Marginal and Conditional $R^2$ for a Mixed-Effects Model

Computes the Nakagawa and Schielzeth (2013) marginal and conditional coefficients of determination from a fitted mixed-effects model, returned in tidy long form. The marginal $R^2$ is the proportion of total variance explained by the fixed effects alone; the conditional $R^2$ is the proportion explained by the fixed and random effects together. The two quantities are the mixed-effects companions to the intraclass correlation returned by icc_lmer: where the ICC isolates the share of variance attributable to a single grouping factor, the marginal and conditional $R^2$ summarize how much of the outcome variance the fixed and random parts of the model account for.

Usage

R2_mixed_effects(
  model,
  conf_level = 0.95,
  ci_method = c("none", "boot"),
  B = 1000,
  seed = NULL,
  ...
)

Arguments

model: A fitted mixed-effects model. A lmer fit (class lmerMod) is the primary target; an lme fit is also accepted.
conf_level: Confidence level. Default 0.95. Used only when a bootstrap interval is requested (see ci_method).
ci_method: Interval method for the two $R^2$ values. The default "none" returns point estimates only. "boot" adds a parametric bootstrap percentile interval (lmerMod models only, via bootMer); see Details.
B: Number of bootstrap replications when ci_method = "boot". Default 1000.
seed: Optional integer seed for the bootstrap. Default NULL, meaning the caller's current RNG state is used and left unchanged. When supplied, the seed is set inside the function and the caller's RNG state is restored on exit.
...: Currently unused.

Value

A data.frame with rows "R2_marginal" and "R2_conditional" in the value column. When ci_method = "boot", lower- and upper-limit rows for each quantity are appended and the confidence level is carried on the object.

Details

Variance decomposition. Writing $\sigma^2_f$ for the variance of the fixed-effect linear predictor, $\sigma^2_r$ for the variance attributable to the random effects, and $\sigma^2_\varepsilon$ for the residual variance, $$R^2_{\mathrm{marginal}} \;=\; \frac{\sigma^2_f}{\sigma^2_f + \sigma^2_r + \sigma^2_\varepsilon}, \qquad R^2_{\mathrm{conditional}} \;=\; \frac{\sigma^2_f + \sigma^2_r}{\sigma^2_f + \sigma^2_r + \sigma^2_\varepsilon}.$$ The fixed-effect variance is $\sigma^2_f = \mathrm{var}(\mathbf{X}\boldsymbol{\beta})$, the variance of the fitted fixed-effect linear predictor across the observations. The residual variance is $\sigma^2_\varepsilon = \mathrm{sigma}(\mathrm{model})^2$.

Random-effect variance. For a random-intercept model the random-effect variance is the sum of the variance components read off VarCorr. For a model with random slopes the variance contributed by a random-effects term depends on the values of the associated covariates, so the sum of the diagonal variance components is not correct on its own. This function uses the Johnson (2014) extension: for each random-effects term with design matrix $\mathbf{Z}$ and estimated covariance matrix $\boldsymbol{\Sigma}$, its contribution is the mean over the observations of the quadratic form $\mathbf{z}_i^\top \boldsymbol{\Sigma}\, \mathbf{z}_i$, that is, $\tfrac{1}{n}\,\mathrm{tr}(\mathbf{Z}\boldsymbol{\Sigma}\mathbf{Z}^\top)$, and $\sigma^2_r$ is the sum of these contributions across all random-effects terms. For a random-intercept term this reduces to the intercept variance component, so the two paths agree.

Scope. The decomposition here is the one appropriate for a Gaussian (identity-link) linear mixed model, which is what lmer and lme fit. Generalized linear mixed models introduce a distribution-specific variance term and are not handled by this function.

The bootstrap interval. The default ci_method = "none" reports the two point estimates alone, so the bootstrap is what to ask for when the marginal and conditional $R^2$ are to be reported with an interval and the refits it costs are affordable. With ci_method = "boot" the interval comes from a parametric bootstrap (bootMer): each of the B replicates (1000 by default) simulates a new response vector from the fitted model, refits the model, and recomputes the two $R^2$ values. The unit of resampling is therefore a whole simulated data set drawn from the estimated model, not a resampled set of cases. Only the percentile interval is offered: the limits are the empirical quantiles of the B bootstrap values (Efron & Tibshirani, 1993); there is no BCa or bootstrap standard error variant. Replicates whose refit fails are dropped, and the interval is computed from the replications that return a value. The default B = 1000 is adequate for the central quantiles a percentile interval uses; raising it tightens the Monte Carlo error of the reported limits. Bootstrap results vary from run to run; supply seed for reproducibility.

References

Efron, B., & Tibshirani, R. J. (1993). An introduction to the bootstrap. New York, NY: Chapman & Hall/CRC.

Johnson, P. C. D. (2014). Extension of Nakagawa & Schielzeth's $R^2_{GLMM}$ to random slopes models. Methods in Ecology and Evolution, 5(9), 944–946. doi:10.1111/2041-210X.12225

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)

# Marginal R2 is the proportion of variance the fixed effects account for;
# conditional R2 adds what the random effects account for, so the gap
# between the two is what the subject-level terms buy.
R2_mixed_effects(fit)
#>  term           value
#>  R2_marginal    0.279
#>  R2_conditional 0.799
#> 
#> Confidence level: 95%

# A bootstrap interval is available through ci_method = "boot". It refits
# the model once per replication, so it is not run here; the call is
#   R2_mixed_effects(fit, ci_method = "boot", B = 1000, seed = 113)
# where B = 1000 is the default and seed is supplied because the limits
# otherwise move from run to run. Raise B when the Monte Carlo error of
# the reported limits needs to be smaller.

Marginal and Conditional \(R^2\) for a Mixed-Effects Model