Posted by
Mike on
Oct 14, 2016; 6:17pm
URL: http://spssx-discussion.165.s1.nabble.com/Undefined-Mauchly-s-Test-tp5733212p5733258.html
On Thursday, October 13, 2016 6:56 PM, Emil Rudobeck
>Mike, thank you for your explanations and references. Even
>without a widely used procedure, I think the methods you
>mentioned are rather thorough for finding out if sphericity is
>violated and I can use them whenever Mauchly’s test is unavailable
>(or in some cases to complement or substitute for Mauchly’s test).
You're welcome. In the situation where N < P, I would suggest
checking the current statistical literature whenever you can because
it appears to me to be an active area of development, especially
in the context of "big data" (e.g., when there are many more measures
than cases).
>I am familiar with the matrices in your link since I use linear
>mixed models (LMM). As long as we’re on the topic, I wanted
>to clarify something: I had read a while ago that repeated
>measures ANOVA assumes the CS structure and it’s one
>of its weaknesses as compared to LMM, which has flexible
>covariance structures. With a better understanding of sphericity,
>I’m curious as to how ANOVA could use CS where in fact
>sphericity (a special case of CS) is all that’s required to meet
>the conditions of the test. Maybe I have misunderstood something.
First, let me point out that many statistics textbooks, especially
in psychology, are terrible at citing sources for the points they
make in their text. I think this is one reason there has been the
"unholy amalgamation of Fisherian and Neyman-Pearson approaches"
that Gerd Gigerenzer has complained about. Textbook authors
often do not appear to understand how the Fisherian framework
and the Neyman-Pearson framework differ nor how acrimonious
the exchanges became between Fisher and Neyman over time
(at one point Fisher said something compared Neyman's
approach to a "communist plot" in statistics -- Fisher went over
the edge and/or held some "unreasonable" beliefs even though
he was genius in other areas). Getting to the point, it is unclear
to me when repeated measures ANOVA as we know it was
first presented (one could bet that Fisher did so in one of the
editions of his book on Methods for Research Methods but I
think a better bet would be one of the editions of Snedecor &
Cochran's "Statistical Methods" -- Snedecor was the one who
converted what Fisher called his "z-test" into what we now refer
to as the "F test" [he provided the first F tables which by-passed
the need to do calculations with logarithms in Fisher's z-test).
so, it is unclear how the assumption of compound symmetry
was asserted as necessary for the repeated measures
ANOVA. I don't have the reference for Box circa 1950s that
showed that if sphericity was violated, his correction to the
degrees of freedom could be used to determine the significance
of the F-test and would become the basis for further correctiosn
by Huynh-Feldt and Greenhouse and Geisser. Given that
Sphericity can be obtained without compound symmetry,
the emphasis on sphericity and downplaying compound
symmetry is understandable.
Second, the concern with compound symmetry may not have
originated with ANOVA. but in the field of psychometrics and
the measurement model being used for the data. I think the
following reference is relevant to this point:
Wilks, S. S. Sample Criteria for Testing Equality of Means,
Equality of Variances, and Equality of Covariances in a Normal
Multivariate Distribution. Ann. Math. Statist. 17 (1946), no. 3,
257--281.
NOTE: Available at:
http://projecteuclid.org/euclid.aoms/1177730940The paper is concerned with developing likelihood tests that
test the following hypotheses:
(1) All means are equal: H(m) tested by L(m)
(2) All variances are equal: H(v) tested by L(v)
(3) All covariances are equal: H(cv) text by L(cv)
The working example that is used are three variables that are
assumed to follow a "parallel" measurement model which
assumes all means are equal, all variances are equal, and
all covariances are equal. An omnibus test that test all
of these conditions, that is, L(m,v,cv) is presented as well
as likelihood tests for specific components, say, whether
all variances are equal and all covariances are equal, that
is, L(v,cv). Note that if the L(v,cv) is nonsignificant, one
has compound symmetry and one can validly do a one-way
repeated measures ANOVA (or as Wilks puts it "analysis of
variance test for a k by n layout where k is the number of
measures and n is the number of cases).-- see section 1.5
in Wilk's paper for some really ugly math in support of the
point that L(m) is equivalent to one-way repeated measures
ANOVA. I think that Wilks is working on extending the
traditional assumption of independent groups ANOVA,
namely, homogeneity of variance, and avoiding the problem
that in the two group situation was known as the Fisher-Behrens
problem, that is, heterogeneous variances (which wasn't yet
solved but Welch, Brown-Forsyth, and others would provide
solutions). One drawback of using L(m) is that it requires
a large sample (see page 265).
In his section 1.7, Wilks compares the test L(v,cv) with Mauchly's
test for sphericity of a normal multivariate. The difference between
the two test is that Mauchley's test was designed to test the hypothesis
that all variances are equal and all covariances are equal to ZERO,
regardless of whether the pop means differ or not. Wilks designates
Mauchly's test as the likelihood L(s) which uses the sample standard
deviation and variances -- Wilks points out that the test actually uses
the sqrt[L(s)] but the two are equivalent. After some realy ugly math,
Wilks concludes this section with the following:
|Stated in other words,
|
| Mauchly's criterion L(s) is a test for the hypothesis that contours
|of equal probability density in the multivariate normal population
|distribution are spheres, while L(v,cv) is a test for the hypothesis
that
|the contours of equal prob'l,bility are k-dimensional ellipsoids
|with k - 1 equal axes in general shorter than the k-th axis vhich is
|equally inclined to the k coordinate axes of the distribution function.
What this translates into is unclear but if one meditates on it like
a Zen koan, I'm sure that enlightenment will eventually come. ;-)
Wilks works through an example for his test and provide additional
derivations. With respect to ANOVA, Wilks work in this article
focuses on the measurement model for the data one is collecting.
Clearly, he shows that the assumed model has an impact on the
tests he is presenting but does not really connect it to ANOVA
outside of point out how L(v,cv) differs from Mauchly's test, or,
how compound symmetry differs from sphericity. Sphericity
is a looser criterion to meet, focusing primarily on equality of
variances, the traditional assumption made for ANOVA.
It seems to me that most researchers don't think about the
measurement model for their data and, thus, don't care
whether their data meet the requirements for compound
symmetry or sphericity. I believe that there probably has
been more work in this area since 1948 but I don't know
what that is outside of the various modifications to the
degrees of freedom to adjust the F-tests for violations
of sphericity.
I'll stop my comments on this point since I have gone on far too
long but, hopefully, with some benefit. HTH.
-Mike Palij
New York University
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