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NAEP Analysis and Scaling → Estimation of NAEP Score Scales → Item Scaling Models → The Generalized Partial Credit Model

NAEP Technical DocumentationThe Generalized Partial Credit Model

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Exemplar Theoretical Item Response Function for the Generalized Partial Credit Item Response Theory Model

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In addition to the multiple-choice and other two-category items, a number of extended constructed-response items are presented in NAEP assessments. The NAEP writing assessments include only extended constructed-response items, but most other NAEP assessments also include some extended constructed-response items. Each of these items is scored on a multipoint scale with potential scores ranging from 0 to 3, from 0 to 4, or from 0 to 5. For some subjects, short constructed-response items are scored on a three-point scale (0 to 2) as well as on a two-category scale (0 to 1). Items that are scored on a multipoint scale are referred to as polytomous items, in contrast with the multiple-choice and constructed-response items, which are scored correct or incorrect and referred to as dichotomous items.

The polytomous items are scaled using a generalized partial credit model (Muraki 1992). The fundamental equation of this model is the probability that a person with score θk on scale k will have, for the jth item, a response xj that is scored in the ith of mj ordered score categories:

The probability that x sub j equals i given theta sub k, a sub j, b sub j, d sub j one to d sub j, m sub j minus one equals the exponential of the sum over v from zero to i of 1.7 times a sub j times the quantity theta sub k minus b sub j plus d sub j, v divided by the sum over g from zero to m sub j minus one of the exponential of the sum over v from zero to g of 1.7 times a sub j times the quantity theta sub k minus b sub j plus d sub j. This is equivalent to P sub ji of theta sub k.


mj is the number of categories in the response to item j;
xj is the response to item j, with possibilities 0, 1, ... , mj - 1;
aj is the slope parameter;
bj is the item location parameter characterizing overall difficulty; and
dj,i is the category i threshold parameter (see below).

Indeterminacies in the parameters of the above model are resolved by setting dj,0 = 0 and setting

The sum from i equal to one to m sub j minus one of d sub j i equals zero..

Muraki (1992) points out that bj - dj,i is the point on the θk scale at which the plots of Pj,i -1(θk) and
Pji(θk) intersect and so characterizes the point on the θk scale at which the response to item j has equal probability of falling in response category i - 1 and falling in response category i.

When mj = 2, so that there are two score categories (0,1), it can be shown that Pji(θk) from the generalized partial credit model for i = 0,1 corresponds respectively to Pj0(θk) and Pj1(θk) of the two-parameter logistic model.

Last updated 12 August 2008 (RF)

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