Explained Variance in Two-Level Models: A New Approach

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.