Explained Variation in Mixed Models

Mixed models are statistical frameworks that accommodate hierarchical or clustered data by modelling both fixed effects, which represent average relationships across all observations, and random effects, which capture variability specific to groups or subjects. Explained variation in this context refers to the proportion of total variance in the response variable that can be attributed to these model components. In simple linear models, the coefficient of determination (R²) provides a clear measure of goodness of fit, but when random effects and correlated residuals are present, the decomposition of variance becomes more intricate. Contemporary practice distinguishes between marginal R², quantifying variance explained by fixed effects alone, and conditional R², encompassing both fixed and random effects. Estimation methods have evolved from variance component decomposition and likelihood-based comparisons to simulation-based approximations for non-Gaussian data. Recent theoretical developments propose single-step formulae that avoid fitting additional null models and incorporate bias corrections for small samples. These advances, implemented in user-friendly software, enable researchers across ecology, epidemiology, social sciences and beyond to assess model performance, compare nested structures and understand the relative contribution of predictors to observed heterogeneity.

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Research from all publishers

Several contributions have advanced the quantification of explained variation in mixed models. One seminal study extended the R² measure originally formulated for generalised linear mixed models to accommodate random slopes, thereby allowing users to quantify the variance explained by both fixed effects and complex random-effect structures. A further development introduced an adjusted coefficient of determination for generalised linear mixed models that accounts for correlation among observations and applies a bias correction without requiring a separate null model, enhancing computational efficiency and interpretability. In addition, game-theoretic techniques have been applied to decompose total explained variance among multiple predictors via Shapley values, providing confidence intervals for the attribution of variance components and offering a transparent means to compare the relative importance of covariates across alternative model specifications.

Explained Variation in Mixed Models publication trend

The graph below shows the total number of articles in explained variation in mixed models across all publications each year (not limited to Nature Index journals).

Technical terms

Mixed model: A statistical model combining fixed effects (common to all observations) and random effects (group-specific deviations).

Fixed effects: Model parameters representing systematic relationships assumed to be constant across clusters.

Random effects: Model parameters capturing variability between clusters or subjects beyond the fixed structure.

Coefficient of determination (R²): A metric indicating the proportion of total variance explained by a model.

Marginal R²: The proportion of variance explained by fixed effects alone in a mixed model.

Conditional R²: The proportion of variance explained by both fixed and random effects together.

Generalised linear mixed model (GLMM): An extension of mixed models allowing for non-normal response distributions via link functions.

Shapley value decomposition: A game-theoretic method for fairly attributing portions of explained variance to individual predictors.

References

1. Extension of Nakagawa & Schielzeth's R2GLMM to random slopes models. Methods in Ecology and Evolution (2014).
2. An adjusted coefficient of determination (R2) for generalized linear mixed models in one go. Biometrical Journal (2023).
3. Shapley Value Confidence Intervals for Attributing Variance Explained. Frontiers in Applied Mathematics and Statistics (2020).