6.2 The Poisson Regression Model
- The Poisson regression model adds observed heterogeneity
\[\begin{aligned} \mu_{i} = E(y_{i}\,|\,\mathbf{x}_{i}) &= \exp(\mathbf{x}\boldsymbol{\beta}) \\ &= \exp(\beta_{0} + \beta_{1}x_{1,i} + \beta_{2}x_{2,i} + \ldots + \beta_{k}x_{k,i}) \end{aligned}\]
\[\ln(\mu_{i}) = \beta_{0} + \beta_{1}x_{1,i} + \beta_{2}x_{2,i} + \ldots + \beta_{k}x_{k,i}\]
where μi is the mean of Poisson distribution for observation i
Predicted rates
- Predicted rates (\(\widehat{\mu}\)) are easy to calculate:
\[E(y) = \widehat{\mu} = \exp(\mathbf{x}\boldsymbol{\beta})\]
- Taking the exponential of xβ forces \(\widehat{\mu}\) to be positive
Predicted probabilities of specific counts
- Also (relatively) easy to calculate predicted probabilities of specific counts (when y = j). E.g., The probability of observing a 0 (j = 0) is often of interest
- For observation i
\[\Pr(y = j\,|\,\mathbf{x}_{i}) = \frac{e^{-\widehat{\mu}_{i}}\,\widehat{\mu}_{i}^{\,j}}{j!}\]
which can differ for each observation i
- Can calculate a probability of any specific count j using the above formula
- Can also calculate the probability of a range of counts by adding up multiple individual probabilities. E.g.,
\[\Pr(y = 1,2,3,4) = \Pr(y = 1) + \Pr(y = 2) + \Pr(y = 3) + \Pr(y = 4)\]
- The Poisson model is didactically useful, but I do not generally recommend the Poisson model for reasons considered in 6.6 (there are better alternatives)