stats on polar coordinate system

From: Scott Seidman (namdiesttocs_at_mindspring.com)
Date: 08/17/04 

Date: 17 Aug 2004 20:26:42 GMT

I have a number of subjects providing data under three conditions
(frequency). The data they provide is a gain value and a phase value,
which is actually one complex number.

I'd like to see if gain and/or phase change as a function of condition,
preferably under some repeated measures model, and would appreciate some
feedback on my approach.

First, I'm sure I need to change from complex numbers to some other
coordinate system. Polar doesn't work so well, so I'm moving to just
using the real part and the imaginary part as two dependent variables.

>From there, I can do a principle component analysis, or just grit my
teeth and do the MANOVA. At the moment I'm opting for the latter.

If I don't use repeated measures, and include the condition as a
category, everything works nicely and all my hypothesis testing seems to
work.

The problem seems to come with the repeated measures model. SYSTAT v11
wants me to specify six levels of repeated measures, but I'm really
dealing with three levels on two dependent variables. Is there any way
to do this repeated measures correctly?

Indeed, the test run with six levels seems to give the right answers, but
the post hoc testing gets pretty messy, giving p-values between all six
of the levels, but I can't seem to find a way to get a joint p-value for
my two dependent variables. I know I can use the general linear model to
do the joint test, but wouldn't that throw out all the pairwise
comparisons that the repeated measures model gives me?

Alternatively, I could do all the subtractions on my own, and just test
the differences separately for difference from (0,0), kind of doing my
own post-hoc test.

The above is the important part, but there is another matter that I'd
appreciate thoughts on. It's mostly my personal demon, but I'll kick it
around for discussion.

Having done the stats on the cartesian data, how are they best
transformed back to polar data? Once I know that there's a difference
between my populations, as tested in the cartesian system, is it fair to
now test directly for separate gain and phase differences, as sort of a
post-hoc test? The twist is that if the origin lies within your data
distribution, phase is largely meaningless--which is something I've been
wrestling with for a while (and also why I've abandoned testing on the
polar coordinates). I'd like to be able to say that as my condition
(which is stimulus frequency) goes down, gain decreases and phase lead
increases. This seems largely to be the case, but for the lowest stim
frequency, the response is so small that phase can be pretty much
anything.