STATA ITEM RESPONSE THEORYREFERENCE MANUAL

RELEASE 14

A Stata Press PublicationStataCorp LLCCollege Station, Texas

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Contents

irt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction to IRT models 1Control Panel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IRT Control Panel 18

irt 1pl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . One-parameter logistic model 27irt 1pl postestimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Postestimation tools for irt 1pl 37

irt 2pl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Two-parameter logistic model 42irt 2pl postestimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Postestimation tools for irt 2pl 53

irt 3pl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Three-parameter logistic model 58irt 3pl postestimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Postestimation tools for irt 3pl 69

irt grm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Graded response model 74irt grm postestimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Postestimation tools for irt grm 85

irt nrm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nominal response model 90irt nrm postestimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Postestimation tools for irt nrm 100

irt pcm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Partial credit model 105irt pcm postestimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Postestimation tools for irt pcm 115

irt rsm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rating scale model 120irt rsm postestimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Postestimation tools for irt rsm 129

irt hybrid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hybrid IRT models 134irt hybrid postestimation . . . . . . . . . . . . . . . . . . . . . . . . . . . Postestimation tools for irt hybrid 146

estat report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Report estimated IRT parameters 152

irtgraph icc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Item characteristic curve plot 159irtgraph tcc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Test characteristic curve plot 171

irtgraph iif . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Item information function plot 176irtgraph tif . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Test information function plot 182

dif . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction to differential item functioning 188diflogistic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Logistic regression DIF 191difmh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . MantelHaenszel DIF 196

Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

Subject and author index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

i

Cross-referencing the documentation

When reading this manual, you will find references to other Stata manuals. For example,

[U] 26 Overview of Stata estimation commands[R] regress[D] reshape

The first example is a reference to chapter 26, Overview of Stata estimation commands, in the UsersGuide; the second is a reference to the regress entry in the Base Reference Manual; and the thirdis a reference to the reshape entry in the Data Management Reference Manual.

All the manuals in the Stata Documentation have a shorthand notation:

[GSM] Getting Started with Stata for Mac[GSU] Getting Started with Stata for Unix[GSW] Getting Started with Stata for Windows[U] Stata Users Guide[R] Stata Base Reference Manual[BAYES] Stata Bayesian Analysis Reference Manual[D] Stata Data Management Reference Manual[FN] Stata Functions Reference Manual[G] Stata Graphics Reference Manual[IRT] Stata Item Response Theory Reference Manual[XT] Stata Longitudinal-Data/Panel-Data Reference Manual[ME] Stata Multilevel Mixed-Effects Reference Manual[MI] Stata Multiple-Imputation Reference Manual[MV] Stata Multivariate Statistics Reference Manual[PSS] Stata Power and Sample-Size Reference Manual[P] Stata Programming Reference Manual[SEM] Stata Structural Equation Modeling Reference Manual[SVY] Stata Survey Data Reference Manual[ST] Stata Survival Analysis Reference Manual[TS] Stata Time-Series Reference Manual[TE] Stata Treatment-Effects Reference Manual:

Potential Outcomes/Counterfactual Outcomes[ I ] Stata Glossary and Index

[M] Mata Reference Manual

iii

Title

irt Introduction to IRT models

Description Remarks and examples References Also see

DescriptionItem response theory (IRT) is used in the design, analysis, scoring, and comparison of tests and

similar instruments whose purpose is to measure unobservable characteristics of the respondents. Thisentry discusses some fundamental and theoretical aspects of IRT and illustrates these with workedexamples.

Binary response modelsirt 1pl One-parameter logistic modelirt 2pl Two-parameter logistic modelirt 3pl Three-parameter logistic model

Categorical response modelsirt grm Graded response modelirt nrm Nominal response modelirt pcm Partial credit modelirt rsm Rating scale model

Multiple IRT models combinedirt hybrid Hybrid IRT model

Remarks and examplesResearchers are often interested in studying abilities, personality traits, and other unobservable

characteristics. Throughout this manual, we most often refer to the unobserved characteristic of interestas the latent trait, but we will sometimes also use the term ability.

Latent traits cannot be measured directly, because they are unobservable, but they can be quantifiedwith an instrument. An instrument is simply a collection of items designed to measure a personslevel of the latent trait. For example, a researcher interested in measuring mathematical ability (latenttrait) may design a test (instrument) consisting of 100 questions (items).

When designing the instrument or analyzing data from the instrument, the researcher is interestedin how each individual item relates to the trait and how the group of items as a whole relates to thistrait. IRT models allow us to study these relationships.

IRT models are used extensively in the study of cognitive and personality traits, health outcomes, andin the development of item banks and computerized adaptive testing. Some examples of applied workinclude measuring computer anxiety in grade school children (King and Bond 1996), assessing physicalfunctioning in adults with HIV (Wu, Hays, Kelly, Malitz, and Bozzette 1997), and measuring thedegree of public policy involvement of nutritional prof