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st: intraclass correlation of slope/coeff ??

From   "Jacki Buros" <>
Subject   st: intraclass correlation of slope/coeff ??
Date   Tue, 29 May 2007 17:33:15 -0400

Hi ..

I have what may be considered a novice question about the ICC but I'm
nonetheless stumped. Hopefully you can help.

I am working with a 3-level hierarchical linear model with the
following levels: patient, artery, and time. Observations are nested
so that each artery within each patient has several observations over
a period of time. My model has a random intercept and a random
coefficient (time) at both patient and artery levels. Time is my only
fixed effect.

My goal is to determine the intra-class correlation coefficient (ICC) for
the change over time at the patient level. In other words, to address
the question: "to what extent does the rate of change cluster within a
patient across arteries?"

I have set this up in Stata using -xtmixed- as:

. xtmixed measurement time || patient: time || artery: time


. xtmixed measurement time || patient: time, cov(uns) || artery: time, cov(uns)

Using the latter of the two commands to consider the possibility of
covariance, I get (as hypothesized for an earlier analysis)
significantly non-zero covariances at both the artery and patient
levels. Going forward with this model, I have the following variance
estimates with which to estimate the desired ICC:

patient: coeff_time, _cons, and cov(timeXcons)
artery: coeff_time, _cons, and cov(timeXcons)
time: residual

Question is, which calculation would best represent the ICC of the
slope within a patient across arteries?

  (1) ICC = Var(coeff_time)patient /

  (2) ICC  = Var(coeff_time)patient / [sum of all other variance and
covariance values]

  (3) ICC  = Var(coeff_time)patient / [sum of all other variance but
not covariance values]

  (4) ICC  = Var(coeff_time)patient / [Var(coeff_time)artery +
Cov(timeXcons)artery + Var(coeff_time)patient + Cov(timeXcons)patient]

My naive sense is that (1) represents the "adjusted" ICC of the slope
-- adjusted for covariance between slope and intercept, and that (4)
represents the "unadjusted" ICC, but I'm not sure. I also have the
problem that my covariance at one of my levels is negative, so maybe I
should use its absolute value? There must be a more formal way to go
about estimating this ..

Any thoughts (and/or references) would be much appreciated ... I have
found only a few references that discuss the analysis of group-level
effects and\or group-level means, and would appreciate any insight you
might have to offer.

Thanks very much for your time.

Jacki Buros
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