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

The Three-Parameter Logistic Model

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Exemplar Theoretical Item Response Function (IRF) for the Three-Parameter Logistic Item Response Theory (IRT) Model

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A three-parameter logistic (3PL) model is used for the multiple-choice items (which are scored correct or incorrect.) The fundamental equation of the 3PL 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, b sub j, and c subj equals c sub j plus the quantity one minus c sub j, divided by the quantity one plus the exponential negative one point seven times a sub j times the quantity theta sub k minus b sub j is equivalent to p sub j one of theta sub k.


xj is the response score for 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;
bj is the threshold parameter of item j, characterizing its difficulty; and
cj where 0 < cj < 1, is the lower asymptote parameter of item j, reflecting the chances of students of very low scale score selecting the correct option.

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

P sub j zero is equivalent to the probability that x sub j equals zero given theta sub k, a sub j, b sub j and c 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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