Use of the Lagrange Multiplier Test for Assessing Measurement Invariance Under Model Misspecification

Area of Science:

• Psychometrics
• Statistical modeling
• Educational measurement

Background:

• Measurement noninvariance poses challenges in item response theory (IRT) models.
• Model misspecification, including local dependence and nonnormal latent variables, can impact test accuracy.
• Evaluating statistical tests for detecting these issues is crucial for reliable IRT analysis.

Purpose of the Study:

• To assess the Type I error rates and power of four Lagrange multiplier (LM) test variants.
• To investigate test performance under different types of model misspecification in IRT.
• To compare empirical and asymptotic power calculations for these tests.

Main Methods:

• Monte Carlo simulation study to evaluate test performance.
• Analysis of four LM test versions: Hessian, cross-product, generalized LM, and generalized jackknife score tests.
• Examination of misspecification scenarios: local item dependence and nonnormal latent variable distribution.

Main Results:

• All tests performed well under mild misspecification.
• Strong misspecification led to performance deterioration, particularly for false positive rates (local dependence) and power (small samples, latent variable misspecification).
• LM test with Hessian and generalized LM tests showed better false positive control; LM test with cross-product offered higher power in small samples.

Conclusions:

• LM tests are valuable for detecting measurement noninvariance in IRT models.
• Test performance is sensitive to the degree and type of model misspecification.
• Asymptotic power is a viable, time-efficient alternative to empirical power calculations.