The R Book

Quasi-likelihood

The precise relationship between the variance and the mean is well established for all the GLM error families (Table 13.1). In some cases, however, we may be uneasy about specifying the precise form of the error distribution. We may know, for example, that it is not normal (e.g. because the variance increases with the mean), but we don't know with any confidence that the underlying distribution is, say, negative binomial.

There is a very simple and robust alternative known as quasi-likelihood, introduced by Wedderburn (1974), which uses only the most elementary information about the response variable, namely the variance–mean relationship (see Taylor's power law, p. 198). It is extraordinary that this information alone is often sufficient to retain close to the full efficiency of maximum likelihood estimators.

Suppose that we know that the response is always positive, the data are invariably skew to the right, and the variance increases with the mean. This does not enable us to specify a particular distribution (e.g. it does not discriminate between Poisson or negative binomial errors), and hence we cannot use techniques like maximum likelihood or likelihood ratio tests. Quasi-likelihood frees us from the need to specify a particular distribution, and requires us only to specify the mean-to-variance relationship up to a proportionality constant, which can be estimated from the data:

An example of the principle at work compares quasi-likelihood with maximum likelihood ...