Models for Rates with Poisson Errors

Abstract

In a recent paper, Frome (1983) described the fitting of models with Poisson errors and data in the form of rates. Some of the models considered are log-linear and hence can be fitted simply by GLIM or by any package that handles log-linear models; some are not and require either special treatment in GLIM or the use of a program that handles iterative weighted least squares. (Why so often 'reweighted'? If the procedure is iterative you can change the weights). It is perhaps worth stressing two desirable aspects of any such program if it is to be easily used: the first is that the user should be able to specify categorical terms (main effects, interactions etc.) without having to generate the corresponding dummy variables explicitly, and the second is the possibility of including a fixed intercept ('offset' in GLIM terminology) in the model. For Poisson models with rates and a log link the offset in GLIM is just ln(c) in Frome's notation. The statement that follows Equation (3.4) in Frome's paper that 'this model is nonlinear in the parameters, and consequently the computational procedures designed for generalized linear models cannot be used' deserves some discussion. The computational procedures for a generalized linear model (GLM) can be easily adapted to cope with nonlinear parameters, as was demonstrated by McCullagh and Nelder (1983, Ch. 10). If a term has the form