Dear WizaRds,
This is mostly a statistics question, but I'm figuring that R is the right
solution (even before I start!)
I have some bio data of heart rate versus time (rats taken from resting to
maximal heart rate). I want to regress heart rate on time. The data have been
normalized such that resting heart rate is zero at time=0, so that all curves
intersect at the origin (and at the origin only). The regression function is
monotonically increasing, but little is known about it.
The data are also strictly ordinal: I have a factor such that lower-order data
*must* be on a curve that is offset beneath a higher-order curve. Don't ask
where the factors come from...but *given* these factors the assumption, or
rather the *constraint*, is that lower orders are better (lower-order rats are
fitter rats with better cardiovascular response). Most important is that these
curves do not intersect because of these factors (a fitter rat can not have a
worse heart-rate
response than a less fit rat!).
Here's a schematic to show you what I mean. It's very rough, but it gets the
point across:
seq(0, 1, len=100)
f<- 1/seq(.3,1,len=6)
windows();
plot(t, sqrt(3*t), type='n', xlab='time', ylab='heart rate');
grid()
for (i in seq(along=f))
lines(t, sqrt(f[i]*t), col=ceiling(2*i))
Now I'm wondering where I should start and I'm think that this is really not
much different from having a y_i ~ x_i | factor_i ... the i^th response curve
just like a dummy variable male/female linear regression. But in some way the
factors are related (there's a dose behind it, if ya see what I mean), so they
are not independent...they're all part of a "system" (some studies have more
"juice" overall, so the whole "system" is varying from one group of rats to the
next).
This means that in some "systems" the curves will be closer or some curves
within the system will be closer together.
Here is my question for you guys: what is the best way to model this kind of
problem, especially given that each "i^th curve" could have as few as 3 data
points? If I can only vaguely assume the type of the regression function
(monotonic, rapidly increasing from the origin kind of like sqrt(t), as above)
should it be a parametric or nonparametric regression? What about those errors?
Gee! It's hard to assume anything (there're so few of 'em and they're probably
heteroskedastic!
Where do I begin? I'd gladly accept all references, pointers, and such like.
Best, and thanks for R and R-Help, all you Core guys!!!
Jack.
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