There is still a question as to what would be considered a large value for these ratios. I ran reliability for one of the datasets and the max/min variance ratio yields 5.0. The ratio for correlations is -6.5, but it seems you're saying that the presence of any negative correlation or covariance is already evidence of sphericity violation.
I don't know if this an widely used approach to sphericity, but at least it's a starting point to get a better idea about the structure of the data. Hopefully others will chime in if there are more rigorous or established methods available.
________________________________
From: SPSSX(r) Discussion [
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[hidden email]]
Sent: Thursday, October 06, 2016 11:43 AM
To:
[hidden email]Subject: Re: Undefined Mauchly's Test
I don't know of any specific procedure(s) for testing sphericity when
there are more variables than subjects but I would suggest using the
RELIABILITY procedure to get descriptive statistics on the two
components of sphericity:
(1) What is the ratio of largest variance to the smallest variance.
If this number is large, it provides evidence that sphericity may
not be present (i.e., heterogeneity of variance). I know that
compound symmetry requires all variances to be the same but
sphericity does not (the correlation and variance/SD are involved).
(2) What is the ratio of the largest correlation to the smallest
correlation. Again if the number is large, or there are negative
correlations, this would be evidence for lack of sphericity.
One could do significance testing between the largest and smallest
variances and/or the correlated correlations to determine whether
they are "significantly" different but that will probably depend upon
the number of subjects/cases you have.
If you can get a sorted covariance matrix graphic, it could also
help in seeing whether there are patterns in covariance patterns
(e.g., banding) but SPSS does not provide this though one could
probably write a macro to do it..
I would think that the presence of any negative covariance would
imply the absence of sphericity.
If others know of more appropriate tests or procedure, I to would
like to know. There may be better general alternatives but the
appropriateness for any actual dataset will depend upon the
characteristics of that dataset.
-Mike Palij
New York University
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----- Original Message -----
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Sent: Thursday, October 06, 2016 2:20 PM
Subject: Undefined Mauchly's Test
Given the paucity of information online, I was wondering if anyone knows the procedural approach to the evaluation of sphericity when Mauchly's test is undefined, which is the case when the number of repeated levels is larger than the number of subjects (insufficient df). I am not sure if sphericity can still be assumed based on the reported values of epsilon larger than 0.75, whether based on Greenhouse-Geisser or Huynh-Feldt. In one particular dataset, epsilon is less than 0.1. Presumably it can be assumed that sphericity is violated when epsilon is that low.
I am aware of using mixed models to overcome the assumptions of sphericity. My concern is with GLM in this case.
Citations would be welcome.
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