GLM for Count Data

Generalized linear models for count data are regression techniques available for modeling outcomes describing a type of discrete data where the occurrence might be relatively rare. A common distribution for such a random variable is Poisson.

The probability that a variable Y with a Poisson distribution is equal to a count value
$${ P(Y = y) = {{\lambda^y e^{-\lambda}} \over {y!}}}  y = 0, 1, 2, ..., \infty $$
where \( \lambda \) is the average count called the rate

The Poisson distribution has a mean which is equal to the variance:
E(Y) = Var(Y) = \(\lambda\)

Other properties of the Poisson distribution:

1. If Y_1 ~ Poisson(λ_1) and Y_2 ~ Poisson(λ_2) where Y_1 and Y_2 are the number of subjects in groups 1 and 2, then Y_1 + Y_2 ~ Poisson(λ_1 + λ_2)
2. This generalizes to the situation of n groups as: Assuming n independent counts of Y_i are measured, if each Y_i has the same expected number of events λ then Y_1 + Y_2 + ... + Y_n ~ Poisson(nλ). In this situation λ is interpretable as the average number of events per group.