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A Note on N in Bayesian Information Criterion for Item Response Models.

Sun-Joo Cho1, Paul De Boeck2,3

  • 1Vanderbilt University, Nashville, TN, USA.

Applied Psychological Measurement
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PubMed
Summary

This study clarifies the sample size (N) in the Bayesian Information Criterion (BIC) for item response models. Researchers should verify BIC calculations, as software may use incorrect N values for fixed or random item models.

Keywords:
Bayesian information criterioninformation functionitem response theory

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Area of Science:

  • Psychometrics
  • Educational Measurement
  • Statistical Modeling

Background:

  • The Bayesian Information Criterion (BIC) is widely used for model selection in item response theory (IRT).
  • The accurate calculation of the penalty term, specifically the sample size (N), is crucial for valid BIC application.
  • Previous applications of BIC in IRT models may have used inconsistent or incorrect values for N.

Purpose of the Study:

  • To derive the correct sample size (N) component for the BIC penalty term in two-parameter logistic (2PL) item response models.
  • To differentiate the appropriate N for fixed-item versus random-item IRT models.
  • To provide guidance on validating BIC outputs from statistical software.

Main Methods:

  • Theoretical derivation of the N parameter within the BIC formula.
  • Distinction between fixed-item and random-item model structures in the context of sample size calculation.
  • Analysis of the implications for BIC computation in standard IRT software.

Main Results:

  • For fixed-item models, N in the BIC penalty term correctly represents the number of persons.
  • For random-item models, N in the BIC penalty term should be the total number of observations (persons × items).
  • Software implementations may not consistently apply these distinctions.

Conclusions:

  • Researchers must be aware of the specific IRT model (fixed vs. random items) when interpreting or calculating BIC.
  • It is recommended to manually calculate BIC or validate software-generated values to ensure the correct N is used.
  • Incorrect N values can lead to erroneous model selection decisions based on BIC.