Confidence Intervals for Multilevel R-Squared

Load Packages

library(lme4)
## Loading required package: Matrix
library(MuMIn)  # for computing multilevel R-squared
library(r2mlm)  # another package for R-squared
## Loading required package: nlme
## 
## Attaching package: 'nlme'
## The following object is masked from 'package:lme4':
## 
##     lmList
## Registered S3 method overwritten by 'parameters':
##   method                         from      
##   format.parameters_distribution datawizard
library(bootmlm)  # for multilevel bootstrapping
library(boot)  # for bootstrap CIs

An Example Multilevel Model

fm1 <- lmer(Reaction ~ Days + (Days | Subject), sleepstudy)

Nakagawa-Johnson-Schielzeth \(R^2\)

r.squaredGLMM(fm1)
## Warning: 'r.squaredGLMM' now calculates a revised statistic. See the help page.
##            R2m       R2c
## [1,] 0.2786511 0.7992199

The marginal \(R^2\) considers the total variance accounted for due to the fixed effect associated with the predictors (Days in this example). See Nakagawa, Johnson, & Schielzeth (2017) for more information.

Right-Sterba \(R^2\)

More fine-grained partitioning, as described in Rights & Sterba (2019)

r2mlm(fm1)

## $Decompositions
##                      total
## fixed           0.27851304
## slope variation 0.08915267
## mean variation  0.43165365
## sigma2          0.20068063
## 
## $R2s
##          total
## f   0.27851304
## v   0.08915267
## m   0.43165365
## fv  0.36766572
## fvm 0.79931937

The fixed part is the same as the marginal \(R^2\).

Confidence Intervals for \(R^2\)

Neither MuMIn::r.squaredGLMM() nor r2mlm::r2mlm() provided confidence intervals (CIs) for the \(R^2\), but general guidelines for effect size reporting would suggest always reporting CIs for point estimates of effect size, just like for any point estimates in statistics. We can use multilevel bootstrapping to get CIs.

To do bootstrap, first defines an R function that gives the target \(R^2\) statistics. We can do it for the marginal \(R^2\):

marginal_r2 <- function(object) {
  r.squaredGLMM(object)[[1]]
}
marginal_r2(fm1)
## [1] 0.2786511

Parametric Bootstrap

The lme4::bootMer() supports basic parametric multilevel bootstrapping

# This takes about 30 sec on my computer
boo01 <- bootMer(fm1, FUN = marginal_r2, nsim = 999)
boo01
## 
## PARAMETRIC BOOTSTRAP
## 
## 
## Call:
## bootMer(x = fm1, FUN = marginal_r2, nsim = 999)
## 
## 
## Bootstrap Statistics :
##      original      bias    std. error
## t1* 0.2786511 0.005759548  0.07545455
## 
## 19 message(s): boundary (singular) fit: see help('isSingular')
## 9 warning(s): Model failed to converge with max|grad| = 0.00203576 (tol = 0.002, component 1) (and others)

Here is the bootstrap distribution

plot(boo01)

Bias-corrected estimate

The above shows that the sample estimate of \(R^2\) was upwardly biased. To correct for the bias, we can use the bootstrap bias-corrected estimate

2 * boo01$t0 - mean(boo01$t)
## [1] 0.2728915

Confidence intervals

You can get three types of bootstrap CIs ("norm", "basic", "perc") with bootMer:

boot::boot.ci(boo01, index = 1, type = c("norm", "basic", "perc"))
## BOOTSTRAP CONFIDENCE INTERVAL CALCULATIONS
## Based on 999 bootstrap replicates
## 
## CALL : 
## boot::boot.ci(boot.out = boo01, type = c("norm", "basic", "perc"), 
##     index = 1)
## 
## Intervals : 
## Level      Normal              Basic              Percentile     
## 95%   ( 0.1250,  0.4208 )   ( 0.1088,  0.4114 )   ( 0.1460,  0.4485 )  
## Calculations and Intervals on Original Scale

Residual Bootstrap

The bootmlm::bootstrap_mer() implements the residual bootstrap, which is robust to non-normality.

