except for random variations, irrespective of the
group for whom the item curve is plotted…
…items that do not yield the same item response function
for two or more groups are violating one of the fundamental
assumptions of item response theory, namely that the item and
the test in which it is contained are measuring the same
unidimensional trait…
Possible DIF
Item Bias

Items may be biased against one gender, linguistic, or
social group
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Can result in people being falsely identified with problems or
missing problems
Two elements in bias detection
Statistical detection of Differential Item Functioning
 Item review
 If source of problems not related to performance, then item is
biased
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DIF detection
 Important
part of test validation
 Helps to ensure measurement equivalence
 Scores on individual items are compared for two
groups:
 Reference
 Focal
 Groups
group under study
matched on total test score (ability)
DIF detection
 DIF
can be uniform or nonuniform
 Uniform
 Probability
of correctly answering item correctly is
consistently higher for one group
 Nonuniform
 Probability
of correctly answering item is higher for
one group at some points on the scale; perhaps lower at
other points
Illustration of IRT with ADL
and IADL Scales
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The latent traits represent the ability to perform self-care
activities and instrumental activities (necessary for
independent living)
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Item difficulty (b): the level of function corresponding to
a 50% chance of endorsing the item
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Item discrimination (a): slope of the item characteristic
curve, or how well it differentiates low from high
functioning people
3 models
 One-parameter
(Rasch) model provides estimates
of item difficulty only
 Two-parameter
model provides estimates of
difficulty and discrimination
 Three-parameter
 IRT
model allows for guessing
does have different methods for dichotomous
and polytomous item scales
IRT models: dichotomous items
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One parameter model
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Probability correct response (given theta)
= 1/[1 + exp(theta – item difficulty)]
Two-parameter model
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Probability correct response (given theta)
= 1/{1 + exp [ – discrimination (theta – item difficulty)]}
Three parameter model:
Adds pseudo-guessing parameter
Two parameter model is most appropriate for
epidemiological research
Steps in applying IRT
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Step One: Assess dimensionality
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Factor analytic techniques
Exploratory factor analysis
 Study ratio of first to second eigenvalues (should be 3:1 or 4:1)
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Also χ2 tests for dimensionality
Calibrate items
Calculate item difficulty and discrimination and examine how
well model fits
 χ2 goodness of fit test
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Compare goodness of fit between one-parameter and two-parameter
models
Examine root mean square residual (values should be < 2.5)
Steps in IRT: continued
 Score
the examinees
 Get item information estimates
 Based
 Study
on discrimination adjusted for ‘standard error’
test information
 If choosing items from a larger pool, can discard
items with low information, and retain items that
give more information where it is needed
Item Information

Item information is a function of item difficulty and
discrimination. It is high when item difficulty is close to
the average level of function in the group and when ICC
slope is steep
The ADL scale example
 Caregiver
ratings of ADL and IADL performance
for 1686 people
 1048
with dementia and 484 without dementia
 1364
had complete ratings
ADL/IADL example
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Procedures
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Assessed dimensionality. Found two dimensions: ADL and
IADL
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Assessed fit of one-parameter and two parameter model for each
scale
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Two-parameter better
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Only 3 items fit one-parameter model
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Sig. improvement in χ2 goodness of fit
Used two-parameter model to get item statistics for 7 ADL items
and 7 IADL items
ADL/IADL
 Got
results for each item: difficulty,
discrimination, fit to model
 Results
for item information and total scale
information
Example of IRT with Relative’s Stress
Scale
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The latent trait (theta) represents the intensity of stress due
to recent life events
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Item severity or difficulty (b): the level of stress
corresponding to a 50% chance of endorsing the item
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Item discrimination (a): slope of the item characteristic
curve, or how well it differentiates low from high stress
cases
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Item information is a function of both: high when (b) is
close to group stress level and (a) is steep
Stress Scale: Item Information
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item information is a function of item difficulty and
discrimination. It is high when item difficulty is close to
group stress level and when ICC slope is steep
item 1
2
3
info .05 .5
.4
4
5
6
7
8
9
10
.05 .9
27
.5
.4
.06 .08
Stress Scale: Item Difficulty
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Item severity or difficulty (b) indicates the level of stress
(on theta scale) corresponding to a 50% chance of
endorsing the item
item 1
2
3
4
5
6
7
8
9
10
diff. 6.2 3.9 3.4 6.2 2.8 1.6 2.3 3.8 9.5 7.9
Stress Scale: Item Discrimination

item discrimination reflected in the slope of the item
characteristic curve (ICC): how well does the item
differentiate low from high stress cases?
item 1
2
3
4
5
6
7
8
9
10
disc 0.2 0.6 0.5 0.2 0.8 4.3 0.7 0.5 0.2 0.2
Example of developing Index of
Instrumental Support
 Community
Sample: CSHA-1
 Needed baseline indicator of social support as it is
important predictor of health
 Concept: Availability and quality of instrumental
support
 Blended IRT and classical methods
Sample
 8089
people
 Randomly divided into two samples:
 Development
and validation
 Procedures
 Item
selection and coding
 7 items
Procedure
 IRT
analyses
 Tested
dimensionality
 Two-parameter
model
 Estimated
item parameters
 Estimated
item and test information
 Scored
individual levels of support
External validation
 Internal
consistency
 Construct
validity
 Correlation
with size of social network
 Correlation
with marital status
 Correlation
with gender
 Predictive
validity
Empirical comparison of IRT and CTT
in scale validation
 Few
studies. So far, proponents of IRT assume it
is better. However,
 IRT
and CTT often select the same items
 High
correlations between CTT and IRT difficulty and
discrimination
 Very
high (0.93) correlations between CTT and IRT
estimates of total score
Empirical comparisons (cont’d)
 Little
difference in criterion or predictive validity of
IRT scores
 IRT
scores are only slightly better
 When
 IRT
item discriminations are highly varied, IRT is better
item parameters can be sample dependent
 Need
to establish validity on different samples, as in CTT
Advantages of IRT
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Contribution of each item to precision of total test score
can be assessed
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Estimates precision of measurement at each level of ability and for
each examinee
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With large item pool, item and test information excellent for testbuilding to suit different purposes
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Graphical illustrations are helpful
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Can tailor test to needs: For example, can develop a criterionreferenced test that has most precision around the cut-off score
Advantages of IRT
Interval
level scoring
More
analytic techniques can be used with the scale
Ability
Good
on different tests can be easily compared
for tests where a core of items is administered, but different
groups get different subsets (e.g., cross-cultural testing, computer
adapted testing)
Disadvantages of IRT
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Strict assumptions
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Large sample size
(minimum 200; 1000 for
complex models)
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More difficult to use than
CTT: computer programs
not readily available
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Models are complex and
difficult to understand
When should you use IRT?
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In test-building with
 Large
item pool
 Large
number of subjects
 Cross-cultural
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To develop short versions of tests
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
testing
(But also use CTT, and your knowledge of the test)
In test validation to supplement information from classical
analyses
Software for IRT analyses
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Rasch or one parameter models:
BICAL (Wright)
 RASCH (Rossi)
 RUMM 2010 http://www.arach.net.au/~rummlab/
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Two or three parameter models
NOHARM (McDonald)
 LOGIST
 TESTFACT
 LISREL
 MULTILOG
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