Coefficients of Determination for Mixed-Effects Models
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In consistency with the law of total variance, the coefficient of determination, also known as R^2, is well-defined for linear regression models to measure the proportion of variation in a variable explained by a set of predictors. Following the same law, we will show that it is natural to extend such a measure for linear mixed models. However, the heteroscedasticity of a generalized linear model challenges further extension. By measuring the change along the variance function responding to different means, we propose to define proper coefficients of determination for generalized linear mixed models, measuring the proportion of variation in the dependent variable modeled by fixed effects, random effects, or both. As in the case of generalized linear models, our measures can be calculated for general quasi-models with mixed effects, which are only modeled with known link and variance functions. When Gaussian models are considered, they reduce to those measures defined for linear mixed models on the basis of the law of total variance.
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