Coefficients of Determination for Mixed-Effects Models

The coefficient of determination is well defined for linear models and its extension is long wanted for mixed-effects models in agricultural, biological, and ecological research. We revisit its extension to define measures for proportions of variation explained by the whole model, fixed effects only, and random effects only. We propose to calculate unexplained variations conditional on individual random and/or fixed effects so as to keep individual heterogeneity brought by available predictors. While naturally defined for linear mixed models, these measures can be defined for a generalized linear mixed model using a distance measured along its variance function, accounting for its heteroscedasticity. We demonstrate the promising performance and utility of our proposed methods via simulation studies as well as applications to real data sets in agricultural and ecological studies.