R-Squared Measures for Mixed-Effects Models

Computes the Rights and Sterba (2019) integrative framework of R-squared measures for a fitted two-level linear mixed-effects (multilevel) model. The model implied outcome variance is fully decomposed into five sources: variance due to level-1 predictors via fixed slopes (\(f_1\)), level-2 predictors via fixed slopes (\(f_2\)), predictors via random slope (co)variation (\(v\)), cluster-specific outcome means via random intercept variation (\(m\)), and level-1 residuals (\(\sigma^2\)). Proportions of the total, within-cluster, and between-cluster outcome variance attributable to combinations of these sources give the family of R-squared measures.

Usage

R2_mixed_effects_decomposition(model)

Arguments

model: A fitted two-level model of class merMod (from lme4, e.g.\ lmer) or lme (from nlme). The model must use numeric predictors (only the cluster variable may be a factor) and must not contain I() terms; create any transformed predictors as their own columns first.

Value

A data.frame (dmar_tbl) with columns term and value. When the level-1 predictors are cluster-mean-centered, the full set of 12 measures is returned, named total_f1, total_f2, total_v, total_m, total_f, total_fv, total_fvm, within_f1, within_v, within_fv, between_f2, and between_m; otherwise the five total-variance measures total_f, total_v, total_m, total_fv, and total_fvm are returned (the within/between split requires cluster-mean-centering). The returned object carries the source-by-target variance decomposition in attr(x, "decomposition").

Details

The companion R2_mixed_effects returns the Nakagawa and Schielzeth marginal and conditional R-squared; this function contains those two as special cases (total_f and total_fvm) within the fuller source decomposition.

The measure superscripts index the variance sources in the numerator and the subscripts index the outcome variance in the denominator: total_* use the total outcome variance, within_* the within-cluster variance (\(f_1 + v + \sigma^2\)), and between_* the between-cluster variance (\(f_2 + m\)). total_fvm is the omnibus measure (all explained sources over the total variance) and, for a random-intercept model, coincides with the Nakagawa and Schielzeth conditional R-squared computed by R2_mixed_effects; total_f coincides with their marginal R-squared. See Rights and Sterba (2019, Table 1) for the definitions.

The measures were derived under the assumption that the fitted model uses cluster-mean-centering of the level-1 predictors (with the cluster means entered as level-2 predictors). When that centering is not detected, only the total-variance measures are returned, matching the reference implementation.

Fitting the model requires lme4 (for a merMod fit) or nlme (for an lme fit) to be installed.

References

Rights, J. D., & Sterba, S. K. (2019). Quantifying explained variance in multilevel models: An integrative framework for defining R-squared measures. Psychological Methods, 24(3), 309–338. doi:10.1037/met0000184

Nakagawa, S., & Schielzeth, H. (2013). A general and simple method for obtaining \(R^2\) from generalized linear mixed-effects models. Methods in Ecology and Evolution, 4(2), 133–142. doi:10.1111/j.2041-210x.2012.00261.x

Author

Ken Kelley kkelley@nd.edu

Examples

fit <- lme4::lmer(Reaction ~ Days + (Days | Subject),
                  data = lme4::sleepstudy)
R2_mixed_effects_decomposition(fit)
#>  term      value 
#>  total_f   0.279 
#>  total_v   0.0892
#>  total_m   0.432 
#>  total_fv  0.368 
#>  total_fvm 0.799