For small and moderate samples, a non-significant Mauchly's test does not mean much at all. That is
why many people will recommend, wisely, that followup test be performed as paired t-tests instead of
using some pooled variance term.
What are you measuring? Is it a good measure, with good scaling expected and no outliers observed?
I don't like analyses where those corrections are made, unless I have a decent understanding of why
they are required, such as, the presence of excess zeroes.
Would some transformation be thought of, by anyone? Analyzing with unnecessarily-unequal variances
is a way to get into unneeded trouble. If the "levels" represent time, it might be appropriate and proper
to test a much more powerful hypothesis that makes use of contrasts (linear for growth, etc.) in order to
overcome the inevitable decline in correlations across time.
You say: more levels than subjects -- Is this because you have very small N or because you have moderate N
but also have too many levels to test a sensible hypothesis across them all?
State your hypotheses. What tests them? A single-d.f. test is what gives best power, whenever one of those
can be used. I favor constructing contrasts -- sometimes in the form of separate variables -- over tests that
include multiple d.f. and multiple hypotheses, all at once. And I would rather remove the causes of
heterogeneity (variances or correlations) beforehand, than have to hope that I have suitably corrected for it.
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