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 λ is the average count called the rate

The Poisson distribution has a mean which is equal to the variance:
E(Y) = Var(Y) = \(\lambda\)
The mean number of events is the rate of incidence multiplied by time passed

Because of this assumption, the Poisson distribution also has the following properties:

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.

Poisson Regression as  a GLM

To specify a generalized linear model we need (1) the distribution of outcome, (2) the link function, and (3) the variance function. For Poisson regression these are (1) Poisson, (2) Log, and (3) identity.

Let's consider a binomial example from a previous class where:
$$ Y_{ij} \approx Binomial(N_{ij}, λ_{ij}) $$
Given a variable Y ~ Binomial(n, p), where n is number of experiements and p the probability of success, this binomial distribution can be approximated well by the Poisson distribution with mean n*p.

So from that we can assume the distribtuion of Y also follows:
$$ Y_{ij} \approx Poisson(N_{ij} λ_{ij}) $$
and thus
$$ log(E(Y_{ij})) = log(N_{ij}) + log(\lambda_{ij}) $$

Since log(N_ij) is calculated from the data, we will proceed by modeling log(λ_ij) as a linear combination of the predictors:
$$ log{λ_ij } = μ + β1Xij _1X_{ij}^1 + β2_2 Xij X_{ij}^2 + . . . + βk_k Xij X_{ij}^k $$

The natural log is attractive as a link function for several reasons:

1. The log link maps relative rates into additive effects
2. With this link, parameters are readily interpretable as rate ratios, these ratios are called relative risks (risk ratios)
3. The log link transforms positive values into the whole real link.

This the Poisson regression models extend the loglinear models; loglinear models are instances of Poisson regression.

Modeling Details

With the assumption
$$ E(Y_i) = N_i*λ_i $$
then using a log link we have:
$$ log(E(Y_{i})) = log(N_{i}) + log(\lambda_{i}) $$

Assuming we have a set of k predictors X¹, X², ..., X^k (continuous and/or ordinal and/or nominal), in Poisson regression one assumes:

$$ log{λ_ij } = μ + β_1X_{ij}^1 + β_2 X_{ij}^2 + . . . + β_k X_{ij}^k $$