Roy's Largest Root

Quote:

Running a MANOVA analysis Pillai's Trace, Wilks' Lambda and Hotelling's
Trace are insignificant.
Roy's Largest Root is significant.

How should I interpret Roy's Largest Root?
- Is it that at least one of the dependent variables significantly differs
over the group-variable?
- Or does it mean that the dependent variables as a group significantly
differ between at least two groups of the group-variable?

Quote:

Burger wrote:
Running a MANOVA analysis Pillai's Trace, Wilks' Lambda and Hotelling's
Trace are insignificant.
Roy's Largest Root is significant.

How should I interpret Roy's Largest Root?
- Is it that at least one of the dependent variables significantly
differs
over the group-variable?
- Or does it mean that the dependent variables as a group significantly
differ between at least two groups of the group-variable?

Olson, C. L. (1976). On choosing a test statistic in multivariate
analyses of variance. Psychological Bulletin, Vol. 83: 579-586.

Quote:

I have finally obtained and read the paper. Still it remains unclear to me
when to use Roy's largest root [I'm not sure what it means "a test on what
is accounted for by the single, largest root". Also, even after reading
several (technical) definitions, I don't understand what an eigenvalue is.].

My hypothesis is that some dependent variables are affected by the
independent variables while others are not. If I understood the article
correctly, my hypothesis is that there is a "concentrated" noncentrality
structure. And Roy's largest root has been found to be the most appropriate
statistic in such a situation, i.e., Roy largest root tests whether at least
one of the dependent variables (the one with the largest eigenvalue) differs
between the groups.

When testing my hypothesis, the overall test is insignificant with Wilks
Lambda. This might still be in line with my hypothesis since I hypothesized
that for some dependent variables the relationship would be insignificant
and thus the overall effect may also be insignificant.
According to Roy's largest root the overall effect is significant.

Hence, my conclusion from a significant Roy's largest root in combination
with an insignificant Wilks Lambda is that this confirms my hypothesis:
there are dependent variables whose means differ between the groups but also
dependent variables whose means do not differ between the groups.

Have I understood it correctly, i.e., is my conclusion correct?

Quote:

Burger wrote:
I have finally obtained and read the paper. Still it remains unclear to
me
when to use Roy's largest root [I'm not sure what it means "a test on
what
is accounted for by the single, largest root". Also, even after reading
several (technical) definitions, I don't understand what an eigenvalue
is.].

My hypothesis is that some dependent variables are affected by the
independent variables while others are not. If I understood the article
correctly, my hypothesis is that there is a "concentrated" noncentrality
structure. And Roy's largest root has been found to be the most
appropriate
statistic in such a situation, i.e., Roy largest root tests whether at
least
one of the dependent variables (the one with the largest eigenvalue)
differs
between the groups.

When testing my hypothesis, the overall test is insignificant with Wilks
Lambda. This might still be in line with my hypothesis since I
hypothesized
that for some dependent variables the relationship would be
insignificant
and thus the overall effect may also be insignificant.
According to Roy's largest root the overall effect is significant.

Hence, my conclusion from a significant Roy's largest root in
combination
with an insignificant Wilks Lambda is that this confirms my hypothesis:
there are dependent variables whose means differ between the groups but
also
dependent variables whose means do not differ between the groups.

Have I understood it correctly, i.e., is my conclusion correct?

Concentrated noncentrality means that all the variables show similar
patterns of mean differences, where "similar" means "strongly
correlated, without regard to the sign of the correlation".

Diffuse noncentrality means the opposite, that some variables show
patterns of mean differences that are different from (uncorrelated
with) the patterns shown by other variables.

In both cases, the correlation referred to is the between-groups
correlation, which depends on only the group means and sample sizes,
and is not the same thing as the either the overall correlation or the
within-groups correlation.

Having variables that show no mean differences would raise the power
of Roy relative to Wilks, but getting a significant Roy and an
insignificant Wilks can certainly not be taken as confirming that the
groups differ on only some of the variables. In addition to the purely
logical problems with that conclusion, your results could equally well
indicate heterscedasticity and/or nonnormality, to which Roy is more
sensitive than Wilks is.

In any case, manova does not address your research question, which is
about the relative sensitivity of the dependent variables to the
independent variables. You are predicting that some (but not all) of
the variables have negligible (but not necessarily zero) effect sizes.
The corresponding null is that all the effect sizes are the same. I am
not aware of a test for such a hypothesis.

Quote:

Thanks, I guess with my hypothesis I should then just look at the F-test of
the individual dependent variables and in this way determine that some of
them differ significantly over the groups while others do not. The overall
test is less useful with my hypothesis.

Quote:

[...] However, it would still be nice to be able
to formally test the hypothesis that all the variables are equally
sensitive to the experimental manipulation, that the differences
between the variables are just error.