Decomposition of the Explained Variation in the Linear Mixed Model

The concept of variation explained is widely used to assess the relevance of factors in the analysis of variance. In the linear model, it is the main contribution to the coefficient of determination which is widely used to assess the proportion of variation explained, to determine model goodness-of-fit and to compare models with different covariables. There has not been an agreement on a similar concept of explained variation for the class of linear mixed models yet. Based on the restricted maximum likelihood equations, we prove a full decomposition of the sum of squares of the dependent variable in the context of the variance components form of the linear mixed model. The decomposition is dimensionless, has an intuitive and simple definition in terms of variance explained, is additive for several random effects and reduces to the decomposition in the linear model. Thereby, we introduce a novel measure for the explained variation which we allocate to contributions of single fixed and random effects (partial explained variations). Partial explained variations are considered to constitute easily interpretable quantities, quantifying relevance of fixed and random effects on a common scale and allowing to rank their importance. Our result lead us to propose a natural extension of the well-known adjusted coefficient of determination to the variance components form of the linear mixed model. We illustrate the usefulness in two public low-dimensional datasets as well as in two high-dimensional datasets in the context of GWAS. The approach is made readily available in the user-friendly $R$-package "varianceExplained".