Quantifying explained variance in multilevel models: An integrative framework for defining R-squared measures

Researchers often mention the utility and need for R-squared measures of explained variance for multilevel models (MLMs). Although this topic has been addressed by methodologists, the MLM R-squared literature suffers from several shortcomings: (a) analytic relationships among existing measures have not been established so measures equivalent in the population have been redeveloped 2 or 3 times; (b) a completely full partitioning of variance has not been used to create measures, leading to gaps in the availability of measures to address key substantive questions; (c) a unifying approach to interpreting and choosing among measures has not been provided, leading to researchers' difficulty with implementation; and (d) software has inconsistently and infrequently incorporated available measures. We address these issues with the following contributions. We develop an integrative framework of R-squared measures for MLMs with random intercepts and/or slopes based on a completely full decomposition of variance. We analytically relate 10 existing measures from different disciplines as special cases of 5 measures from our framework. We show how our framework fills gaps by supplying additional total and level-specific measures that answer new substantive research questions. To facilitate interpretation, we provide a novel and integrative graphical representation of all the measures in the framework; we use it to demonstrate limitations of current reporting practices for MLM R-squareds, as well as benefits of considering multiple measures from the framework in juxtaposition. We supply and empirically illustrate an R function, r2MLM, that computes all measures in our framework to help researchers in considering effect size and conveying practical significance. (PsycINFO Database Record (c) 2019 APA, all rights reserved).