Item response theory: Difference between revisions

Revision as of 18:06, 24 July 2005

Revision as of 21:51, 28 July 2005

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==Overview==

'''Item Responseresponse Theory'''theory <b>(IRT)</b>''' designatesis a body of related [[psychometric]] [[theory]] that provides a foundation for stating and testing the [[hypothesis]] that, upon interaction between a given person and assessment item in a particular empirical context, the [[probability]] of a discrete item response is governed by the ___location of the item and person with respect to a latent [[trait]]. For this reason, IRT may be regarded as roughly synonymous with '''Latent Trait Theory'''. The term <i>latent</i> is used to emphasise that discrete item responses are taken to be <i>observable manifestations</i> of the trait or [[attribute]], the existence of which is hypothesized and must be inferred from the manifest responses. In addition to person and item locations, it is often posited that responses by persons to particular items are influenced by other properties, such as item [[discrimination]].

IRT models are used as a basis for [[statistical estimation]] of [[parameters]] that represent the "locations" of persons and items on a latent continuum or, more correctly, the [[magnitude]] of the latent trait attributable to the persons and items. For example, in attainment testing, estimates may be of the magnitude of a person's ability within a specific ___domain, such as reading comprehension. Once estimates of relevant parameters have been obtained, statistical tests are usually conducted to gauge the extent to which the parameters predict item responses. Stated somewhat differently, such tests are used to ascertain the degree to which the parameters account for the structure of and statistical patterns within the response data, either as a whole, or by considering specific subsets of the data such as response vectors pertaining to individual items or persons. This approach permits the central hypothesis to be subjected to empirical [[testing]], as well as providing information about the psychometric properties of a given assessment, and therefore also the quality of estimates.

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Much of the literature on IRT centers around item response models. A given model constitutes a mathematized hypothesis that the probability of a discrete response to an item is a function of a ''person parameter'' (or, in the case of multidimensional item response theory, a vector of person parameters) and one or more ''item parameters''. For example, in the 3 Parameter Logistic Model (3PLM), the probability of a correct response to an item <i>i</i> is:

:<math>

p_i({\theta})=c_i + \frac{(1-c_i)}{1+e^{-Da_i({\theta}-b_i)}}

</math>

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Item response theory advances the concept of item and test information to replace reliability. Information is also a ''function'' of the model parameters. For example, according to [[Fisher information]] theory, the item information supplied in the case of the [[Rasch model]] for dichotomous response data is simply the probability of a correct response multiplied by the probably of an incorrect response, or,

:<math>

I({\theta})=p_i({\theta})*q_i({\theta}).\,

<math>

I({\theta})=p_i({\theta})*q_i({\theta})

</math>

The Standard[[standard Errorerror of estimation]] (SE)is the reciprocal of the test information of at a given trait level, is the

:<math>

The Standard Error of estimation (SE)is the reciprocal of the test information of at a given trait level, is the

\mbox{SE}({\theta})=1/sqrt(I({\theta})).\,

<math>

SE({\theta})=1/sqrt(I({\theta}))

</math>

Thus more information implies less error of measurement.

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For other models, such as the two and three parameters models, the discrimination parameter plays an important part in the function. The item information function for the two parameter model is

:<math>

I({\theta})=a_i^2*p_i({\theta})*q_i({\theta}).\,

<math>

I({\theta})=a_i^2*p_i({\theta})*q_i({\theta})

</math>

In general, item information functions tend to look "bell-shaped." Highly discriminating items have tall, narrow information functions; they contribute greatly but over a narrow range. Less discriminating items provide less information but over a wider range.

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==External links==

* [http://work.psych.uiuc.edu/irt/tutorial.asp IRT Tutorial]

[[Category:Psychometrics]]

* [http://edres.org/irt/ An introduction to IRT]

*

* [http://www.rasch.org/ All about Rasch Measurement]

[http://work.psych.uiuc.edu/irt/tutorial.asp IRT Tutorial]

*

[http://edres.org/irt/ An introduction to IRT]

*

[http://www.rasch.org/ All about Rasch Measurement]

[[de:Probabilistische Testtheorie]]

[[Category:Psychometrics]]