An extended quasi-likelihood function

SUMMARY Wedderburn's original definition of quasi-likelihood for generalized linear models is extended to allow the comparison of variance functions as well as those of linear predictors and link functions. The relationship between generalized linear models and the use of transformations of the response variable is explored, and the ideas are illustrated by three examples. generalized linear models by allowing the full distributional assumption about the random component in the model to be replaced by a much weaker assumption in which only the first and second moments were defined. In making this extension Wedderburn widened the scope of generalized linear models in a way very similar to that of Gauss when he replaced the assumption of normality in classical linear models by that of equal variance. For generalized linear models with distributions in the exponential family, likelihood ratio and score tests are used for testing hypotheses concerning nested subsets of covariates in the linear predictor and for assessing hypothesized link functions. These methods are also applicable with Wedderburn's form of quasi-likelihood. However neither of these methods is suitable for the comparison of different variance functions. In this paper we introduce an extended quasi-likelihood function which allows for the comparison of various forms of all the components of a generalized linear model, i.e. the linear predictor, the link function, and the variance function. We then apply the ideas to the analysis of several sets of data.