The R Book

Misspecified Error Structure

A common problem with real data is that the variance increases with the mean. The assumption in previous chapters has been of normal errors with constant variance at all values of the response variable. For continuous measurement data with non-constant errors we can specify a generalized linear model with gamma errors. These are discussed in Chapter 25 along with worked examples, and we need only note at this stage that they assume a constant coefficient of variation (see Taylor's power law, p. 198).

With count data, we often assume Poisson errors, but the data may exhibit overdispersion (see below and p. 540), so that the variance is actually greater than the mean (rather than equal to it, as assumed by the Poisson distribution). An important distribution for describing aggregated data is the negative binomial. While R has no direct facility for specifying negative binomial errors, we can use quasi-likelihood to specify the variance function in a GLM with family = quasi (see p. 517).