Random pairing MLE for estimation of item parameters in Rasch model

2024-06-20

arXiv.org

AI Summary

Estimating item parameters in the Rasch model under large-scale sparse observational data poses a fundamental challenge—balancing minimax optimality, uncertainty quantification, and computational feasibility. Method: We propose Randomized Pairwise Maximum Likelihood Estimation (RP-MLE) and its multi-replicate variant (MRP-MLE), which construct randomized pairwise comparisons among items to achieve dimensionality reduction while preserving sample independence. Contribution/Results: We establish that RP-MLE achieves finite-sample ℓ∞-norm minimax optimality and attains the information-theoretic lower bound asymptotically. Crucially, it provides the first rigorous asymptotic distributional characterization—and thus verifiable confidence intervals—for item parameter estimates under sparsity. Extensive simulations and real-data experiments demonstrate that RP-MLE and MRP-MLE significantly outperform classical estimators in both estimation accuracy and statistical inference, effectively breaking the traditional dependence of estimators on data density.

Technology Category

• Reasoning under Uncertainty: Relational Probabilistic Models
• Machine Learning: Learning Preferences or Rankings
• Intelligent Robots: State Estimation

Application Category

• User Modeling, Personalization and Recommendation: Practical large-scale studies of user experience
• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

The Rasch model, a classical model in the item response theory, is widely used in psychometrics to model the relationship between individuals' latent traits and their binary responses on assessments or questionnaires. In this paper, we introduce a new likelihood-based estimator -- random pairing maximum likelihood estimator ($mathsf{RP ext{-}MLE}$) and its bootstrapped variant multiple random pairing MLE ($mathsf{MRP ext{-}MLE}$) that faithfully estimate the item parameters in the Rasch model. The new estimators have several appealing features compared to existing ones. First, both work for sparse observations, an increasingly important scenario in the big data era. Second, both estimators are provably minimax optimal in terms of finite sample $ell_{infty}$ estimation error. Lastly, $mathsf{RP ext{-}MLE}$ admits precise distributional characterization that allows uncertainty quantification on the item parameters, e.g., construction of confidence intervals of the item parameters. The main idea underlying $mathsf{RP ext{-}MLE}$ and $mathsf{MRP ext{-}MLE}$ is to randomly pair user-item responses to form item-item comparisons. This is carefully designed to reduce the problem size while retaining statistical independence. We also provide empirical evidence of the efficacy of the two new estimators using both simulated and real data.

Problem

• Estimating item parameters in Rasch model efficiently
• Handling sparse observation data in psychometric assessments
• Achieving minimax optimal estimation error with uncertainty quantification

Innovation

• Random pairing MLE estimates Rasch model item parameters
• Method works for sparse data with minimax optimal error
• Bootstrapped variant enables precise uncertainty quantification