Explained Variance in Two-Level Models: A New Approach

While the proportion of explained variance is well-defined in linear models, Snijders and Bosker (1994) 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. We present a solution that allows researchers to decompose variance components from null models into parts explained and unexplained by level-1 predictors. We also offer an extension that incorporates level-2 predictors. Our 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. We illustrate our method with an example analyzing sibling similarities in lifecycle income.