9.1 Poisson Regression

9.1.1 Review of GLM

To review, there are three important components to the GLM:

1. A random component: The random component of the GLM contains the response variable \(\mathbf{Y}\) and its probability distribution (e.g. the binomial distribution of \(\mathbf{Y}\) in the binary regression model).
2. A Linear Predictor: The linear predictor typically takes the form of \(\mathbf{X}\boldsymbol{\beta}\) where \(\mathbf{X}\) is an \(n \times q\) matrix of observations and \(\boldsymbol{\beta}\) is an \(q \times 1\) column vector.
3. Link Function: The link function, typically specified as \(g()\), is used to relate each component of \(\mathbb{E}(\mathbf{Y})\) to the linear predictor, \(g[\mathbb{E}(\mathbf{Y})]=\mathbf{X}\boldsymbol{\beta}\).

9.1.2 Poisson Regression as GLM

Poisson regression can also be formulated as a GLM:

\[ \mathrm{log}(\mu) = \beta_{0} + \beta_{1}x_{1} \] or equivalently,

\[ \mu = \mathrm{exp}(\beta_{0} + \beta_{1}x_{1}) = \mathrm{exp}(\beta_{0})\mathrm{exp}(\beta_{1}x_{1}). \]

1. A random component: The distribution of \(\mathbf{Y}\) is assumed to be Poisson, \(Y_{i} \sim \mathrm{Poisson}(\mu_{i})\).
2. A Linear Predictor: The systematic component takes the form of \(\mathbf{X}\boldsymbol{\beta}\).
3. Link Function: For Poisson regression the log link is used.