Variability explained by covariates in linear mixed-effect models for longitudinal data

Variability explained by covariates or explained variance is a well-known concept in assessing the importance of covariates for dependent outcomes. In this paper we study R-2 statistics of explained variance pertinent to longitudinal data under linear mixed-effect models, where the R-2 statistics are computed at two different levels to measure, respectively, within- and between-subject variabilities explained by the covariates. By deriving the limits of R-2 statistics, we find that the interpretation of explained variance for the existing R-2 statistics is clear only in the case where the covariance matrix of the outcome vector is compound symmetric. Two new R-2 statistics are proposed to address the effect of time-dependent covariate means. In the general case where the outcome covariance matrix is not compound symmetric, we introduce the concept of compound symmetry projection and use it to define level-one and level-two R-2 statistics. Numerical results are provided to support the theoretical findings and demonstrate the performance of the R-2 statistics. The Canadian Journal of Statistics 38: 352-368; 2010 (C) 2010 Statistical Society of Canada