Marginal likelihood inference for a model for item responses and response times

Area of Science:

• Psychometrics
• Statistical Modeling
• Educational Measurement

Background:

• Hierarchical models are crucial for understanding complex data structures in testing.
• Accurate estimation and model fit testing are essential for reliable assessment.
• Existing methods may not fully capture the nuances of response accuracy and speed.

Purpose of the Study:

• To develop marginal maximum-likelihood procedures for parameter estimation in a hierarchical model.
• To introduce methods for testing the fit of this model, including assumptions about item parameters and response patterns.
• To evaluate the feasibility and performance of these procedures through simulations and empirical application.

Main Methods:

• A hierarchical model combining dichotomous response and response time models.
• Fisher's identity for efficient estimation of item parameters.
• Lagrange multiplier tests for assessing subpopulation invariance (differential item functioning), response function shape, and conditional independence.
• Simulation studies to assess feasibility, power, and Type I error rates.
• Application to a real-world dataset from a computerized adaptive test.

Main Results:

• Demonstrated feasibility of the proposed estimation and testing procedures.
• Quantified the power and Type I error rates of the model fit tests.
• Successfully applied the methods to analyze item performance in a language comprehension test.
• Fisher's identity provides an efficient way to estimate item parameters.

Conclusions:

• The developed marginal maximum-likelihood procedures are effective for parameter estimation and model fit testing in hierarchical models.
• These methods offer a robust framework for analyzing both accuracy and speed in test item responses.
• The approach is applicable to various testing contexts, including computerized adaptive testing.