Random effects coefficient of determination for mixed and meta-analysis models

Area of Science:

• Statistics
• Biostatistics
• Econometrics

Background:

• Mixed models are statistical tools incorporating random effects to analyze complex data structures.
• Quantifying the influence of random effects is crucial for model selection and interpretation.
• Existing methods for assessing fixed effects do not directly apply to random effects.

Purpose of the Study:

• To introduce and define a novel coefficient: the random effects coefficient of determination.
• To establish this coefficient's utility in assessing the strength and significance of random effects in mixed models.
• To differentiate the proposed coefficient from existing fixed effects coefficients of determination.

Main Methods:

• Development of the random effects coefficient of determination formula.
• Derivation of explicit formulas for specific mixed models: random intercept, growth curve, and meta-analysis.
• Application and illustration of the coefficient using three real-world datasets.

Main Results:

• The random effects coefficient of determination ranges from 0 to 1, indicating the proportion of variance explained by random effects.
• A coefficient close to 0 suggests weak random effects, simplifying the model to standard linear regression.
• A coefficient close to 1 implies substantial random effects, potentially allowing estimation via dummy variables.

Conclusions:

• The random effects coefficient of determination provides a robust measure for evaluating random effects in mixed models.
• This coefficient aids in determining whether random effects significantly contribute to explaining outcome variability.
• The findings support informed model selection, distinguishing between complex mixed models and simpler regression approaches.