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  3. Extending Bias Adjustments For R-squared To Multilevel Models.
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Extending bias adjustments for R-squared to multilevel models.

Yingchi Guo1, Jason D Rights1

  • 1Department of Psychology, University of British Columbia.

Psychological Methods
|May 18, 2026

View abstract on PubMed

Summary
This summary is machine-generated.

Researchers often use R-squared to explain variance in regression models. This study introduces adjusted R-squared for multilevel models to correct upward bias, improving model accuracy.

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Area of Science:

  • Statistics
  • Multilevel Modeling
  • Regression Analysis

Background:

  • R-squared is widely used to assess model fit by quantifying explained variance.
  • Classic R-squared estimators in single-level regression are known to be upwardly biased.
  • Adjusted R-squared is common in single-level models but rarely used in multilevel contexts due to complexity and lack of attention to bias.

Purpose of the Study:

  • To provide a pedagogical overview of adjusted R-squared logic and its extension to multilevel models.
  • To evaluate the susceptibility of multilevel R-squared measures to upward bias using analytical and simulation methods.
  • To propose and assess adjustments for correcting bias in multilevel R-squared measures.

Main Methods:

  • Extension of single-level adjusted R-squared logic to multilevel regression.
  • Analytical evaluation using expectation algebra to assess bias in R-squared measures.
  • Empirical evaluation through simulation studies to quantify bias and the effectiveness of adjustments.

    • Main Results:

    • Proposed adjustments effectively reduce upward bias in multilevel R-squared measures compared to unadjusted versions.
    • Identified factors influencing the discrepancy between adjusted and unadjusted R-squared values.
    • Demonstrated the practical computation of adjusted measures with software and an empirical example.

    Conclusions:

    • Adjusted R-squared measures offer a less biased estimation of explained variance in multilevel models.
    • Understanding factors affecting bias is crucial for accurate model interpretation.
    • The proposed adjustments and computational guidance facilitate improved statistical practice in multilevel research.