How to extract metafor::rma.mv() equivalent of lme4::VarCorr() for multilevel mixed effect meta-regression with random slopes

I am attempting to calculate pseudo R squared's for a multilevel mixed-effect meta-regression that includes random slopes in the metafor package (i.e., rma.mv() object) using a similar approach to the following citation:

https://besjournals.onlinelibrary.wiley.com/doi/10.1111/2041-210X.12225

Digging into the code of this citation that uses the lme4/nlme packages to demonstrate how to perform the calculations, it seems like a metafor equivalent of a VarCorr() output of a lme4/nlme fit model is needed.

I am struggling to figure out how to obtain this summary of the variances (or SD) at each level of the random effects for a rma.mv() object. Any insight here is appreciated!

This is the code from the citation used to calculate the pseudo R squared's:

# Fit Model
orangemod.rs <- 
    lmer(circumference ~ ageYears + (ageYears | Tree), data = Orange)


# First we need the design matrix (X), the number of observations (n)
# and the fixed effect estimates (Beta)

  X <- model.matrix(orangemod.rs)
  n <- nrow(X)
  Beta <- fixef(orangemod.rs)

# First the fixed effects variance (eqn 27 of Nakagawa & Schielzeth): 

  Sf <- var(X %*% Beta)

# Second, the list of covariance matrices of the random effects.
# Here the list has only a single element because there is only
# one level of random effects.

  (Sigma.list <- VarCorr(orangemod.rs))

# Use equation 11 in the paper to estimate the random effects variance 
# component.
# Using sapply ensures that the same code will work when there are multiple
# random effects (i.e. where length(Sigma.list) > 1)

  Sl <- 
    sum(
      sapply(Sigma.list,
        function(Sigma)
        {
          Z <-X[,rownames(Sigma)]
          sum(diag(Z %*% Sigma %*% t(Z)))/n
        }))

# As this model is an LMM, the additive dispersion variance, Se, is
# equivalent to the residual variance. The residual standard deviation
# is stored as an attribute of Sigma.list:
  
  Se <- attr(Sigma.list, "sc")^2

# Finally, the distribution-specific variance, Sd, is zero for LMMs:

  Sd <- 0

# Use eqns 29 and 30 from Nakagawa & Schielzeth to estimate marginal and
# conditional R-squared:

  total.var <- Sf + Sl + Se + Sd
  (Rsq.m <- Sf / total.var) 
  (Rsq.c <- (Sf + Sl) / total.var) 

All of these steps can be easily reproduced for the rma.mv() object besides the step with VarCorr() as this function doesn't accept rma.mv() objects:

# Second, the list of covariance matrices of the random effects.
# Here the list has only a single element because there is only
# one level of random effects.

  (Sigma.list <- VarCorr(orangemod.rs))

# Groups   Name        Std.Dev. Corr  
# Tree     (Intercept)  1.8966        
#          ageYears     8.7757  -1.000
# Residual             10.0062      

Here is a simple model that illustrates the point and highlights two of the components of the rma.mv() model that I think might have the necessary information but I'm not exactly sure how to get it into the final VarCorr() format.

library(metafor)
library(tidyverse)

data("dat.konstantopoulos2011")
df=dat.konstantopoulos2011%>%
  as.data.frame()

#fit random slope model
m1=rma.mv(yi ~ year, vi, random = list(~year|study, ~year|district, ~1|school), data = df,
          cvvc = TRUE)

lme4::VarCorr(m1)
nlme::VarCorr(m1)

#potentially useful components
m1$V
m1$vvc


What I'm looking to recreate

# Groups   Name        Std.Dev. Corr  
# study    (Intercept)  [VALUE]        
#          ageYears     [VALUE]  [VALUE]
# district (Intercept)
#          ageYears              [VALUE]
# school   (Intercept)
# Residual             [VALUE]      

The variance-covariance matrix of the random effects corresponding to ~year|study can be extracted with m1$G. If you only need the variances, then m1$tau2 will do. The var-cov matrix for ~year|district is m1$H and the variances are in m1$gamma2. The variance corresponding to ~1|school is m1$sigma2.

Note that if you want random slopes, then the model you are fitting doesn't provide that. You would need to use:

rma.mv(yi ~ year, vi, 
       random = list(~year|study, ~year|district, ~1|school),
       struct=c("GEN","GEN"), data = df)

See: https://wviechtb.github.io/metafor/reference/rma.mv.html#specifying-random-effects