# This takes about 30 sec on my computer
boo02 <- bootstrap_mer(fm1, FUN = marginal_r2, nsim = 999, type = "residual")
## boundary (singular) fit: see help('isSingular')
## boundary (singular) fit: see help('isSingular')
## boundary (singular) fit: see help('isSingular')
## boundary (singular) fit: see help('isSingular')
## boundary (singular) fit: see help('isSingular')
## boundary (singular) fit: see help('isSingular')
## boundary (singular) fit: see help('isSingular')
## boundary (singular) fit: see help('isSingular')
## boundary (singular) fit: see help('isSingular')
## boundary (singular) fit: see help('isSingular')
## boundary (singular) fit: see help('isSingular')
## boundary (singular) fit: see help('isSingular')
## boundary (singular) fit: see help('isSingular')
## boundary (singular) fit: see help('isSingular')
## boundary (singular) fit: see help('isSingular')
## boundary (singular) fit: see help('isSingular')
## boundary (singular) fit: see help('isSingular')
## boundary (singular) fit: see help('isSingular')
boo02
## 
## ORDINARY NONPARAMETRIC BOOTSTRAP
## 
## 
## Call:
## bootstrap_mer(x = fm1, FUN = marginal_r2, nsim = 999, type = "residual")
## 
## 
## Bootstrap Statistics :
##      original      bias    std. error
## t1* 0.2786511 0.005397786  0.08418536

In this example, the results are similar. The boostrap bias-corrected estimate of \(R^2\), and the three basic CIs, can similarly be computed as in parametric bootstrap.

Confidence Intervals

In addition to the three CIs previously discussed, which are first-order accurate, we can also obtain CIs that are second-order accurate: (a) bias-corrected and accelerated (BCa) CI and (b) studentized CI (also called the bootstrap-\(t\) CI). For (a), it requires the influence value of the \(R^2\) function, whereas for (b), it requires an estimate of the sampling variance of the \(R^2\) estimate.

Influence value

# Based on the group jackknife
inf_val <- bootmlm::empinf_mer(fm1, marginal_r2, index = 1)

Sampling variance with the numerical delta method

This part is a bit more technical; skip this if you’re not interested in the studentized CI.

To obtain an approximate sampling variance of the \(R^2\), it would be easier to use the r2mlm::r2mlm_manual() function to compute \(R^2\). We first write a function that computes \(R^2\) using input of the fixed and random effects:

manual_r2 <- function(theta, data,
                      # The following are model-specific
                      wc = 2, bc = NULL, rc = 2,
                      cmc = FALSE) {
  n_wc <- length(wc)
  n_bc <- length(bc) + 1
  dim_rc <- length(rc) + 1
  n_rc <- dim_rc * (dim_rc + 1) / 2
  gam_w <- theta[seq_len(n_wc)]
  gam_b <- theta[n_wc + seq_len(n_bc)]
  tau <- matrix(NA, nrow = dim_rc, ncol = dim_rc)
  tau[lower.tri(tau, diag = TRUE)] <- theta[n_wc + n_bc + seq_len(n_rc)]
  tau2 <- t(tau)
  tau2[lower.tri(tau2)] <- tau[lower.tri(tau)]
  s2 <- tail(theta, n = 1)
  r2mlm_manual(data,
               within_covs = wc,
               between_covs = bc,
               random_covs = 2,
               gamma_w = gam_w,
               gamma_b = gam_b,
               Tau = tau2,
               sigma2 = s2,
               clustermeancentered = cmc,
               bargraph = FALSE)$R2s[1, 1]
}
theta_fm1 <- c(fixef(fm1)[2], fixef(fm1)[1],
               VarCorr(fm1)[[1]][c(1, 2, 4)], sigma(fm1)^2)
manual_r2(theta_fm1, data = fm1@frame)
## [1] 0.278513

Now computes the numerical gradient

grad_fm1 <- numDeriv::grad(manual_r2, x = theta_fm1, data = fm1@frame)

We also need the asymptotic covariance matrix of the fixed and random effects

vcov_fixed <- vcov(fm1)
vcov_random <- vcov_vc(fm1, sd_cor = FALSE, print_names = FALSE)
vcov_fm1 <- bdiag(vcov_fixed, vcov_random)
# Need to re-arrange the first two columns
vcov_fm1 <- vcov_fm1[c(2, 1, 3:6), c(2, 1, 3:6)]

