Categorical Data Analysis Using The SAS® System, 2nd Edition

12.2. Methodology for Poisson Regression

Suppose that a response variable Y is distributed as Poisson and has expected value μ. Recall that the variance of a Poisson variable is also μ. If you have a single explanatory variable x, you can write a regression model for μ as

g(μ) = α + xβ

where g is a link function, in terms of a GLM (generalized linear model). Usually, g is taken to be the log function. If so, you have a loglinear model

log(μ) = α + xβ

You can rewrite this model as

μ = e^αe^xβ

If you increase the explanatory variable x by one unit, it has a multiplicative effect of e^β on μ. Since this model is specified as a GLM, with a log link and a Poisson distribution, you can fit it with the GENMOD procedure and use the usual deviance and likelihood ...