Poisson Regression

Introduction

The Poisson regression model is a standard model for count data where the response variable is given in the form of event counts such as the number of insurance claims within a given period of time or the number of cases with a specific disease in epidemiology. Let (Y _i , x _i ) denote n independent observations, where x _i is a vector of explanatory variables and Y _i is the response variable. It is assumed that the response given x _i follows a Poisson distribution which has probability function

$$P({Y }_{i} = r) = \left \{\begin{array}{@{}l@{\quad }l@{}} \frac{{\lambda }_{i}^{r}} {r!} {e}^{-{\lambda }_{i}}\quad &\text{ for }r \in \{ 0,1,2,\ldots \,\} \\ 0 \quad &\text{ otherwise}. \end{array} \right .$$

Mean and variance of the Poisson distribution are given by E(Y _i ) = var(Y _i ) = λ _i . Equality of the mean and variances is often referred to as the equidispersion propertyof the Poisson distribution. Thus, in contrast to the normal distribution, for which mean and...