Measuring explained variation in linear mixed effects models

We generalize the well-known R(2) measure for linear regression to linear mixed effects models. Our work was motivated by a cluster-randomized study conducted by the Eastern Cooperative Oncology Group, to compare two different versions of informed consent document. We quantify the variation in the response that is explained by the covariates under the linear mixed model, and study three types of measures to estimate such quantities. The first type of measures make direct use of the estimated variances; the second type of measures use residual sums of squares in analogy to the linear regression; the third type of measures are based on the Kullback-Leibler information gain. All the measures can be easily obtained from software programs that fit linear mixed models. We study the performance of the measures through Monte Carlo simulations, and illustrate the usefulness of the measures on data sets.