Extending bias adjustments for R-squared to multilevel models

In both single-level and multilevel regression analyses, researchers commonly report R-squared to quantify the proportion of outcome variance that is explained by a model or its component parts. It is well-established that the classic estimator of R-squared for single-level regression (via ordinary least squares estimation) is biased upward, and thus an adjusted version of R-squared is commonly reported. In the multilevel modeling literature, however, there is inherently more complexity in defining R-squared, and the possibility of upward bias in estimation has received very little attention, and thus researchers almost never report adjusted versions of measures. The purposes of this article are to (a) provide a pedagogical overview of the logic and computation of adjusted R-squared in single-level regression and show how this same logic can be extended to multilevel models; (b) evaluate the degree to which R-squared measures developed specifically for multilevel contexts are susceptible to upward bias, both analytically via expectation algebra and empirically via simulation; and (c) provide and evaluate a set of adjustments that correct for this bias. We ultimately show that the proposed adjustments do, in fact, yield less bias than the unadjusted versions of measures and discuss factors affecting the discrepancy between adjusted and unadjusted versions, as well as the differential impact such factors have across different measures. We also discuss software with which researchers can compute adjusted measures and illustrate their computation with an empirical example. (PsycInfo Database Record (c) 2026 APA, all rights reserved).