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

Abstract

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.

Investigating Confidence Intervals of Item Parameters When Some Item Parameters Take Priors in the 2PL and 3PL Models

“…To formulate the sandwich estimator, Yuan et al combined the cross-product method in the IRT literature and the Louis (1982) information form. For the Louis (1982) method, a refined computational approach was proposed by Liu and Chalmers (2021). Their approach greatly enhanced the computational efficiency of the Louis procedure, demonstrating the refined approach's potential across IRT and other latent variable models.…”

Uncertainty modelling and computation in Bayesian operational modal analysis

“…The identity also performs well with real-world data from various areas, such as public health (Ibrahim, 1990;Hogan and Laird, 1997;Ibrahim et al, 1999), ecology (Hughes, 1997;Booth and Hobert, 1999), and social statistics (Glas, 1999;Monroe, 2019;Liu and Chalmers, 2021).…”