Questions about ANOVAs and Sphericity

[rohanp16]

Hello everyone !! :)

I had a few questions about ANOVAs and sphericity. I am conducting a 2 way ANOVA with one factor (A) having two levels and the other factor (B) having 21 levels.

Q1) In the 'Within Subjects effects' Table I see there is a main effect of factor B. One should then go to 'planned comparisons' to see where the effect lies right ? However SPSS throws out a 'multivariate tests' output just below the 'planned comparisons' table. What does this table mean ? Below the table it says

"Each F tests the multivariate effect of Factor B. These tests are based on the linearly independent pairwise comparisons among the estimated marginal means"

What does this table mean ? I'm 90% sure there is no need to look at this table, but REALLY want to know what it means.

Q2) I have a significant interaction effect (A*B) and I used the following syntax to see where the difference lied,

  /EMMEANS=TABLES(A*B) COMPARE(B) ADJ(BONFERRONI)
  /EMMEANS=TABLES(B*A) COMPARE(A) ADJ(BONFERRONI)

The output that SPSS throws out, I should first look at the 'Multivariate tests' table and only if that is significant do I see which direction the differences lie in the 'Planned comparisons' table right ?

Q3) While conducting the Mauchly's Test of Sphericity, I realize one requires more than 2 levels of a variable for sphericity to be an issue. So for factor A I get a value of 1 for Mauchly's W which means sphericity is not an issue (and a . in the 'significance' column). However for my other factor, I get a 0.000 and no significance value (i.e - once again there is . in the significance column). Does that mean sphericity is an issue or is not an issue ? If it was an issue, why didn't it just spit out some value <0.05 in the significance column?

Thanks a lot, take care :)

[Swank, Paul R]

ANOVA with repeated measures has an additional assumption in addition to the typical ones of independence, normality, and homogeneity of variance. It is referred to as sphericity. It means that the variance-covariance matrix must assume a certain patter for the usual F tests to be appropriate. Another approach to repeated measures is to treat it as a multivariate model, typically with some type of transform on the dvs. That is the multivariate test. So one decides whether to use the univariate (ANOVA) result or the multivariate result. It should be pointed out however, that 1) there are adjustments that can be made for the univariate tests if sphericity is a problem and 2) although the multivariate test does not rely on a sphericity assumption, it does have an assumption of multivariate normality that the univariate test does not have.

Dr. Paul R. Swank,
Professor and Director of Research
Children's Learning Institute
University of Texas Health Science Center-Houston


-----Original Message-----
From: SPSSX(r) Discussion [mailto:[hidden email]] On Behalf Of rohanp16
Sent: Tuesday, April 12, 2011 3:17 AM
To: [hidden email]
Subject: Questions about ANOVAs and Sphericity

Hello everyone !! :)

I had a few questions about ANOVAs and sphericity. I am conducting a 2 way
ANOVA with one factor (A) having two levels and the other factor (B) having
21 levels.

Q1) In the 'Within Subjects effects' Table I see there is a main effect of
factor B. One should then go to 'planned comparisons' to see where the
effect lies right ? However SPSS throws out a 'multivariate tests' output
just below the 'planned comparisons' table. What does this table mean ?
Below the table it says

"Each F tests the multivariate effect of Factor B. These tests are based on
the linearly independent pairwise comparisons among the estimated marginal
means"

What does this table mean ? I'm 90% sure there is no need to look at this
table, but REALLY want to know what it means.

Q2) I have a significant interaction effect (A*B) and I used the following
syntax to see where the difference lied,

  /EMMEANS=TABLES(A*B) COMPARE(B) ADJ(BONFERRONI)
  /EMMEANS=TABLES(B*A) COMPARE(A) ADJ(BONFERRONI)

The output that SPSS throws out, I should first look at the 'Multivariate
tests' table and only if that is significant do I see which direction the
differences lie in the 'Planned comparisons' table right ?

Q3) While conducting the Mauchly's Test of Sphericity, I realize one
requires more than 2 levels of a variable for sphericity to be an issue. So
for factor A I get a value of 1 for Mauchly's W which means sphericity is
not an issue (and a . in the 'significance' column). However for my other
factor, I get a 0.000 and no significance value (i.e - once again there is .
in the significance column). Does that mean sphericity is an issue or is not
an issue ? If it was an issue, why didn't it just spit out some value <0.05
in the significance column?

Thanks a lot, take care :)

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[Rich Ulrich]

>
> Q3) While conducting the Mauchly's Test of Sphericity, I realize one
> requires more than 2 levels of a variable for sphericity to be an issue. So
> for factor A I get a value of 1 for Mauchly's W which means sphericity is
> not an issue (and a . in the 'significance' column). However for my other
> factor, I get a 0.000 and no significance value (i.e - once again there is .
> in the significance column). Does that mean sphericity is an issue or is not
> an issue ? If it was an issue, why didn't it just spit out some value <0.05
> in the significance column?


I would guess that "sphericity is an issue."  The only reason that
occurs to me, for why you don't see a tiny p-value for it, is that
(perhaps) it was non-computable owing to a zero value for some variance
or a 1.0  for some correlation.

Sphericity implies that all of the covariances are similar.
If the 21 levels of B represent time or space, I would expect that
contiguous levels have higher correlations, which is one of the most
common violations.

Further, Mauchley's test is not particularly powerful.  Followup tests
can be biassed even when Mauchley's gives a p-value of .3 or .4.

A followup style that is frequently recommended for repeated measures
is a set of paired t-tests, so that each test uses its appropriate error.
If I had your data, I might check to compare those t-tests with the
similar tests from the requested contrasts... I don't expect them to
be the same.  Differences in the tests would illustrate the severity
of the problem for these data in particular.

