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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

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

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 θ_k on scale k will have, for the j^th item, a response x_j that is scored in the i^th of m_j ordered score categories:

where

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 d_j,0 = 0 and setting

Muraki (1992) points out that b_j - d_j,i is the point on the θ_k scale at which the plots of P_{j,i -1}(θ_k) and
P_ji(θ_k) intersect and so characterizes the point on the θ_k 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 m_j = 2, so that there are two score categories (0,1), it can be shown that P_ji(θ_k) from the generalized partial credit model for i = 0,1 corresponds respectively to P_j0(θ_k) and P_j1(θ_k) of the two-parameter logistic model.

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