Now apply the multivariate delta method

crossprod(grad_fm1, vcov_fm1) %*% grad_fm1
## 1 x 1 Matrix of class "dgeMatrix"
##             [,1]
## [1,] 0.005919918

We now need a function that computes both \(R^2\) and the asymptotic sampling variance of it.

marginal_r2_with_var <- function(object,
                                 wc = 2, bc = NULL, rc = 2) {
  dim_rc <- length(rc) + 1
  vc_mat <- matrix(seq_len(dim_rc^2), nrow = dim_rc, ncol = dim_rc)
  vc_index <- vc_mat[lower.tri(vc_mat, diag = TRUE)]
  theta_obj <- c(fixef(object)[wc], fixef(object)[c(1, bc)],
                 VarCorr(object)[[1]][vc_index], sigma(object)^2)
  r2 <- manual_r2(theta_obj, data = object@frame)
  grad_obj <- numDeriv::grad(manual_r2, x = theta_obj, data = object@frame)
  # Need to re-arrange the order of the fixed effects
  names_wc <- names(object@frame)[wc]
  names_bc <- c("(Intercept)", names(object@frame)[bc])
  vcov_fixed <- vcov(object)[c(names_wc, names_bc), c(names_wc, names_bc)]
  vcov_random <- vcov_vc(object, sd_cor = FALSE, print_names = FALSE)
  vcov_obj <- bdiag(vcov_fixed, vcov_random)
  v_r2 <- crossprod(grad_obj, vcov_obj) %*% grad_obj
  c(r2, as.numeric(v_r2))
}
marginal_r2_with_var(fm1)
## [1] 0.278513044 0.005919918

Five Bootstrap CIs

Now, we can do bootstrap again, using the new function that computes both the estimate and the asymptotic sampling variance

# This takes quite a bit longer due to the need to compute variances
boo03 <- bootstrap_mer(fm1, FUN = marginal_r2_with_var, nsim = 999,
                       type = "residual")
## boundary (singular) fit: see help('isSingular')
## Warning in checkConv(attr(opt, "derivs"), opt$par, ctrl = control$checkConv, :
## Model failed to converge with max|grad| = 0.00828807 (tol = 0.002, component 1)
## boundary (singular) fit: see help('isSingular')
## boundary (singular) fit: see help('isSingular')
## Warning in checkConv(attr(opt, "derivs"), opt$par, ctrl = control$checkConv, :
## Model failed to converge with max|grad| = 0.00967386 (tol = 0.002, component 1)
## boundary (singular) fit: see help('isSingular')
## boundary (singular) fit: see help('isSingular')
## Warning in checkConv(attr(opt, "derivs"), opt$par, ctrl = control$checkConv, :
## Model failed to converge with max|grad| = 0.00453875 (tol = 0.002, component 1)
## boundary (singular) fit: see help('isSingular')
## Warning in checkConv(attr(opt, "derivs"), opt$par, ctrl = control$checkConv, :
## Model failed to converge with max|grad| = 0.00280723 (tol = 0.002, component 1)
## boundary (singular) fit: see help('isSingular')
## Warning in checkConv(attr(opt, "derivs"), opt$par, ctrl = control$checkConv, :
## Model failed to converge with max|grad| = 0.00764723 (tol = 0.002, component 1)
## Warning in checkConv(attr(opt, "derivs"), opt$par, ctrl = control$checkConv, :
## Model failed to converge with max|grad| = 0.0024541 (tol = 0.002, component 1)
## boundary (singular) fit: see help('isSingular')
## boundary (singular) fit: see help('isSingular')
## boundary (singular) fit: see help('isSingular')
## boundary (singular) fit: see help('isSingular')
## boundary (singular) fit: see help('isSingular')
## boundary (singular) fit: see help('isSingular')
## Warning in checkConv(attr(opt, "derivs"), opt$par, ctrl = control$checkConv, :
## Model failed to converge with max|grad| = 0.0049756 (tol = 0.002, component 1)
## Warning in checkConv(attr(opt, "derivs"), opt$par, ctrl = control$checkConv, :
## Model failed to converge with max|grad| = 0.00228953 (tol = 0.002, component 1)
## boundary (singular) fit: see help('isSingular')
## Warning in sqrt(diag(m)): NaNs produced
## Warning in sqrt(diag(m)): NaNs produced

