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

NAEP Technical DocumentationThe Two-Parameter Logistic Model

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

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 c_j 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 θ_k will respond correctly to item j as

where

	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

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