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

From: Christine A. <adrion_at_ibe.med.uni-muenchen.de>
Date: Thu, 31 Jan 2008 05:48:16 -0800 (PST)

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


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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