Re: Homogeneity of variance test for correlated samples?

On Thu, 14 May 2009 06:28:40 -0700 (PDT), Bruce Weaver
<bweaver@xxxxxxxxxxxx> wrote:

On May 13, 10:26 pm, Rich Ulrich <rich.ulr...@xxxxxxxxxxx> wrote:
On Wed, 13 May 2009 14:24:18 -0700 (PDT), Bruce Weaver



<bwea...@xxxxxxxxxxxx> wrote:
On May 12, 8:16 am, luiz.alberto.s.melo...@xxxxxxxxx wrote:
On 11 maio, 17:49, Bruce Weaver <bwea...@xxxxxxxxxxxx> wrote:

If I'm not mistaken, the common tests for homogeneity of variance
(e.g., Levene's test, Bartlett's test) are intended for use with
independent groups.  What tests are available & recommended for
testing homogeneity of variance with correlated samples?  (Not much
comes up when I Google this.)

Before someone jumps to the wrong conclusion, let me add that I am not
asking because I want to test the homogeneity of variance assumption
before proceeding with a repeated measures ANOVA.  I am asking because
a colleague is actually interested in the variances/SDs.

See Pitman's test:
Pitman EJG. A note on normal correlation. Biometrika 1939;31(1-2):
9-12.
core.ecu.edu/psyc/wuenschk/StatHelp/Pitman.doc

Luiz

Pitman's test is a t-test that compares the variances of two
correlated samples.  Does anyone know of a test of homogeneity of
variance for k correlated samples, with k > 2?

Homogeneity of variances *and*  of correlations
(and you don't have much for repeated measures, if you don't
have both)  is what is called "sphericity", and there is the usual
test by Mauch.

Bruce, you are right and I am wrong. Compound symmetry
is what also requires equal variances. The first hit on Googling
is my own stats-FAQ, where I quoted a nice posting from 1997.


Keep in mind that the test has horrible power characteristics.
One rule of thumb for social sciences was that, yes, you
should worry about non-sphericity if the p-value is less
than 0.50 (fifty, not oh-five).  At the other extreme, for Ns
of 1000s, it is east to reject at 0.05, whether or not it should
matter.

Thanks Rich. But...

Sphericity is homogeneity of variance of all possible paired
differences for the repeated measures factor. Homogeneity of variance
and homogeneity of covariance for the levels of a repeated measures
factor is called "compound symmetry" (CS). As I understand it, CS is
a more general condition. I.e., if CS holds, then sphericity also
holds. But, you can have sphericity without CS. I usually see
Mauchly's test referred to as a test of sphericity, so I'm not
convinced that it tests homogeneity of variance and covariance.

For a nice note on this, see

http://psychologicalstatistics.blogspot.com/2006/05/what-is-all-this-stuff-about.html

That's a very nice note. He includes a number of details that
I have mentioned from time to time. And he shares my
prejudice for using specific contrasts, over using the "corrected"
tests.



Bear in mind too that I do not want to test assumptions before
preceding with repeated measures ANOVA. I asked the question because a
colleague has a situation in which he wishes to test hypotheses about
the variances in different conditions. I'll have to find out a bit
more about how many levels there are to his repeated measures factor.
If there are not too many, perhaps he cam make do with pairwise
contrasts via Pitman's test.


Googling also finds a reference for testing correlated variances,
CP Han in Biometrika, 1968.

--
Rich Ulrich
.

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