Generalized Residuals for General Models for Contingency Tables With Application to Item Response Theory

Abstract

Generalized residuals are a tool employed in the analysis of contingency tables to examine possible sources of model error. They have typically been applied to log-linear models and to latent-class models. A general approach to generalized residuals is developed for a very general class of models for contingency tables. To illustrate their use, generalized residuals are applied to models based on item response theory (IRT) models. Such models are commonly applied to analysis of standardized achievement or aptitude tests. To obtain a realistic perspective on application of generalized residuals, actual testing data are employed.

Keywords:

• Marginal distributions
• Marginal estimation
• Maximum likelihood
• Normal approximations
• Rasch model
• Three-parameter logistic model
• Two-parameter logistic model

Acknowledgments

We are grateful to Neil Dorans, Hongwen Guo, Andreas Oranje, Frederic Robin, Matthias von Davier, the editor, the associate editor, and the two anonymous reviewers for their helpful advice; to Jill Carey, Behroz Maneckshana, Anthony Giunta, Rui Gao, Kevin Larkin, Aleta Sclan, and Yi-Hsuan Lee for their help with the data; and to Ruth Greenwood for her help with copyediting. The research reported here was supported by the Institute of Education Sciences, U.S. Department of Education, through Grant R305D120006 to Educational Testing Service as part of the Statistical and Research Methodology in Education Initiative. The opinions expressed in this article are those of the authors and do not represent views of the Institute, the U.S. Department of Education, or Educational Testing Service. Note: Sandip Sinharay was at ETS when he performed most of the work for this article. He is currently at CTB/McGraw-Hill.