Generalized Linear Models—The Missing Link

We consider generalized linear models, including an extension due to Thompson and Baker (1981), within the larger framework of multiparameter exponential family models. This general approach shows that the link function is not a necessary feature of a computer algorithm for calculating maximum likelihood estimates for such models by iteratively reweighted least squares. It is argued that the link function is a useful component of model fitting and interpretation in situations where there is a natural link to an underlying linear model (e.g., logistic regression). However in many instances there is no single link function (e.g., multinomial regression) or else unlinked parameters exist (e.g., bioassay with a spontaneous response rate). We attempt to show by a number of examples that a general approach via exponential family models is preferable in such situations.