I am interested in examining the relationship of
Y are continuous variables. I expect the relationship is linear. I know I can do a simple linear regression here, however I'm curious if I could do something a bit different to make a more informed inference.
Y values are themselves estimated parameters from another test. I have an estimated mean and standard deviation for each
Y value (from normal distributions). Could I use the whole probability distribution from each
Y observation rather than just the mean estimate when examining
Y's relationship with
X? The standard deviations for different
Y values can vary quite a bit, so I think it could be interesting to see how their distributions may overlap and combine and whether that will affect the statistical relationship between
Does anyone know what kind of test I can do to examine this? I'm analyzing my data in R.
Here's a brief sketch trying to illustrate my situation. Note that each
Y observation is a distribution (not just the one I pointed out).
If it helps, I could also make
X a distribution rather than a single value, but it would not be normally distributed. For each observation
Y, there are 100s to 1000s of
X observations that I'm averaging to get the single
X value to examine in this relationship. Rather than averaging them to get a single
X value, I could work with the whole distribution of observations.
If that broadens the range of statistical tests that are available then I can do that too. If it doesn't make a difference, I can just use average values as well.