Item Response Theory in Health Measurement
Outline 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 This is the 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 context: 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. SF-36 bathing vs. able to do vigorous activities)
Some limitations Item difficulty, discrimination, and ability are confounded
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
Item Response Theory Complete theory of measurement and item selection Theoretically, item characteristics are not sample
dependent; estimates of ability are not item dependent Item scores are presented on the same scale as ability Puts all individual scores on standardized, interval level
scale; easy to compare between tests and individuals
Item Response Theory Assumes that a normally distributed latent trait underlies
performance on a measure Assumes unidimensionality
I.e., all items measure the same construct Assumes local independence
Items are uncorrelated with each other when ability is held constant
Given unidimensionality, any response to an item is a monotonically increasing function of the latent trait (see the item characteristic curves in next slide)
Illustration of IRT with ADL and IADL Scales
The latent traits represent the ability to perform self-care activities and instrumental activities (necessary for independent living)
Item difficulty (b): the level of function corresponding to a 50% chance of endorsing the item
Item discrimination (a): slope of the item characteristic curve, or how well it differentiates low from high functioning people
Example of differing item characteristic curves(Note: parameter = 2.82 for the steep curve, 0.98 for the shallow curve)
IRT can show distribution of respondents along theta and can alsoshow distribution of item difficulties (lower chart)
And can also show you the theta location of different response levels (here 0 to 3 scale)
Differential Item FunctioningAssuming 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 andthe 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 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
DIF detection Important part of test validation Helps to ensure measurement equivalence Scores on individual items are compared for two
groups:ReferenceFocal group under study
Groups 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
NonuniformProbability of correctly answering item is higher for
one group at some points on the scale; perhaps lower at other points
3 models One-parameter (Rasch) model provides estimates
of item difficulty only Two-parameter model provides estimates of
difficulty and discrimination Three-parameter model allows for guessing IRT does have different methods for dichotomous
and polytomous item scales
IRT models: dichotomous items One parameter model
Probability correct response (given theta)= 1/[1 + exp(theta – item difficulty)]
Two-parameter model Probability correct response (given theta)
= 1/{1 + exp [ – discrimination (theta – item difficulty)]}Three parameter model:
Adds pseudo-guessing parameterTwo parameter model is most appropriate for
epidemiological research
Steps in applying IRT Step One: Assess dimensionality
Factor analytic techniques Exploratory factor analysis Study ratio of first to second eigenvalues (should be 3:1 or 4:1)
Also χ2 tests for dimensionality Calibrate items
Calculate item difficulty and discrimination and examine how well model fits
χ2 goodness of fit test 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 on discrimination adjusted for ‘standard error’ Study 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 people1048 with dementia and 484 without dementia1364 had complete ratings
ADL/IADL example Procedures
Assessed dimensionality. Found two dimensions: ADL and IADL
Assessed fit of one-parameter and two parameter model for each scale
Two-parameter better Only 3 items fit one-parameter model 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
The latent trait (theta) represents the intensity of stress due to recent life events
Item severity or difficulty (b): the level of stress corresponding to a 50% chance of endorsing the item
Item discrimination (a): slope of the item characteristic curve, or how well it differentiates low from high stress cases
Item information is a function of both: high when (b) is close to group stress level and (a) is steep
Stress Scale: Item Information 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 4 5 6 7 8 9 10
info .05 .5 .4 .05 .9 27 .5 .4 .06 .08
Stress Scale: Item Difficulty 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 coding7 items
Procedure IRT analyses
Tested dimensionalityTwo-parameter modelEstimated item parametersEstimated item and test informationScored individual levels of support
External validation Internal consistency Construct validity
Correlation with size of social networkCorrelation with marital statusCorrelation 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
discriminationVery 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 betterWhen item discriminations are highly varied, IRT is better
IRT item parameters can be sample dependent Need to establish validity on different samples, as in CTT
Advantages of IRT Contribution of each item to precision of total test score
can be assessed Estimates precision of measurement at each level of ability and for
each examinee
With large item pool, item and test information excellent for test-building to suit different purposes
Graphical illustrations are helpful Can tailor test to needs: For example, can develop a criterion-
referenced 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 on different tests can be easily compared
Good 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 Strict assumptions Large sample size
(minimum 200; 1000 for complex models)
More difficult to use than CTT: computer programs not readily available
Models are complex and difficult to understand
When should you use IRT? In test-building with
Large item poolLarge number of subjects
Cross-cultural testing To develop short versions of tests
(But also use CTT, and your knowledge of the test) In test validation to supplement information from classical
analyses
Software for IRT analyses Rasch or one parameter models:
BICAL (Wright) RASCH (Rossi) RUMM 2010 http://www.arach.net.au/~rummlab/
Two or three parameter models NOHARM (McDonald) LOGIST TESTFACT LISREL MULTILOG