I'm looking for a statistical or mathematical way to test the difference between two slopes. Others have asked related questions but my problem is quite particular.
I'm running a Poisson regression of this form with X the focal predictor and Z the moderator.
Y = exp(b0 + b1X - b2X^2) [assume b1 and b2 are positive so that the function is concave]
Y = exp(b0 + b1X - b2X^2 + b3ZX + b4ZX^2 + b5Z)
I am running this regression for 2 samples of a single group. My goal is to find out whether one group is significantly more sensitive to the introduction of the moderator than the other group. Any suggestions on how to test that would be awesome.
Here is what I am currently doing
After running this regression, I calculate the turning points and then the slope (derivative) of the full Poisson model at various distances 'a' from the turning point (where the slope is obviously 0). Based on this information I can estimate a simple slope line that gives me an idea of how much the introduction of the moderator Z affects the concave shape of the main effect X and X^2 on Y. I can determine this at different values for Z. The goal is to be able to explain something like "a 1 standard deviation increase of the moderator Z in sample 1, has a much stronger effect than the same increase in sample 2. The concave function between X and Y steepens thus significantly more in sample 1."
The 2 samples are not of equal size, and have different means and standard deviations for the moderator Z. I want to find out whether the estimated simple slope lines are statistically different.