A Generalized Partial Credit Model: Application of an Em Algorithm

Abstract

The Partial Credit model with a varying slope parameter has been developed, and it is called the Generalized Partial Credit model. The item step parameter of this model is decomposed to a location and a threshold parameter, following Andrich's Rating Scale formulation. The EM algorithm for estimating the model parameters was derived. The performance of this generalized model is compared with a Rasch family of polytomous item response models based on both simulated and real data. Simulated data were generated and then analyzed by the various polytomous item response models. The results obtained demonstrate that the rating formulation of the Generalized Partial Credit model is quite adaptable to the analysis of polytomous item responses. The real data used in this study consisted of NAEP Mathematics data which was made up of both dichotomous and polytomous item types. The Partial Credit model was applied to this data using both constant and varying slope parameters. The Generalized Partial Credit model, which provides for varying slope parameters, yielded better fit to data than the Partial Credit model without such a provision. Index terms: item response model polytomous item response model the Partial Credit model the Rating Scale model the Nominal Response model NAEP

Modelling non-ignorable missing-data mechanisms with item response theory models

“…In the example given below, both items with dichotomous and polytomous responses will be considered. These responses will be analysed with the generalized partial credit model (GPCM: Muraki, 1992). In the GPCM the probability, pðx ik ¼ jju i ; a k ; b k Þ; that respondent i responds to item k in category j, j ¼ 1; : : : ; m k ; is denoted by…”

An investigation of enhancement of ability evaluation by using a nested logit model for multiple-choice items

“…On account of order of multiple-choice options, GPCM and NRM, presented ordered multiple-choice items and unordered multiple-choice items respectively, were used to generate responses. Item parameters were generated randomly from the following distributions: slope parameter α~Uniform (0.5, 2) and intercept parameters δ v~U niform (-2, 2) for the GPCM, followed by the imposition of constraints δ 1 < δ 2 < δ 3 (Muraki, 1992), and slop parameter λ v~U niform (-2, 2) and intercept parameter ξ v~U niform (-2, 2) for the NRM, followed by the imposition of constraints and (Suh& Bolt, 2010). Besides, all simulated items in this part included four options, namely one correct option and three distractors.…”

“…In consideration of order of options in multiplechoice items, there are mainly two ways of modeling when polytomous models were used. One is transforming unordered categories into ordered ones and fitting ordered polytomous models, for example GPCM (Muraki, 1992). The other is fitting unordered polytomous models like Bock's (1982) nominal response model (NRM) and Thissen and Steinberg' s (1984) multiple-choice model (MCM) which was general model of NRM.…”

A nonparametric item analysis of a selected item subset of the Learning Process Questionnaire

“…Master's partial credit model is based on the Rasch or one-parameter model for dichotomous items, which means a constant slope parameter for all items so that the raw score is a sufficient statistic for estimating respondents' trait level (Wright & Masters, 1982). In this model, rather than estimating the category boundaries for each item as in the graded response model, the relative difficulty or threshold associated with a response moving from one item category to the next is estimated (also see Muraki 1992Muraki , 1997, for recent modifications). Andrich's rating scale model like the partial credit model assumes a constant slope parameter for all items, which means that the raw total score is a sufficient statistic for estimating the trait level.…”