Multinomial and Poisson Nonlinear Models

Abstract

In this chapter, we consider two additional statistical models that are very useful for analyzing data sets in agronomy, medicine, and social sciences: The multinomial model and the Poisson model. The binomial model considered in the preceding chapter deals with response variables with two categories: The occurrence or nonoccurrence of an event is observed. The multinomial model considers multicategory response variables. This topic is discussed in the first part of this chapter. Our objective is not to provide a complete course on the multinomial model (see McCul-lagh and Nelder [MN89] and Aitkin et al. [AAFH89] for an introduction to this topic). We begin with two examples of multicategory response data and describe the modeling of the probability functions and the quasi-likelihood method for estimating the parameters. Then we demonstrate how to estimate the parameters, calculate confidence intervals, and perform tests, taking full advantage of the nls2's facilities: The relationships between the probability functions and the independent variables do not need to be log-linear; the statistical inference for any parameter function is carried out easily using the same nls2 functions as for the nonlinear regression model. The second part of the chapter is concerned with count data. We describe how the data from the cortisol assay example presented in Chapter 1 can be fitted using a Poisson nonlinear model. We show how we can easily adapt for overdispersed data using nls2. 7.1 Multinomial Model We illustrate the use of the multinomial model with two examples that have already been analyzed in the literature. The first example, concerning the study of pneumoconiosis among coal miners, has been analyzed by McCullagh and Nelder [MN89, page 178] and by Aitkin et al. [MAH89, page 225] and the second one, concerning a cheese tasting experiment by McCullagh and