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:
|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
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.