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NAEP Analysis and Scaling → Estimation of NAEP Score Scales → Item Scaling Models → The Two-Parameter Logistic Model

The Two-Parameter Logistic Model

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Exemplar Theoretical Item Response Function for the Two-Parameter Logistic Item Response Theory Model

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A two-parameter logistic (2PL) model is used for the short constructed-response items that were scored as correct or incorrect. The form of the 2PL model is the same as that of the 3PL model, with the cj parameter fixed at zero. The fundamental equation of the 2PL model defines the probability that a person whose score on scale k is characterized by the unobservable variable θk will respond correctly to item j as

The probability that x sub j equals one given theta sub k, a sub j, and b sub j equals one divided by the quantity one plus the exponential of the quantity negative one point seven times a sub j times the quantity theta sub k minus b sub j.  This is equivalent to uppercase p sub j one of theta sub k


xj is the response score to item j, 1 if correct and 0 if not;
aj where aj > 0, is the slope parameter of item j, characterizing its sensitivity to scale score; and
bj is the threshold parameter of item j, characterizing its difficulty.

The probability of an incorrect response to the item is further defined as

P sub j zero is equivalent to the probability taht x sub j equals zero given theta sub k, a sub j and b sub j, which is equal to one minus P sub j one of theta sub k

Last updated 14 July 2008 (DB)

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National Center for Education Statistics -
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