R-squared Measures for Multilevel Models with Three or More Levels

Area of Science:

• Statistics
• Quantitative Psychology

Background:

• Multilevel models (MLMs) with three or more levels are increasingly used.
• Existing R-squared measures for these complex models are limited.
• Understanding explained variance and predictor centering effects is crucial.

Purpose of the Study:

• To extend existing R-squared frameworks for MLMs to handle three or more levels.
• To clarify analytic relationships between total and level-specific R-squared measures.
• To explicate the impact of predictor centering on R-squared computation and interpretation.

Main Methods:

• Extension of the Rights and Sterba two-level MLM R-squared framework to higher levels.
• Mathematical and pedagogical clarification of total and level-specific R-squared relationships.
• Demonstration of R-squared computation under various predictor centering strategies.

Main Results:

• A general set of R-squared measures for MLMs with three or more levels is proposed.
• Preexisting three-level measures are shown to be special cases of the new framework.
• The framework accommodates all predictor centering strategies for R-squared calculation.

Conclusions:

• The developed R-squared measures provide a comprehensive approach to quantifying explained variance in complex MLMs.
• The study offers clarity on the interplay between R-squared, model levels, and centering strategies.
• Associated software (r2mlm R package) is provided for practical application and visualization.