TY - JOUR

T1 - A note on computing Louis’ observed information matrix identity for IRT and cognitive diagnostic models

AU - Liu, Chen Wei

AU - Chalmers, Robert Philip

N1 - Funding Information:
The first author acknowledges the grant support from the Taiwan Ministry of Science and Technology, Grant Number MOST 109‐2410‐H‐003‐034 and the National Taiwan Normal University (NTNU), Taiwan.
Publisher Copyright:
© 2020 The British Psychological Society

PY - 2021/2

Y1 - 2021/2

N2 - Using Louis’ formula, it is possible to obtain the observed information matrix and the corresponding large-sample standard error estimates after the expectation–maximization (EM) algorithm has converged. However, Louis’ formula is commonly de-emphasized due to its relatively complex integration representation, particularly when studying latent variable models. This paper provides a holistic overview that demonstrates how Louis’ formula can be applied efficiently to item response theory (IRT) models and other popular latent variable models, such as cognitive diagnostic models (CDMs). After presenting the algebraic components required for Louis’ formula, two real data analyses, with accompanying numerical illustrations, are presented. Next, a Monte Carlo simulation is presented to compare the computational efficiency of Louis’ formula with previously existing methods. Results from these presentations suggest that Louis’ formula should be adopted as a standard method when computing the observed information matrix for IRT models and CDMs fitted with the EM algorithm due to its computational efficiency and flexibility.

AB - Using Louis’ formula, it is possible to obtain the observed information matrix and the corresponding large-sample standard error estimates after the expectation–maximization (EM) algorithm has converged. However, Louis’ formula is commonly de-emphasized due to its relatively complex integration representation, particularly when studying latent variable models. This paper provides a holistic overview that demonstrates how Louis’ formula can be applied efficiently to item response theory (IRT) models and other popular latent variable models, such as cognitive diagnostic models (CDMs). After presenting the algebraic components required for Louis’ formula, two real data analyses, with accompanying numerical illustrations, are presented. Next, a Monte Carlo simulation is presented to compare the computational efficiency of Louis’ formula with previously existing methods. Results from these presentations suggest that Louis’ formula should be adopted as a standard method when computing the observed information matrix for IRT models and CDMs fitted with the EM algorithm due to its computational efficiency and flexibility.

KW - EM algorithm

KW - cognitive diagnostic models

KW - item response theory

KW - observed information matrix

KW - standard errors

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U2 - 10.1111/bmsp.12207

DO - 10.1111/bmsp.12207

M3 - Article

C2 - 32757460

AN - SCOPUS:85087551018

SN - 0007-1102

VL - 74

SP - 118

EP - 138

JO - British Journal of Mathematical and Statistical Psychology

JF - British Journal of Mathematical and Statistical Psychology

IS - 1

ER -