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. Translational Abstract R-squared measures are useful indications of effect size that are ubiquitously reported for single-level regression models. For multilevel models (MLMs), wherein observations are nested within clusters (e.g., students nested within schools), researchers likewise often mention the utility and necessity of R-squared measures; however, they find it difficult to relate, interpret, and choose among alternative existing measures, especially in the context of random slopes. Though methodologists have addressed this topic, the MLM R-squared literature suffers from several shortcomings: (a) the relationships among existing measures have not been established, leading to certain measures being redeveloped multiple times; (b) previous sets of measures have not considered all of the different ways that variance can be explained in MLMs, 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; and (d) existing software rarely incorporates available measures. In this article, we develop an integrative framework of R-squared measures for MLMs with random intercepts and/or slopes that addresses each of these shortcomings. We show that 10 existing measures are special cases of those from our framework, and show how our framework fills gaps by also supplying novel total and level-specific measures that answer important research questions. To facilitate interpretation, we introduce a unified graphical representation of all of the measures in our framework. We also demonstrate limitations of current R-squared reporting practices, and explain how considering our full framework overcomes these. We supply and illustrate new software that computes all of our measures to help researchers in considering effect size in MLM applications.