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

The partial credit model (PCM) with a varying slope parameter is developed and called the generalized partial credit model (GPCM). The item step parameter of this model is decomposed to a location and a threshold parameter, following Andrich's (1978) rating scale formulation. The EM algorithm for estimating the model parameters is derived. The performance of this generalized model is compared on both simulated and real data to a Rasch family of polytomous item response models. Simulated data were generated and then analyzed by the various polytomous item response models. The results demonstrate that the rating formulation of the GPCM is quite adaptable to the analysis of polytomous item responses. The real data used in this study consisted of the National Assessment of Educational Progress (Johnson & Allen, 1992) mathematics data that used both dichotomous and polytomous items. The PCM was applied to these data using both constant and varying slope parameters. The GPCM, which provides for varying slope parameters, yielded better fit to the data than did the PCM.

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This study examined the application of the MML-EM algorithm to the parameter estimation problems of the normal ogive and logistic polytomous response models for Likert-type items. A rating-scale model was developed based on Samejima's (1969) graded response model. The graded response model includes a separate slope parameter for each item and an item response parameter. In the rating-scale model, the item response parameter is resolved into two parameters: the item location parameter, and the category threshold parameter characterizing the boundary between response categories. For a Likert-type questionnaire, where a single scale is employed to elicit different responses to the items, this item response model is expected to be more useful for analysis because the item parameters can be estimated separately from the threshold parameters associated with the points on a single Likert scale. The advantages of this type of model are shown by analyzing simulated data and data from the General Social Surveys. Index terms: EM algorithm, General Social Surveys, graded response model, item response model, Likert scale, marginal maximum likelihood, polytomous item response model, rating-scale model. An item response model expresses a probabilistic relationship between an examinee's performance on a test item and the examinee's latent trait. Several types of item response models have been proposed. The applicability of dichotomous item response models (Birnbaum, 1968; Lord, 1980; Lord & Novick, 1968; Rasch, 1960/1980) to cognitive item response data has been extensively studied. Several polytomous item response models have been formulated based on these dichotomous models

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