Generalized Linear Mixed Models (GLMMs) can serve as a unifying statistical framework for both descriptive and explanatory Rasch models. This chapter briefly traces the evolution from linear regression to generalized linear models (GLMs) through mixed models to GLMMs that can integrate fixed and random effects to address grouped and hierarchical data structures. The chapter explains how GLMMs extend GLMs by modeling random effects, and this makes them particularly well-suited for item response modeling where persons and items are treated as distinct sources of variation. The dichotomous Rasch model is expressed within the GLMM framework, and equivalence is shown between Rasch parameter estimation and GLMM formulations. An applied example using the Learning Stimulation Scale (introduced in Chap. 1 ) demonstrates the estimation process in R with the EIRM package. The example includes a description of data preparation, model specification, parameter estimation, and the interpretation of the Wright Map. There are several key steps described in detail including the reshaping of data to a long format, coding covariates, specifying fixed and random effects, and the interpretation of results. The chapter emphasizes that Rasch’s invariance properties depend on good model-data fit, and this is explored further in Chap. 4 . By embedding Rasch models within GLMMs, researchers can extend measurement to explanatory contexts using covariates, and also preserve the theoretical foundations of Rasch measurement. Generalized Linear Mixed Models (GLMMs) can serve as a unifying statistical framework for both descriptive and explanatory Rasch models. This chapter briefly traces the evolution from linear regression to generalized linear models (GLMs) through mixed models to GLMMs that can integrate fixed and random effects to address grouped and hierarchical data structures. The chapter explains how GLMMs extend GLMs by modeling random effects, and this makes them particularly well-suited for item response modeling where persons and items are treated as distinct sources of variation. The dichotomous Rasch model is expressed within the GLMM framework, and equivalence is shown between Rasch parameter estimation and GLMM formulations. An applied example using the Learning Stimulation Scale (introduced in Chap. 1 ) demonstrates the estimation process in R with the EIRM package. The example includes a description of data preparation, model specification, parameter estimation, and the interpretation of the Wright Map. There are several key steps described in detail including the reshaping of data to a long format, coding covariates, specifying fixed and random effects, and the interpretation of results. The chapter emphasizes that Rasch’s invariance properties depend on good model-data fit, and this is explored further in Chap. 4 . By embedding Rasch models within GLMMs, researchers can extend measurement to explanatory contexts using covariates, and also preserve the theoretical foundations of Rasch measurement.

Rasch Measurement Theory and Generalized Linear Mixed Models

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发表日期：

2026年

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George Engelhard,
Stefanie A. Wind

摘要

Generalized Linear Mixed Models (GLMMs) can serve as a unifying statistical framework for both descriptive and explanatory Rasch models. This chapter briefly traces the evolution from linear regression to generalized linear models (GLMs) through mixed models to GLMMs that can integrate fixed and random effects to address grouped and hierarchical data structures. The chapter explains how GLMMs extend GLMs by modeling random effects, and this makes them particularly well-suited for item response modeling where persons and items are treated as distinct sources of variation. The dichotomous Rasch model is expressed within the GLMM framework, and equivalence is shown between Rasch parameter estimation and GLMM formulations. An applied example using the Learning Stimulation Scale (introduced in Chap. 1 ) demonstrates the estimation process in R with the EIRM package. The example includes a description of data preparation, model specification, parameter estimation, and the interpretation of the Wright Map. There are several key steps described in detail including the reshaping of data to a long format, coding covariates, specifying fixed and random effects, and the interpretation of results. The chapter emphasizes that Rasch’s invariance properties depend on good model-data fit, and this is explored further in Chap. 4 . By embedding Rasch models within GLMMs, researchers can extend measurement to explanatory contexts using covariates, and also preserve the theoretical foundations of Rasch measurement. Generalized Linear Mixed Models (GLMMs) can serve as a unifying statistical framework for both descriptive and explanatory Rasch models. This chapter briefly traces the evolution from linear regression to generalized linear models (GLMs) through mixed models to GLMMs that can integrate fixed and random effects to address grouped and hierarchical data structures. The chapter explains how GLMMs extend GLMs by modeling random effects, and this makes them particularly well-suited for item response modeling where persons and items are treated as distinct sources of variation. The dichotomous Rasch model is expressed within the GLMM framework, and equivalence is shown between Rasch parameter estimation and GLMM formulations. An applied example using the Learning Stimulation Scale (introduced in Chap. 1 ) demonstrates the estimation process in R with the EIRM package. The example includes a description of data preparation, model specification, parameter estimation, and the interpretation of the Wright Map. There are several key steps described in detail including the reshaping of data to a long format, coding covariates, specifying fixed and random effects, and the interpretation of results. The chapter emphasizes that Rasch’s invariance properties depend on good model-data fit, and this is explored further in Chap. 4 . By embedding Rasch models within GLMMs, researchers can extend measurement to explanatory contexts using covariates, and also preserve the theoretical foundations of Rasch measurement.