Random pairing MLE for estimation of item parameters in Rasch model

TL;DR

This paper introduces the random pairing MLE estimators for the Rasch model, which are effective for sparse data, optimally accurate, and enable uncertainty quantification, with demonstrated empirical success.

Contribution

The paper proposes the novel RP-MLE and MRP-MLE estimators for Rasch model item parameters, capable of handling sparse data and providing finite-sample optimality and uncertainty quantification.

Findings

Estimators work well with sparse observations.

Both estimators are minimax optimal in finite samples.

Empirical results confirm effectiveness on simulated and real data.

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 to assessments or questionnaires. In this paper, we introduce a new likelihood-based estimator -- random pairing maximum likelihood estimator ($\mathrm{RP\text{-}MLE}$) and its bootstrapped variant multiple random pairing MLE ($\mathrm{MRP\text{-}MLE}$) which 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, both admit precise distributional characterization that allows uncertainty quantification on the item parameters, e.g., construction of confidence intervals for the item parameters. The main idea underlying $\mathrm{RP\text{-}MLE}$ and $\mathrm{MRP\text{-}MLE}$ 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.