6.8 The quasi-Poisson model

This model (Hoef and Boveng 2007) generalizes the Poisson model by allowing the variance to depend linearly on the mean \(\lambda\), through a dispersion parameter \(\theta\):

\[\begin{equation} Var(y)=\theta \lambda \tag{6.74} \end{equation}\]

Setting \(\theta=1\) returns the Poisson model, while \(\theta>1\) accommodates overdispersion. Note that this is a quasi-likelihood specification: it states only how the variance relates to the mean, and does not correspond to any probability distribution.

Then, we estimate the regression function (6.70) using the IWLS iterative weighted least squares, which is the WLS \(\beta_{WLS}=\bigg(X^tWX\bigg)^{-1}X^tWz\) discussed in 5.2.4.1 used iteratively, where the working response \(z\) is obtained at each iteration from the current estimates by:

\[\begin{equation} z_i=x_i\beta+\frac{y_i-\lambda_i}{\lambda_i} \tag{6.75} \end{equation}\]

and the weight matrix \(W\) is diagonal, given by:

\[\begin{equation} W=diag\Bigg(\frac{\lambda_1^2}{\theta\lambda_1},...,\frac{\lambda_n^2}{\theta\lambda_n}\Bigg)=\frac{1}{\theta}diag\big(\lambda_1,...,\lambda_n\big) \tag{6.76} \end{equation}\]

Written this way, an important property becomes visible. Since \(\theta\) enters \(W\) only as the scalar factor \(\frac{1}{\theta}\), it cancels between the two terms of \(\bigg(X^tWX\bigg)^{-1}X^tWz\). The quasi-Poisson model therefore returns exactly the same coefficient estimates as the Poisson model; what it changes is the variance matrix, which is inflated by \(\theta\), so that every standard error is multiplied by \(\sqrt\theta\). This is what makes the model useful: the Poisson estimates stay consistent under overdispersion, but their standard errors are too small, and \(\theta\) repairs them.

The dispersion parameter itself is estimated from the Pearson statistic (6.73):

\[\begin{equation} \widehat\theta=\frac{1}{n-k}\sum\limits_{i=1}^{n}\frac{\big(y_i-\widehat\lambda_i\big)^2}{\widehat\lambda_i} \tag{6.77} \end{equation}\]