Table Of Contents

- Manual
- Getting Started
- Starting the Program
- Retrieving Data
- Manipulating Data
- The Variable List
- The Variable List Menu
- Filter Observations/Selecting
- Add New Variables
- Delete Variables
- Edit Metadata
- Set Replicate Weights
- New Variable Reserve
- Edit Value Labels
- Dummy Code Categorical Variable
- Collapse Categories of Categorical Variable
- Set Missing Values
- The Expression Evaluator

- Saving and Re-running Actions

- Sampling
- Procedures
- Measurement Models
- MML Models for Test Data
- Other Available Procedures

- Graphics
- Tools
- Estimation Methods
- Optimization Techniques
- Variance Estimation

- Post-hoc Procedures
- More user input instructions
- The User Interface
- Input Instructions
- Options
- Output Precision

- Glossary of Terms and Symbols

- Getting Started

Generalized Partial Credit Model

The generalized partial credit model (GPCM) is an item response theory (IRT) model developed for situations where item responses are contained in two or more ordered categories. Items are conceptualized as a series of ordered steps where examinees receive partial credit for successfully completing a step. Steps correspond to the various levels of performance needed to complete an item. The GPCM is formulated based on the assumption that each probability of choosing the *k*-th category over the *k* - 1-th category is governed by the dichotomous response model.

The generalized partial credit model (GPCM) is an IRT model developed to analyze partial credit data, where responses are scored 0,1,…, *k*, where *k* is the highest score category for the item.

Masters (1982) developed the partial credit model (PCM) by formulating a polytomous rating response model based on the Rasch (1960) dichotomous model. In Masters’ formulation, the probability of choosing the *k*th category over the *k* - 1th category of item *i* is governed by the dichotomous response model. Let *P _{jk}*(q) denote the specific probability of selecting the

where *k* = 1,2,..., *m _{j}*

Note that in this model, all items are assumed to have uniform discriminating power.

Muraki (1992) extented Masters’ PCM by relaxing the assumption of uniform discriminating power of test items based on the two-parameter (2PL) logistic response model. In Muraki’s formulation, the probability of choosing category *k* over category *k* - 1 is given by the conditional probability:

where *k* = 1,2,...,*m _{j}*. The above equation can be written as

After normalizing each *P _{jk}*(q) so that S

where *D,* is a scaling constant set to 1.7 to approximate the normal ogive model, *a _{j}* is a slope parameter ,

Indeterminacies in the parameters of the GPCM are resolved by setting *d _{j}*,0 = 0 and setting . Muraki (1992) points out that

The figure below illustrates the item category response functions (what is known as item characteristic curves for dichotomous items) for a GPCM item with parameters * a*=1 and

Masters, G. N. (1982). A Rasch model for partial credit scoring. *Pscychometrika, 47,* 149-174.

Muraki, E. (1992). A generalized partial credit model: Application of an EM algorithm. *Applied Psychological Measurement, 16,* 159-176.

Currently the parameters of the Generalized Partial Credit Model must be estimated through some other statistical software package and imported into AM. Future versions of AM software will allow the user to estimate both item and ability parameters.

NAEP uses the generalized partial credit model to analyze polytomous item where responses are contained in two or more ordered categories. Indeterminacies in the parameters of the GPCM are resolved by setting

*d _{j}*,0 = 0 and setting .