using lmer() to estimate intraclass correlation coefficients (ICCs)

Dear all

I'm using lmer() to obtain variance component estimates for the estimation of 
intraclass correlation coefficients (ICCs). However, I'm struggling with the 
specification of a proper model.

I'd like to illustrate the problem with an example. There are n = 30 subjects. 
For each subject, k = 3 ratings are done with each of two different methods, 
say A and B. With method B, the mean of the ratings per subject are the same, 
but the variance of the ratings per subject is much lower. An artificial data 
set reflecting this could look as follows:

n <- 30
k <- 3
dat <- data.frame(subject = factor(rep(1:n, k * 2)),
                  method = factor(rep(c("A", "B"), each = n * k)))
set.seed(123)
ratingsA <- rnorm(k * n)
ratingsB <- rep(apply(matrix(ratingsA, ncol = n), 1, mean), 3) + rnorm(k * n, 
sd = 0.1)
dat$rating <- c(ratingsA, ratingsB)

A dot plot of the ratings:

library(lattice)
dotplot(rating ~ method | subject, dat)

Now, if I want to get an estimate of the ICC for method A, I could fit a 
random effects model for that part of the data set. The ICC is defined as the 
ratio of the subject-to-subject variance to the total variance (I assume there 
are no rater effects):

library(lme4)
summary(modA <- lmer(rating ~ (1 | subject), dat, subset = method == "A"))
(ICCA <- 0.00000 / (0.00000 + 0.79631))

This is what I expected since the ratings were drawn randomly from a standard 
normal distribution. Similarly, for method B, the random effects model would 
look like this:

summary(modB <- lmer(rating ~ (1 | subject), dat, subset = method == "B"))
(ICCB <- 0.0218857 / (0.0218857 + 0.0097336))

The ICC is about 0.7, i.e. there is an appreciable intraclass correlation.

My question is: If I want to accommodate for the fact that the 2 * 3 = 6 
ratings per subject are not independent, how could I use lmer() to specifiy a 
model for the full data set in order to obtain separate variance components 
for both the subject-to-subject variance and the residual variance for each of 
the two methods (so that I can get estimates for the ICCs)?

Any hints are appreciated.

Best wishes
Thomas Zumbrunn

Thomas Zumbrunn
SCC/CTU, University Hospital Basel
Schanzenstr. 55, CH-4031 Basel
Tel +41 (0)61 556 52 92 (Mo-Mi)
Fax +41 (0)61 265 94 10