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Re: st: tests of sphericity

From   Joseph Coveney <[email protected]>
To   Statalist <[email protected]>
Subject   Re: st: tests of sphericity
Date   Thu, 02 Oct 2003 19:13:38 +0900

David Airey wrote in response to a posting by Ricardo Ovaldia:


> My understanding is that the epsilon corrects for the
> sphericity in the data but is not necessary to use it
> if the data meets the sphericity assumption (i.e.
> passes Mauchly's test). In which case we can use the
> uncorrected F test. I do not like adjusting the df
> unless it is necessary.
> Thank you,
> Ricardo.

I think you are right and that nobody has written a program for
Mauchley's test of sphericity. The test is available in other
statistical packages. In searching a little I did not find a readable
account on what Mauchly's test actually does or how I might program it
myself. Do you know the original reference?


The original reference is

J. W. Mauchley, Significance test of sphericity of a normal n-variate
distribution.  Annals of Mathematical Statistics 11:204-9, 1940.

The formula for Mauchly's chi-square test statistic and an ad hoc SAS IML
macro for a particular dataset are provided in N. H. Timm and T. A.
Mieczkowski, _Univariate & Multivariate General Linear Models:  Theory and
Applications using SAS Software_ (Cary, North Carolina:  SAS Institute,
1997), pp. 110-15.  This source gives a "locally best invariant" test for
sphericity that is supposed to be more powerful than Mauchly's for small
samples.  It also describes a test for the multisample circularity

All of the tests use matrix manipulations, but nothing more involved than
what Stata can do readily (determinants, traces, transposes,
multiplications).  Nearly everything that I've ever read about the matter,
however, advises against Mauchly's test as a precondition.

I have a follow-up question in this vein:  if the assumptions for a
statistical test are plausable (are reasonably expected to be satisfied on
the basis of prior knowledge or on the basis of information independent of
the data under evaluation), especially for assumptions that a test is
sensitive to, does the test statistic that results from a statistical test
that is performed in one way conditional upon passing such a pretest (and
performed in another way conditional upon failing) follow the distribution
of the test statistic that is found in the back of the textbooks?  In other
words, given that I don't have any particular reason to suspect that the
assumption of circularity would be violated for a dataset, if I commit to
use epsilon-based degree-of-freedom adjustments if the p-value associated
with Mauchly's chi-square is 0.05 (or some other threshold) or less, and
commit not to use such adjustments if the Mauchly test gives a p-value
greater than 0.05, then will the subsequent repeated-measures ANOVA's
resulting F-statistic follow an F distribution?  It doesn't have to be
Mauchly/ANOVA; it could be Levene (Bartlett, Brown-Forsythe)/Student t test
with / without Welch (Satterthwaite) adjustment, and so on for other
analogous pretest/test pairs where there is a test statistic that is in some
sense conditional on the outcome of a pretest on the same data.

Joseph Coveney

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