## Warning in sqrt(diag(m)): NaNs produced

## Warning in sqrt(diag(m)): NaNs produced

## Warning in sqrt(diag(m)): NaNs produced

## Warning in sqrt(diag(m)): NaNs produced

## Warning in sqrt(diag(m)): NaNs produced

## Warning in sqrt(diag(m)): NaNs produced

## Warning in sqrt(diag(m)): NaNs produced

## Warning in sqrt(diag(m)): NaNs produced

## Warning in sqrt(diag(m)): NaNs produced

## Warning in sqrt(diag(m)): NaNs produced
## boundary (singular) fit: see help('isSingular')
## boundary (singular) fit: see help('isSingular')
boo03
## 
## ORDINARY NONPARAMETRIC BOOTSTRAP
## 
## 
## Call:
## bootstrap_mer(x = fm1, FUN = marginal_r2_with_var, nsim = 999, 
##     type = "residual")
## 
## 
## Bootstrap Statistics :
##        original        bias    std. error
## t1* 0.278513044  0.0100127766 0.083704256
## t2* 0.005919918 -0.0002646922 0.001150678

And then obtain five types of CIs

boot::boot.ci(boo03, L = inf_val)
## BOOTSTRAP CONFIDENCE INTERVAL CALCULATIONS
## Based on 998 bootstrap replicates
## 
## CALL : 
## boot::boot.ci(boot.out = boo03, L = inf_val)
## 
## Intervals : 
## Level      Normal              Basic             Studentized     
## 95%   ( 0.1044,  0.4326 )   ( 0.0890,  0.4067 )   ( 0.0822,  0.4385 )  
## 
## Level     Percentile            BCa          
## 95%   ( 0.1503,  0.4680 )   ( 0.1408,  0.4466 )  
## Calculations and Intervals on Original Scale

Bootstrap CI With Transformation

Given that \(R^2\) is bounded, it may be more accurate to first transform the \(R^2\) estimates to an unbounded scale, obtain the CIs on the transformed scale, and then back transform it to between 0 and 1. This can be done in boot::boot.ci() as well with the logistic transformation:

boot::boot.ci(boo03, L = inf_val, h = qlogis,
              # Need the derivative of the transformation
              hdot = function(x) 1 / (x - x^2),
              hinv = plogis)
## BOOTSTRAP CONFIDENCE INTERVAL CALCULATIONS
## Based on 998 bootstrap replicates
## 
## CALL : 
## boot::boot.ci(boot.out = boo03, L = inf_val, h = qlogis, hdot = function(x) 1/(x - 
##     x^2), hinv = plogis)
## 
## Intervals : 
## Level      Normal              Basic             Studentized     
## 95%   ( 0.1440,  0.4629 )   ( 0.1449,  0.4572 )   ( 0.1184,  0.4187 )  
## 
## Level     Percentile            BCa          
## 95%   ( 0.1503,  0.4680 )   ( 0.1408,  0.4466 )  
## Calculations on Transformed Scale;  Intervals on Original Scale

Note that the transformation only affects the Normal, Basic, and Studentized CIs.

Conclusion

This post demonstrates how to use multilevel bootstrapping to obtain CIs for \(R^2\). The post only focuses on marginal \(R^2\), but CIs for other \(R^2\) measures can be similarly obtained. The studentized CI is the most complex as it requires obtaining the sampling variance of \(R^2\) for each bootstrap sample. So far, to my knowledge, there has not been studies on which CI(s) perform best, so simulation studies are needed.

Further readings on multilevel bootstrap:

Hok Chio (Mark) Lai 黎學昭
Hok Chio (Mark) Lai 黎學昭
Assistant Professor of Psychology (Quantitative Methods)

My research interests include statistics, multilevel and latent variable models, and psychometrics.

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