The proportion of explained variance is well defined in linear models, but Snijders and Bosker demonstrated that this concept is ill defined in linear multilevel models. Whenever a researcher adds a level 1 predictor to the model, the level 2 variance may increase because the level 2 variance also depends on the level 1 variance. This problem is more pronounced when there are few observations per cluster. The authors present a solution that allows researchers to decompose variance components from null models into parts explained and unexplained by level 1 predictors. The authors also offer an extension that incorporates level 2 predictors. This approach is based on multivariate multilevel modeling and provides a complete decomposition of the gross (or null model) variance components. The approach is also implemented in the user-written Stata program twolevelr2, and the online supplement contains worked code for implementation in R. The authors illustrate this method with an example analyzing sibling similarities in lifetime income.
Holm et al. (Fri,) studied this question.