[R] How to calculate Intraclass-coefficient in 2-level Linear Mixed-Effects models?

[Christine A.]

Dear R-users,

consider a 2-level linear mixed effects model (LME) with random intercept
AND random slope for level 1 AND 2. Does anybody know how to calculate
Intraclass-coefficient (ICC) for highest (innermost) level 2 ??? In the
literature, I did not find an example for these kind of komplex models.

For 1-level Random-Intercept models it would be easy: 
ICC = variance due to the clustering variable / (variance due to the
clustering variable + variance remaining) = sigma^2_RandomIntercept /
(sigma^2_RIntercept + sigma^2_error)

My data are similar to Pixel-Data from R-package nlme:
=> Please assume an adequate model is the following (although you really do
not need the Random Slope on Level "Side within Dog" here... and an adequate
model is also different concerning fixed effects):
-----------------------------------------------------------------------------------
m1 <- lme(pixel ~ day + I(day^2), data = Pixel, 
                         random = list(Dog = ~ day, Side = ~ day))

Linear mixed-effects model fit by REML
 Data: Pixel 
     AIC    BIC  logLik
  904.27 927.71 -443.13

Random effects:
----------------
 Formula: ~day | Dog
 Structure: General positive-definite, Log-Cholesky parametrization
              StdDev    Corr  
(Intercept) 31.494   (Intr)
day           1.0720   -0.786

 Formula: ~day | Side %in% Dog
 Structure: General positive-definite, Log-Cholesky parametrization
               StdDev         Corr  
(Intercept) 15.09          (Intr)
day          0.0000249        0     
Residual    14.534       

Fixed effects: pixel ~ day 
              Value      Std.Error    DF   t-value   p-value
(Intercept) 1093.22   10.9566    81  99.777    0.0000
day           -0.15       0.4912     81  -0.303    0.7629

 Correlation: 
    (Intr)
day -0.668

----------------------------------------------------------------------------------------------
There are 2 nested grouping factors: "Dog" (=Level 1) and "Side within Dog"
(=Level 2).
A LME could be used to calculate the intraclass correlation coefficient
(ICC) as a measure of concordance of pixel intensity in between left and
right Side . 

Can anybody help me with the correct formula, i.e. do you have to include
variance component estimator for level 1 ("Dog"), too?
Is it something like this:

ICC = (sigma2_RandomSlope-Dog + sigma2_RandomSlope-SidewithinDog) /
(sigma2_RandomSlope-Dog + sigma2_RandomSlope-SidewithinDog + sigma2_error)




Any hints would be appreciated!
Many thanks in advance.

Christine Adrion,
University of Munich



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