Comparison of Recent Acceleration Techniques for the EM Algorithm in One- and Two-Parameter Logistic IRT Models

Marie Beisemann,Philipp Doebler,Ortrud Wartlick

doi:10.3390/psych2040018

Marie Beisemann, Philipp Doebler + Show 1 more

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https://doi.org/10.3390/psych2040018

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Journal: Psych	Publication Date: Nov 10, 2020
Citations: 3	License type: CC BY 4.0

Affiliation: TU Dortmund University

Abstract

The expectation–maximization (EM) algorithm is an important numerical method for maximum likelihood estimation in incomplete data problems. However, convergence of the EM algorithm can be slow, and for this reason, many EM acceleration techniques have been proposed. After a review of acceleration techniques in a unified notation with illustrations, three recently proposed EM acceleration techniques are compared in detail: quasi-Newton methods (QN), “squared” iterative methods (SQUAREM), and parabolic EM (PEM). These acceleration techniques are applied to marginal maximum likelihood estimation with the EM algorithm in one- and two-parameter logistic item response theory (IRT) models for binary data, and their performance is compared. QN and SQUAREM methods accelerate convergence of the EM algorithm for the two-parameter logistic model significantly in high-dimensional data problems. Compared to the standard EM, all three methods reduce the number of iterations, but increase the number of total marginal log-likelihood evaluations per iteration. Efficient approximations of the marginal log-likelihood are hence an important part of implementation.

Highlights

Item response theory (IRT) models have long been a staple in psychometric research, with applications in the investigation of test properties in smaller to medium samples, and in large-scale assessments [3,4]
While conditional maximum likelihood is feasible for the 1PL model, marginal maximum likelihood (MML) estimation for IRT models is the predominant estimation approach, for the 2PL model
It is important to keep in mind that the runtime of all three acceleration techniques is determined by a trade-off between a reduced number of iterations to the fixed point and an increased number of F and L evaluations per iteration

Summary

Introduction

Item response theory (IRT) models have long been a staple in psychometric research, with applications in the investigation of test properties in smaller to medium samples (e.g., in personality, intelligence, or creativity research [1,2]), and in large-scale assessments [3,4]. Popular—and the topic of the present work—are logistic IRT models for binary data, the one-parameter (1PL; : Rasch) and two-parameter (2PL) models. While conditional maximum likelihood is feasible for the 1PL model, marginal maximum likelihood (MML) estimation for IRT models is the predominant estimation approach, for the 2PL model. MML estimation is usually performed with the help of the expectation–maximization (EM) algorithm The EM algorithm is well suited to carry out ML estimation for situations in which one can consider parts of the “complete data” to be unobserved. In IRT models, the unobserved data are identical to latent ability. The EM algorithm maximizes the expected complete-data log-likelihood using an iterative two-step approach [5,9] of estimating the current posterior distribution of latent variables given the data and current or initial parameter estimates by determining

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