## page was renamed from Statistics/ItemResponseTheory
= Item Response Theory =

'''Item response theory''' ('''IRT''') is a framework for predicting a response given some [[Statistics/FactorAnalysis|latent factor]].

<<TableOfContents>>

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== Description ==

The framework is used to predict a particular response to a given item. Often there is a predetermined 'correct' response, so the framework is used to predict an individual's ability to correctly response. This is the case of the '''Rasch model''' introduced below.

In general the framework is applicable to predicting any particular response to an item.

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== Formulation ==

Let ''i'' index individuals and ''j'' index items. Individuals also have a 1-dimensional [[Statistics/FactorAnalysis|latent ability]], usually notated as ''θ,,i,,'' but sometimes as ''η,,i,,'' instead.

The '''Rasch model''' is a [[Statistics/LogisticModel|logistic model]] that predicts correct responses (''x,,ij,, = 1'') based on latent ability and '''item difficulty''' ''α,,j,,''. This model is variably considered an example of an '''item characteristic curve''' ('''ICC''') or a '''1-parameter logistic''' ('''1PL''') model.

{{attachment:rasch.svg}}

An evolution of the Rasch model, sometimes called a '''2-parameter logistic''' ('''2PL''') model, introduces a new term to allow for varying slopes. ''λ,,j,,'' reflects the '''discrimination''' of an item. High discrimination suggests that responses are strongly related to latent ability, whereas low discrimination suggests that responses are very similar across individuals with high or low ability.

{{attachment:twoparameter.svg}}

Further evolutions add '''guessing''' and '''inattention''' (ceiling) parameters. ('''3PL''', '''4PL''', etc.) In these cases, the parameters are more commonly notated ''a'', ''b'', ''c'', ''d'', and so on. See [[https://demonstrations.wolfram.com/ComparingTheNormalOgiveAndLogisticItemCharacteristicCurves/|this live demo]] of a 3PL model.

Generalizations of the model include:
 * there is a category of polytomous models for categorical responses
   * '''graded response models''' ('''GRM''') for ordinal responses
 * multidimensional models for multiple latent factors

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== Normal Ogive Models ==

'''Normal ogive models''' trade the logit [[Statistics/GeneralizedLinearModel#Design|link function]] for [[Statistics/ProbitModel|probit]]. Note that the parameters in this specification are almost always notated as:
 * ''β,,j,,'' for item difficulty
 * ''α,,j,,'' for item discrimination

{{attachment:ogive.svg}}



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