A Framework of R-Squared Measures for Single-Level and Multilevel Regression Mixture Models

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Abstract

Psychologists commonly apply regression mixture models in single-level (i.e., unclustered) and multilevel (i.e., clustered) data analysis contexts. Though researchers applying nonmixture regression models typically report R-squared measures of explained variance, there has been no general treatment of R-squared measures for single-level and multilevel regression mixtures. Consequently, it is common for researchers to summarize results of a fitted regression mixture by simply reporting class-specific regression coefficients and their associated p values, rather than considering measures of effect size. In this article, we fill this gap by providing an integrative framework of R-squared measures for single-level regression mixture models and multilevel regression mixture models (with classes at Level-2 or both levels). Specifically, we describe 11 R-squared measures that are distinguished based on what the researcher chooses to consider as outcome variance and what sources the researcher chooses to contribute to predicted variance. We relate these measures analytically and through simulated graphical illustrations. Further, we demonstrate how these R-squared measures can be decomposed in novel ways into substantively meaningful sources of explained variance. We describe and demonstrate new software tools to allow researchers to compute these R-squared measures and decompositions in practice. Using 2 empirical examples, we show how researchers can answer distinct substantive questions with each measure and can gain insights by interpreting the set of measures in juxtaposition to each other.

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