In Poisson regression, the response variable y follows a Poisson distribution, and the logarithm of its expected value is modelled by a linear combination of unknown parameters. A Poisson regression model is also known as a log-linear model.
In statistics, Poisson regression is used to model count data and contingency tables.
Common examples of counting variables include weight (a number of kilograms), time (a number of years), and class size (a number of people).
In Poisson regression outcome variable Y is a count. But we can also have y/t, the rate, as the response variable, where t is an interval representing time, space or some other grouping.
Explanatory variables, X = (X1, X2, … Xk), can be continuous or a combination of continuous and categorical variables.
For simplicity, with a single explanatory variable, we can write:
Log (μ) = α + βX
This is equivalent to μ = exp (α + βX )
exp (α) = effect on the mean of Y, that is μ, when X = 0
exp(β) = with every unit increase in X, the predictor variable has multiplicative effect of exp(β) on the mean of Y, that is μ