[R-sig-ME] Question about conditional R-squared values (and its relationship to the intra-class correlation)

[Joshua Rosenberg]

Hi all, I know that R-squared values for mixed effects models are an area
of active development / some controversy. I'm wondering if anyone could
help me to understand how *conditional R-squared values*, as described by
Nakagawa and Schielzeth (2013) and as implemented in the MuMIn (and
piecewiseSEM) packages. In particular, my (very naive) thought was that the
square root of the conditional r-squared value (my understanding: the
r-squared value for both together the fixed and random effects) minus the
marginal r-squared value (the r-squared value for only the fixed effects)
would / could equal the ICC. Please excuse me if this is completely belying
a very limited understanding.

For example, for this example using the sleepstudy data, the conditional
minus the marginal, or what I thought would represent something akin to the
proportion of variance explained only by the random effects, or, in this
example, .424. The intra-class correlation for the random intercept is
.483. Can anyone help me clear about how these two values can (or could /
whether they should) be related?

library(lme4)
library(MuMIn)
library(sjstats)

m1 <- lmer(Reaction ~ Days + (1 | Subject), sleepstudy)

r_squared_vals <- MuMIn::r.squaredGLMM(m1)
# Just the conditional r-squared value (conditional - marginal)
r_squared_vals[2] - r_squared_vals[1] # .424

ICC_val <- sjstats::icc(fm1)
# The intra-class correlation (ICC) for the
ICC_val # 0.483


I ask in part because I'm interested in calculating the partial r-squared
values using the r2glmm
<https://www.google.com/search?q=r2glmm&oq=r2glmm&aqs=chrome.0.69i59j69i60l3.1051j0j7&sourceid=chrome&ie=UTF-8>
package
(and the r2beta() function) and am curious if there can be some similar
proportion of variance explained interpretation for the random effects.
Again, I'm sorry if this is not clear, obvious, or not a direction worth
pursuing fo well-understood reasons.

Thanks for considering!
Josh

-- 
Joshua Rosenberg, Ph.D. Candidate
Educational Psychology
&
 Educational Technology
Michigan State University
http://jmichaelrosenberg.com

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