Dynamic estimation in the extended marginal Rasch model with an application to mathematical computer-adaptive...

Area of Science:

• Statistics
• Psychometrics
• Educational Measurement

Background:

• Traditional statistical models struggle with dynamic datasets and missing values.
• The extended Rasch model offers a framework for item response theory but requires robust estimation methods for longitudinal data.

Purpose of the Study:

• To introduce a general response model adaptable to various restrictions, including the extended Rasch model.
• To develop a dynamic Bayesian estimation procedure for the extended Rasch model capable of handling time-varying data with missing values.
• To facilitate longitudinal comparisons in educational and psychological assessments.

Main Methods:

• A general response model framework is proposed.
• A dynamic Bayesian estimation procedure is developed for the extended Rasch model.
• A data augmentation method is employed to create an augmented person-by-item matrix, preserving sufficient statistics for longitudinal analysis.

Main Results:

• The proposed dynamic Bayesian procedure effectively handles time-varying data with numerous missing values.
• The data augmentation method ensures comparability over time by reproducing sufficient statistics.
• Longitudinal comparisons can be reliably performed using summary statistics like proportion correct and sum scores.

Conclusions:

• The introduced general response model and its extension, the extended Rasch model, provide a flexible framework for data analysis.
• The dynamic Bayesian estimation procedure with data augmentation is a powerful tool for longitudinal analysis of complex datasets.
• The method is validated through an example using data from a computer-adaptive mathematical practice environment.