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Gauss–Hermite Quadrature in Marginal Maximum Likelihood Estimation of Item Parameters

  • Seock-Ho KimEmail author
  • Yu Bao
  • Erin Horan
  • Meereem Kim
  • Allan S. Cohen
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 140)

Abstract

Although many theoretical papers on the estimation method of marginal maximum likelihood of item parameters for various models under item response theory mentioned Gauss–Hermite quadrature formulas, almost all computer programs that implemented marginal maximum likelihood estimation employed other numerical integration methods (e.g., Newton–Cotes formulas). There are many tables that contain quadrature points and quadrature weights for the Gauss–Hermite quadrature formulas; but these tabled values cannot be directly used when quadrature points and quadrature weights are specified by the user of computer programs because the standard normal distribution is frequently employed in the marginalization of the likelihood. The two purposes of this paper are to present extensive tables of Gauss–Hermite quadrature for the standard normal distribution and to present examples that demonstrate the effects of using various numbers of quadrature points and quadrature weights as well as different quadrature formulas on item parameter estimates. Item parameter estimates obtained from more than 20 quadrature points and quadrature weights with either Gauss–Hermite quadrature or the Newton–Cote method were virtually identical.

Keywords

Gauss–Hermite quadrature Item response theory Marginal maximum likelihood estimation Parameter estimation 

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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Seock-Ho Kim
    • 1
    Email author
  • Yu Bao
    • 1
  • Erin Horan
    • 1
  • Meereem Kim
    • 1
  • Allan S. Cohen
    • 1
  1. 1.Department of Educational PsychologyThe University of GeorgiaAthensUSA

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