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

Chen Wei Liu*, Robert Philip Chalmers

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

4 Citations (Scopus)


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.

Original languageEnglish
Pages (from-to)118-138
Number of pages21
JournalBritish Journal of Mathematical and Statistical Psychology
Issue number1
Publication statusPublished - 2021 Feb


  • EM algorithm
  • cognitive diagnostic models
  • item response theory
  • observed information matrix
  • standard errors

ASJC Scopus subject areas

  • Statistics and Probability
  • Arts and Humanities (miscellaneous)
  • General Psychology


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