R$^{2}$s for Correlated Data: Phylogenetic Models, LMMs, and GLMMs

Area of Science:

• Ecology and Evolutionary Biology
• Statistical Modeling
• Quantitative Biology

Background:

• Reporting R-squared is crucial for quantifying model-explained variance.
• Standard R-squared definitions are problematic for models with correlated data (e.g., phylogenetic, mixed models) due to covariances.
• Existing methods struggle to define variance explained by predictors or covariances (e.g., phylogeny).

Purpose of the Study:

• To investigate and compare three novel R-squared metrics for correlated data models.
• To address conceptual challenges in defining R-squared for phylogenetic and mixed models.
• To enable comparison of contributions from predictor variables and variance components (phylogeny, random effects).

Main Methods:

• Developed and evaluated three R-squared metrics: R2resid, R2pred, and R2lik.
• R2resid weights residuals by model-estimated variances/covariances.
• R2pred uses model predictions to compute variance between observed and predicted values.
• R2lik is based on model likelihood, reflecting information content.
• Assessed metrics using simulations for phylogenetic generalized least squares and logistic regression, and for linear/generalized linear mixed models.

Main Results:

• R2resid, R2pred, and R2lik demonstrated similar performance in describing variance explained by model components.
• R2pred most directly answers how much variance is explained by the overall model.
• R2resid is suitable for comparing models across different datasets due to sample size independence.
• R2lik is best for assessing the importance of components within a single model and dataset, aligning with significance tests.

Conclusions:

• The three proposed R-squared metrics effectively quantify variance explained in models with correlated data.
• R2pred offers the most intuitive measure of overall model fit.
• R2resid and R2lik provide complementary insights for model comparison and component importance assessment, respectively.