I ran a Poisson regression inflated in one in Rstudio and got decimal coefficients to which applied exp(), and the result of this operation is less than 1.
I understand that to interpretate coefficients usually is used exp() to see the effect in the count variable (y).
In this question I saw this example:
A 1 unit increase in x leads to an exp(β1) times increase in y. Say your 'old' y was 10, and β1=2, then exp(β1)=7.4, and y would be 10 times 7.4, ie 74. If there were another observation that was 1 unit higher still, that would be 74 * 7.4, etc.
My question is about the case when β1 = -1.2 and then, exp(β1) = 0.3, for example. Since β1 is negative, we would say that the effect in the count variable by x1 is negative (decrease the counts), but then, following the cited example, one could say that "A 1 unit increase in x1 leads to an 0.3 times decrease in y", but as i see this, actually multiplicate by 0.3 leads to a lower number, (10x0.3=3) but is a negative lower number, so it would be the "new y" a higher number than the "old y" and x1 would have a positive impact.
Writing this question I remember that arguments of exp() lowers than 0 would always give decimals after the exp() is applied, so maybe is always like this? the decrease given can be deducted by de - sign, or also by the decimal result of exp(β1)? It's kinda a silly question, but i would like to get a confirmation, I'm pretty new with this regression.
Also, if a β2 is almost 0, and then exp(β2) = 1, it's okay to say that x2 doesn't have impact in the count variable? Because is like *1.
Thnku!!