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Polytomous Item Response Theory Models
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Polytomous Item Response Theory Models



September 2005 | 120 pages | SAGE Publications, Inc
Polytomous Item Response Theory Models provides a unified, comprehensive introduction to the range of polytomous models available within item response theory (IRT). It begins by outlining the primary structural distinction between the two major types of polytomous IRT models. This focuses on the two types of response probability that are unique to polytomous models and their associated response functions, which are modeled differently by the different types of IRT model. It then describes, both conceptually and mathematically, the major specific polytomous models, including the Nominal Response Model, the Partial Credit Model, the Rating Scale model, and the Graded Response Model. Important variations, such as the Generalized Partial Credit Model are also described as are less common variations, such as the Rating Scale version of the Graded Response Model. Relationships among the models are also investigated and the operation of measurement information is described for each major model. Practical examples of major models using real data are provided, as is a chapter on choosing an appropriate model. Figures are used throughout to illustrate important elements as they are described.
 
Series Editor's Introduction
 
Acknowledgments
 
1. Introduction
Measurement Theory

 
Item Response Theory

 
Applying the IRT Model

 
Reasons for Using Polytomous IRT Models

 
Polytomous IRT Models

 
Two Types of Probabilities

 
Two Types of Polytomous Models

 
Category Boundaries

 
Item Category Response Functions

 
 
2. Nominal Response Model
The Mathematical Model

 
Information

 
Relationship to Other IRT Models

 
Variations

 
A Practical Example

 
 
3. Polytomous Rasch Models
Partial Credit Model

 
Category Steps

 
The Mathematical Model

 
Information

 
Relationship to Other IRT Models

 
Variations

 
PCM Summary

 
Rating Scale Model

 
The Mathematical Model

 
Model Parameters

 
Sufficient Statistics and Other Considerations

 
Information

 
Expected Values and Response Functions

 
Response Functions and Information

 
Relationship to Other IRT Models

 
PCM Scoring Function Formulation and the NRM

 
Variations

 
Generalized Partial Credit Model

 
Discrimination and Polytomous Rasch Models

 
Summary of Polytomous Rasch Models

 
Three Practical Examples

 
 
4. Samejima Models
Framework

 
From Response Process to Specific Model

 
The Homogeneous Case: Graded Response Models

 
The Mathematical Model

 
Information

 
Information for Polytomous Models

 
Relationship to Other IRT Models

 
From Homogeneous Class to Heterogeneous Class and Back

 
A Common Misconception

 
Variations

 
Summary of Samejima Models

 
Potential Weaknesses of the Cumulative Boundary Approach

 
Possible Strengths of the Cumulative Boundary Approach

 
A Practical Example

 
 
5. Model Selection
General Criteria

 
Mathematical Approaches

 
Fit Statistic Problems

 
An Example

 
Differences in Modeled Outcome

 
Conclusion

 
 
Acronyms and Glossary
 
Notes
 
References
 
Index
 
About the Authors

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