Nonlinear quasi-likelihood models: Applications to continuous proportions

We discuss some practical aspects of fitting and interpreting nonlinear quasi-likelihood regression models. The examples considered are for data in the form of continuous rates or proportions distributed on the unit interval. However, many of the concepts and procedures illustrated should be helpful with other models as well. Our approach to estimation for multi-parameter models is by the method of fitting expected values discussed by Jennrich and Moore (1975) and Cox (1984a). This allows the use of standard nonlinear regression computer programs. An important feature of this approach is that it does not require the existence of a link function. Thus nonlinear models are easily accommodated, including models with a parametric link function, as well as models with a link to a nonlinear predictor having multiplicative terms. After a brief discussion of the properties of these models and the method of fitting, we provide a detailed illustration of fitting a number of link-linear as well as nonlinear models to a set of data considered by Wedderburn (1974). The analysis makes use of a diagnostic tool known as the biplot to help select a multiplicative interaction model. A second example of a useful nonlinear model is provided by an experiment in toxicology. As part of the discussion of these alternative models, we illustrate the choice of link and variance functions using an extended quasi-likelihood developed by Neider and Pregibon (1987).