statisticalrethinkin..

11.2. POISSON 295FIGURE 11.7. Locations of populationsin islands data. Map reproducedfrom Kline and Boyd 2010,supplemental.e most basic Poisson model contains no prediction variables, just a constant mean:n i ∼ Poisson(λ)where n i is the value of Total.Tools on row i and λ is the mean count.Before fitting this model, though, let’s establish the link function to use. We don’t technicallyneed a link right now, but it’ll be easier to move on to the later models, if we establishthe link now. Also, it will help with estimation. It’s most common to use a log-link with aPoisson model, such that the mean λ is modeled:n i ∼ Poisson(λ i )log λ i = αis log-link is the reason that Poisson models are oen called log-linear models. e linearmodel part of the GLM is the logarithm of the observed mean. is actually implies, as you’llsee in a bit, that on the real count scale the posited relationship will not be linear.If you solve the second line above for λ i , you find that this log-link implies that λ i =exp(α). We really do need a link here, otherwise the mean will be allowed to go below zero.If the linear model is exponentiated, as it is here, this guarantees that λ will be positive, evenif the linear model itself is negative. Go ahead and try it, if you don’t immediately intuit thistruth.Now here’s the code to fit the model:m10.8