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The effective sample size in Bayesian information criterion for level-specific fixed and random-effect selection in a

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Summary
This summary is machine-generated.

New Bayesian Information Criterion (BIC) formulas are derived for multi-level models, addressing discrepancies in existing methods. These enhanced BIC criteria offer superior model selection for complex hierarchical data structures.

Keywords:
Bayesian information criterionlevel‐specific fixed effectslinear mixed modelsmixed modelmodel selectionmulti‐level modelrandom effect

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Area of Science:

  • Statistics
  • Multilevel Modeling
  • Econometrics

Background:

  • Existing Bayesian Information Criterion (BIC) implementations in statistical software for multi-level models are inconsistent.
  • Discrepancies arise from varied sample size specifications in the BIC penalty term for multi-level models.
  • Uncertainty exists regarding the correct application of BIC for selecting models with level-specific fixed and random effects.

Purpose of the Study:

  • To derive accurate BIC penalty terms for selecting fixed and random effects in two-level nested multi-level models.
  • To propose new BIC versions, denoted as BIC_A and BIC_B, addressing full rank and redundant random effects, respectively.
  • To evaluate the performance of the new BIC criteria against existing methods for multi-level model selection.

Main Methods:

  • Derivation of BIC penalty terms for level-specific fixed and random effects in a two-level nested design.
  • Decomposition of the penalty term into cluster-level and parameter-level components for BIC_A.
  • Derivation of BIC_B for scenarios with redundant random effects.
  • Numerical and simulation studies to validate the derived formulae and compare performance.

Main Results:

  • Formulas for BIC_A and BIC_B were derived and validated against empirical values.
  • BIC_A decomposes the penalty into average sample size per cluster and total parameters times the number of clusters.
  • Simulation studies demonstrated that the new BIC criteria (BIC_A or BIC_B) outperform standard BIC versions using total sample size or number of clusters.
  • The proposed BIC criteria performed at least as well as existing methods across various multi-level conditions.

Conclusions:

  • The derived BIC formulae (BIC_A and BIC_B) provide a more accurate and reliable method for multi-level model selection.
  • The new BIC criteria are recommended as a superior global selection criterion for complex hierarchical data.
  • The practical application of the new BIC is illustrated using a textbook example dataset.