Determine statistical difference of slopes of quadratic relationship in a Poisson regression 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.
Baseline: 

Y = exp(b0 + b1X - b2X^2)
  [assume b1 and b2 are positive so that the function is concave]

Full model: 

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.
 A: The proper statistical way of calculating such difference would be to add a group indicator variable interacted with the slopes that you would like to test and compare it with the model where the group indicator variable is not present. This procedure helps you to work with the full data, and the sample difference between them will be encoded in the standard errors of the group parameter's estimates.
Here is the explanation of such procedure. Suppose the first group is named group $a$ and the second one is group $b$. You should compare your full model:
$$
g(\mu) = \beta_0 + \beta_1x + \beta_2x^2 +\beta_3zx^2 + \beta_4zx + \beta_5z\tag{full model}
$$
with group interactions effect:
$$
g(\mu) = \beta_0 + \beta_1x + \beta_2x^2 +\beta_3zx^2 + \beta_4zx + \beta_5z + \beta_6zx^2\cdot\mathbb{1}_{g=b} + \beta_7zx\cdot\mathbb{1}_{g=b} + \beta_8z\cdot\mathbb{1}_{g=b} + \beta_9\mathbb{1}_{g=b}
$$
$\beta_9$ should be included in the model if there is a difference in the mean levels of the response variable between the two groups ($a$ or $b$).
The base line of the proposed model is then in reference to group $a$. Note that if there is a difference between groups exists, the estimates for the curvature parameters are updated by the terms where $\mathbb{1}_{g=b}$ is true.
So who do you test the model? There are a number of approaches:


*

*Wald test for $\beta_6$, $\beta_7$, $\beta_8$ and $\beta_9$

*likelihood ratio tests between the full model and the proposed model

*score tests

*etc..

