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Permutation randomization methods for testing measurement equivalence and detecting differential item functioning in

Terrence D Jorgensen1, Benjamin A Kite2, Po-Yi Chen2

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Researchers often use chi-squared (χ2) tests and alternative fit indices (AFIs) to assess measurement invariance. This study shows permutation tests offer a more reliable solution for testing invariance across groups, controlling Type I errors effectively.

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Area of Science:

  • Psychometrics
  • Multigroup Factor Analysis
  • Statistical Modeling

Background:

  • Traditional methods for testing measurement invariance in multigroup factor analysis rely on nonsignificant chi-squared (χ2) difference tests or fixed cutoffs for alternative fit indices (AFIs).
  • Large sample sizes can lead to inflated Type I error rates with traditional methods, detecting negligible misspecifications.
  • Existing cutoffs for AFIs have demonstrated inconsistent Type I error rates, questioning their reliability in invariance testing.

Purpose of the Study:

  • To evaluate the Type I error rates of traditional χ2 and AFI cutoff methods for assessing measurement invariance across groups.
  • To propose and validate permutation tests as a more robust alternative for testing measurement invariance.
  • To compare the performance of permutation tests against traditional methods under various conditions of model misspecification and sample size.

Main Methods:

  • Simulations were conducted to compare the Type I error rates of permutation tests, Δχ2 tests, and ΔAFIs.
  • Permutation tests were implemented by randomly shuffling group assignments to generate an empirical null distribution for fit measures.
  • The study examined permutation tests for configural, metric, and scalar invariance, including permutation of the maximum modification index for multiple indicator testing.

Main Results:

  • Permutation tests demonstrated superior control of Type I error rates compared to Δχ2 and AFIs, particularly when models had negligible misspecifications but true invariance.
  • For metric and scalar invariance, Δχ2 and permutation tests showed comparable power and Type I error rates.
  • ΔAFIs exhibited inflated Type I error rates in smaller samples, while permutation tests controlling for multiple indicators effectively managed familywise error rates.

Conclusions:

  • Permutation tests provide a more reliable and consistent approach to testing measurement invariance across groups than traditional χ2 and AFI cutoff methods.
  • The proposed permutation test methodology offers better control over Type I errors, especially in large samples or with minor model misspecifications.
  • Researchers are encouraged to adopt permutation tests for more accurate assessment of measurement invariance in multigroup factor analysis.