Measures of explained variance and pooling in multilevel models

Andrew Gelman and Iain Pardoe


Explained variance (R-squared) has been generalized in various ways to multilevel models for hierarchical data structures in which individuals are grouped into units, and there are variables measured on individuals and each grouping unit. The models are based on regression relationships at different levels, with the first level corresponding to the individual data, and subsequent levels corresponding to between-group regressions of individual predictor effects on grouping unit variables. We present an approach to defining R-squared at each level of the multilevel model, rather than attempting to create a single summary measure of fit, by comparing variances within the model. In simple regression, our measure generalizes the classical adjusted R-squared. We also discuss a related variance comparison to summarize the degree to which estimates at each level of the model are pooled together based on the level-specific regression relationship, rather than estimated separately. This pooling factor is related to the concept of shrinkage in simple hierarchical models. We illustrate the methods on a dataset of radon in houses within counties using a series of multilevel models.

Key Words: Adjusted R-squared; Bayesian inference; Hierarchical model; Multilevel regression; Partial pooling; Shrinkage.

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Last updated: October 6, 2004

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© 2004, Iain Pardoe, Lundquist College of Business, University of Oregon

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