Extending bias adjustments for R-squared to multilevel models

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.