Changing the units for an independent variable in a Poisson model I'm trying to model a dose-response relationship between the incidence of cancer (dependent variable, counts) and the radiation dose (independent variable in gray). Can I change the unit of the independent variable from gray to milligray (gray × 1000) and have the same response? 
I have built two models, 1 with independent variable as gray, and the second model with gray × 1000 as the independent variable. Should the model have similar parameter estimates, just different by a factor of 1000? 
I have tried this, and my parameter for dose-response in two separate models goes from 1.000764 per milligray to 2.14 per gray.
 A: The linear predictor is on a log scale, and $\frac{\log 2.14}{\log 1.000764} \approx 1000$. The response is modelled by a Poisson distribution whose mean increases by a factor of $1.000764$ for a 1 mGy increase in radiation dose, or equivalently a factor of $1.000764^{1000}=2.14$ for a 1 Gy increase.
A: I suspect something else is wrong if your parameter changes by a different factor than your units. Here's a simple simulation in R to demonstrate that, as you suspect, multiplying your predictor by 1000 should change the scale of your parameter by the same factor (dividing it):
y=rpois(10000,1);x=rnorm(10000,5);x1000=x*1000    #Generates random count and continuous data
summary(glm(y~x,family='poisson'))$coefficients   #fits Poisson GLM of counts ~ continuous IV

               Estimate  Std. Error   z value  Pr(>|z|)
(Intercept)  0.07375024 0.050409708  1.463017 0.1434629
x           -0.01275800 0.009894814 -1.289363 0.1972721

summary(glm(y~x1000,family='poisson'))$coefficients       #same as above, using multiplied IV

                Estimate   Std. Error   z value  Pr(>|z|)
(Intercept)  0.073750238 5.040971e-02  1.463017 0.1434629 #intercept stats unchanged
x1000       -0.000012758 9.894814e-06 -1.289363 0.1972721 #Estimate and SE differ by x10^(-3)

