[R] MANOVA power, degrees of freedom, and RAO's paradox

[Oliver Bossdorf]

Hi,

I have a nested unbalanced data set of four correlated variables. When I 
do univariate analyses, my factor of interest is significant or 
marginally significant with all of the variables. Small effect size but 
always in the same direction. If I do a MANOVA instead (because the 
variables are not independent!) then my factor is far from being 
significant. How does that come about?

I have found a mention of a so-called Rao's paradox, which seems to deal 
with exactly this phenomenon. Does anyone know more about it, e.g. a 
reference?

The next strange thing is that if do the MANOVA in R, then both 
hypothesis and error degrees of freedom are multiplied by the number of 
variables. When I do it in SAS, however, only the hypothesis d.f. are 4 
x univariate, while the error d.f. are as in univariate, minus 3. This 
is irritating, in particular since no indication is given in the 
handbooks as to how degrees of freedom are calculated in a MANOVA? Can 
anyone tell me more about this? Are there different philosphies that are 
responsible for the differences between R and SAS?

I would be grateful for any help.

Regards, Oliver