Poisson regression is used when y is based on count data.

Some examples of count data ● Number of visits to the hospital in a year ● Number of days in unemployment in a month ● Number of text messages sent per day ● Number of goals scored in a game

With a linear regression, one assumes that the value of y can be predicted based on the values of one or more x-variables – as well as their residuals. A requirement is that the residuals are normally distributed. A Poisson regression is a type of generalised linear model which instead assumes a Poisson distribution. A Poisson distribution describes the probability that a number of events will take place within a given interval of time (or space), conditioned upon the fact that the events occur with a constant speed and that every event is independent from any preceding events. When performing a Poisson regression, one uses a link function that allows for a linear combination of the x-variables to predict the logarithm of y.

Poisson regression is used to predict the rate that y changes given the values of the independent variables. As for any regression analysis, we get a coefficient – here, it is called log incidence rate – that shows the effect of x on y. Usually, we focus on something called the incidence rate ratio (IRR). We can calculate the incidence-rate ratio by taking the exponent of the coefficient.

Overdispersion or zero-inflation?

Poisson regression assumes that the mean is equivalent to the variance. If the variance is greater than the mean, we get something called overdispersion. In this case we can apply a so-called negative binomial regression. Another common problem is that we have a lot of observations with the value 0. In this case, it might be necessary to apply a zero-inflated Poisson regression model. These specifications will be briefly explored in this guide as well.

Other names for Poisson regression

Poisson regression is sometimes called, e.g., log-linear regression.