[R-sig-ME] Computing reliability for the least squares estimates of each level1 coefficient across the set of J level-2 units

[Blazej Mrozinski]

Good evening,
Please accept my apology for cross-posting (I've been advised to avoid it)
but I can't find my answer anywhere.
4 days ago I posted my question to stackOverflow (with screenshots of
formulas, that are skipped here): https://stackoverflow.com/
questions/48438755/how-to-compute-reliability-estimates-
from-lmer-lme-results.
<https://stackoverflow.com/questions/48438755/how-to-compute-reliability-estimates-from-lmer-lme-results>

I'm coming from a commercial MLM (HLM7) software and would like to (...drop
it eventually) replicate some numbers in R.

Specifically I'm looking for a function or formula computing the
**reliability for the least squares estimates of each level1 coefficient
across the set of *J* level-2 units**

Below is an example based on the simple `sleepstudy` data. What I'm looking
for is a way to compute reliability values not only in this very example,
but also in situations where there are more level1 variables.

>From HLM7 manual (Raudenbush, Bryk (2002), p.11) a definition of
reliability is given:
Reliability Estimates (overall or average reliability for the least squares
estimates of each level 1 coefficent across the set of J level-2 units)
calculated according to
Equation 3.58 in Hierarchical Linear Models (2nd ed.)

I used the `sleepstudy` data from `lme4` package to compute a random
intercept and slope model with `lme4::lmer`:

    library(lme4)
    m <- lmer(Reaction ~ Days + (Days|Subject), data = sleepstudy)
    summary(m)

And with HLM7 software

Fixed and random effects estimates are pretty similar (differences in
rounding occur), but HLM7 will also provide it's reliability estimates:

     ----------------------------------------------------
      Random level-1 coefficient   Reliability estimate
     ----------------------------------------------------
      INTRCPT1, G0                        0.730
          DAYS, G1                        0.815
     ----------------------------------------------------

And this is something I'd like to be able to get from `lmer()` results.

Is this possible with a built-in formula? Some other package function?
Or maybe someone could help me in extracting appropriate values from lmer
result object and compute it "by hand" ?
Thank you very much!

Kind Regards,
Blazej Mrozinski

	[[alternative HTML version deleted]]