Quasi-poisson regression model

The Quasi-Poisson Regression is a generalization of the Poisson regression, which is used to model an overdispersed count variable (see Robert 1974). The Poisson model assumes that the variance equals the mean, which is not always the case. A Quasi-Poisson model, which assumes that the variance is a linear function of the mean, is more appropriate when the variance is greater than the mean. The Quasi-Poisson model is characterized by the first two moments (mean and variance) (Robert 1974) and the second parameter, often known as the overdispersion scaling parameter, ϕ, which is used in the estimation of the conditional variance. In addition, this model does not necessarily have a distributional form. However, it is demonstrated in the literature that how to generate a distribution for this model where reparameterization was required (Efron 1986). The model that was estimated with this correction now assumes a Poisson error distribution with mean and variance.

The model estimated with this correction now assumes a Poisson error distribution with mean μ and variance ϕμ. If there is overdispersion in the data, the scaling parameter ϕ will be greater than one; if there is equidispersion, the scaling parameter ϕ will be equal to one, and the resulting model will be equivalent to the standard Poisson regression model. Finally, if the data are underdispersed, ϕ will be less than one. The calculation of the scaling parameter is given by

Because the conditional variances are larger than their corresponding conditional means in the overdispersed model, the standard errors (based on the conditional variances) are a factor of ϕ larger than the standard errors in the standard Poisson model.