What a log-link coefficient multiplies
In a Poisson regression with a log link and no interaction, what does e^{\beta_j} multiply? Why is that multiplier not an odds ratio?
Worked answer
The log link says the log of the mean count is linear in the predictors. Adding 1 to predictor j adds \beta_j to the log mean, which multiplies the mean count by e^{\beta_j}, with the other predictors held fixed. An odds ratio is the multiplier for the odds of a binary outcome under a logit link. A count does not have odds in that sense. Reading e^{\beta_j} from a log link as an odds ratio changes the claim from a ratio of means to a ratio of odds.