Especially if the levels represent time, space, or any other specific
ordering, I would plot the means in order to examine and explain what
is going on.

--
Rich Ulrich




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[Bruce Weaver]

Here is a nice note on sphericity that you may find helpful.

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

HTH.

rohanp16 wrote

Hello everyone !! :)

I had a few questions about ANOVAs and sphericity. I am conducting a 2 way ANOVA with one factor (A) having two levels and the other factor (B) having 21 levels.

Q1) In the 'Within Subjects effects' Table I see there is a main effect of factor B. One should then go to 'planned comparisons' to see where the effect lies right ? However SPSS throws out a 'multivariate tests' output just below the 'planned comparisons' table. What does this table mean ? Below the table it says

"Each F tests the multivariate effect of Factor B. These tests are based on the linearly independent pairwise comparisons among the estimated marginal means"

What does this table mean ? I'm 90% sure there is no need to look at this table, but REALLY want to know what it means.

Q2) I have a significant interaction effect (A*B) and I used the following syntax to see where the difference lied,

  /EMMEANS=TABLES(A*B) COMPARE(B) ADJ(BONFERRONI)
  /EMMEANS=TABLES(B*A) COMPARE(A) ADJ(BONFERRONI)

The output that SPSS throws out, I should first look at the 'Multivariate tests' table and only if that is significant do I see which direction the differences lie in the 'Planned comparisons' table right ?

Q3) While conducting the Mauchly's Test of Sphericity, I realize one requires more than 2 levels of a variable for sphericity to be an issue. So for factor A I get a value of 1 for Mauchly's W which means sphericity is not an issue (and a . in the 'significance' column). However for my other factor, I get a 0.000 and no significance value (i.e - once again there is . in the significance column). Does that mean sphericity is an issue or is not an issue ? If it was an issue, why didn't it just spit out some value <0.05 in the significance column?

Thanks a lot, take care :)

[rohanp16]

1) The 'Multivariate Tests' table doesn't compute results for my Factor B (having 21 levels) and it says

"Cannot produce multivariate test statistics because of insufficient residual degrees of freedom"

I'm guessing this occurs because of the fact that I have more levels of that factor than participants ?

Could this insufficient residual degrees of freedom be the reason Mauchly's Test of Sphericity is unable to provide a result for this factor B ?

2) The code for the syntax I printed to follow up interaction effects, gives out a 'Multivariate Tests' table rather than a Univariate one, how can this be changed ?

Thanks a lot, much appreacted Rich, Paul and Bruce

[ANDRES ALBERTO BURGA LEON]

Hello:


When you run a post-hoc test in ANOVA and are interested in a one-tailed test  it posible to use the reported sig. as sig./2  as a p value for a one tailed test?

Kindly

Andrés

Mg. Andrés Burga León
Coordinador de Análisis e Informática
Unidad de Medición de la Calidad Educativa
Ministerio de Educación del Perú
Calle El Comercio s/n (espalda del Museo de la Nación)
Lima 41
Perú
Teléfono 615-5840

[Swank, Paul R]

As long as the difference is in the predicted directon.

 

Dr. Paul R. Swank,

Professor and Director of Research

Children's Learning Institute

University of Texas Health Science Center-Houston

 

From: SPSSX(r) Discussion [mailto:[hidden email]] On Behalf Of ANDRES ALBERTO BURGA LEON
Sent: Friday, April 15, 2011 4:27 PM
To: [hidden email]
Subject: One tailed post-hoc test in ANOVA

 

Hello:


When you run a post-hoc test in ANOVA and are interested in a one-tailed test  it posible to use the reported sig. as sig./2  as a p value for a one tailed test?

Kindly

Andrés

Mg. Andrés Burga León
Coordinador de Análisis e Informática
Unidad de Medición de la Calidad Educativa
Ministerio de Educación del Perú
Calle El Comercio s/n (espalda del Museo de la Nación)
Lima 41
Perú
Teléfono 615-5840

[Rich Ulrich]

>
> When you run a post-hoc test in ANOVA and are interested in a
> one-tailed test it posible to use the reported sig. as sig./2 as a p
> value for a one tailed test?
>

As Paul S. says, so long as the result is in the proper
direction, that is the way you would compute it.

However, you should keep in mind that some audiences and
many journals are hostile to one-tailed tests.  There should
be a *strong* expectation, a-priori, that the result will be
in that direction.  The one time that I remember reporting
a one-tailed result is one where we had written the test into
our grant proposal, years before, justifying its use on this
secondary hypothesis by the predicted direction *and* the
lack of power that we expected.

ALSO.
There is a theoretical possibility for two-tailed tests where
tails are not the same size.  - In statistical theory, these are
introduced when teaching about the concept of a test that is "UMP",
or "Uniformly Most Powerful."  A test that is UMP has more "power"
to reject the null than any other test of the same hypothesis.
A one-tailed t-test is the UMP test for a difference of means
between (say) normal samples.

Only a one-tailed test will ever be UMP.  A two-tailed t-test
with equal-sized tails will have less power for one set of the
alternatives (A > B,  or B > A) than a t-test with unequal tails.
And so on, generally.

It is a very useful convention which says that we assign half
the p-value to each side.  But would you consider it fair, for
your variable, to take 0.001 as the cut-off for a test in one
direction, and 0.049  as the cut-off in the other direction?
 - That would mean that you would be happy to treat a test result
that is 0.01 (say) in the wrong direction as a possibly curious
and suggestive, but insubstantial, result. - I ask myself this
question, when probing my own feelings about a particular test.

--
Rich Ulrich




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