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Re: st: reliability with -icc- and -estat icc-


From   "JVerkuilen (Gmail)" <jvverkuilen@gmail.com>
To   statalist@hsphsun2.harvard.edu
Subject   Re: st: reliability with -icc- and -estat icc-
Date   Tue, 26 Feb 2013 20:55:17 -0500

On Tue, Feb 26, 2013 at 8:47 PM, Nick Cox <njcoxstata@gmail.com> wrote:
> I believe you on the first point.

To elaborate, there are a number of relationships among the ICC and
alpha reliability. One way to estimate alpha is to compute:

     alpha_hat = Q/(Q-1)*(1 - tr(S)/1' S 1)

where S is the sample covariance matrix and Q is the number of items.
tr(.) is the matrix trace.

This is essentially setting up the data wide. With only three
observations I don't think it's possible to compute the covariances.

Another estimator of alpha is given by setting up the data long. It is:

     alpha_hat = Q*ICC_hat/(1 + (Q-1)*ICC_hat)

where ICC_hat is the estimated intraclass correlation. Estimating the
ICC in this problem is reasonable though it's more restrictive than
the first one in some senses in that it is biased downward when the
group variances are different. Under the assumption that the true
covariance matrix is compound symmetric they are the same.
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