On the coefficient of determination for mixed regression models

Abstract

For mixed regression models, we define a variance decomposition including three terms, explained individual variance, unexplained individual variance and noise variance. In contrast to traditional variance decomposition, it is thus the unexplained, not the explained, variance that is split. It gives rise to a coefficient of individual determination (CID) defined as the estimated fraction of explained individual variance. We argue that in many applications CID is a valuable complement to R 2 , since it excludes noise variance (which can never be explained) and thus has one as a natural upper bound. A general theory for coefficients determination is presented, including various choices of regression models, weight functions and parameter estimates. In particular we focus on models where CID is computable, such as univariate mixed Poisson and logistic regression models, as well as multivariate mixed linear regression models. Large sample properties and confidence intervals are derived and finally, the theory is exemplified using Poisson regression on a Swedish motor traffic insurance data set.