Q30. Poisson regression model statement
[4 marks]
For 80 independent lakes, researchers record the number of fish species, distance from the coast in kilometres, and surface area in hectares. They fit:
glm(species ~ distance + log(area), family = poisson)Write down explicitly the model corresponding to this code. Define your notation and state the independence assumption.
Definitions. For lake i=1,\ldots,80, let Y_i be its species count, d_i its distance from the coast in kilometres, and A_i>0 its area in hectares. Write \mu_i=\mathbb{E}(Y_i\mid d_i,A_i). The covariates are fixed; logarithms are natural. A GLM combines a response distribution with a link between this conditional mean and a linear predictor.
The model is \boxed{ Y_i\mid d_i,A_i\overset{\mathrm{ind}}{\sim} \operatorname{Poisson}(\mu_i), \qquad \log\mu_i=\beta_0+\beta_1d_i+\beta_2\log A_i. } Thus the default link is the log link, and \mu_i=\exp\{\beta_0+\beta_1d_i+\beta_2\log A_i\} =e^{\beta_0}e^{\beta_1d_i}A_i^{\beta_2}. Under the Poisson assumption, \mathbb{E}(Y_i\mid d_i,A_i) =\operatorname{Var}(Y_i\mid d_i,A_i)=\mu_i. The responses from different lakes are independent conditional on their covariates. They need not be identically distributed, because their means can differ.
Key result. A model statement needs a distribution and an independence assumption, as well as the equation linking the mean to the covariates.