The standard errors of the coefficients are then calculated by multiplying the Poisson model coefficient The parameter estimates are the same, however they won't necessarily be as efficient. This is analogous to fitting ordinary least squares on symmetrical, yet not normally distributed data. Likelihood estimates from the Poisson model. Mean and variance ($var(y)=\phi\mu$), the estimates of the regression coefficients are identical to those of the maximum Instead, if the Newton-Ralphson iterative reweighting least squares algorithm is applied using a direct specification of the relationship between This approach does not utilize an underlying error distribution to calculate the maximum likelihood (there is no quasi-Poisson distribution). Quasi-Poisson models - these introduce the dispersion parameter ($\phi$) into the model.There are a number of ways of overcoming this limitation, the effectiveness of which depend on the causes of overdispersion. It turns out that overdispersion is very common for count data and it typically underestimates variability, standard errors and thus deflated p-values. The degree to which the variability is greater than the mean (and thus the expected degree of variability) is called dispersion.Įffectively, the Poisson distribution has a dispersion parameter (or scaling factor) of 1. The variance increases more rapidly than does the mean. The distribution of counts might be somewhat clumped which can result in higher than expected variability (that is $\sigma\gt\mu$). For example, when there are other unmeasured influences on the response variable, Under certain circumstances, this might not be the case. Whilst the expectation that the mean=variance ($\mu=\sigma$) is broadly compatible with actual count data (that variance increases at the same rate as the mean), The canonical link function for the Poisson distribution is a log-link function. In the Poisson distribution, the variance has a 1:1 relationship with the mean. Repeated observations from a Poisson distribution located close to zero will yield a much smaller spread of observations than will samples drawn fromĪ Poisson distribution located a greater distance from zero. Put differently, the variance is a function of the mean. Like count data (number of individuals, species etc), the Poisson distribution encapsulates positive integers and is bound by zero at one end.Ĭonsequently, the degree of variability is directly related the expected value (equivalent to the mean of a Gaussian distribution). Into the scale of the linear predictor (which is $-\infty,\infty$).Īs implied in the name of this group of analyses, a Poisson rather than Gaussian (normal) distribution is used to represent the errors (residuals). The role of the link function is to transform the expected values of the response y (which is on the scale of (0,$\infty$), as is the Poisson distribution from which expectations are drawn) The linear predictor is typically a linear combination of effects parameters (e.g. Poisson regression is a type of generalized linear model (GLM) that models a positive integer (natural number) response against a linear predictor via a specific link function.