8 - Multinomial Response Models

Summary

In many regression problems the response is restricted to a fixed set of possible values, the so-called response categories. Response variables of this type are called polytomous or multicategory responses. In economical applications, the response categories may refer to the choice of different brands or to the choice of the transport mode (Example 1.3). In medical applications, the response categories may represent different side effects of medical treatment or several types of infection that may follow an operation. Most rating scales have fixed response categories that measure, for example, the medical condition after some treatment in categories like good, fair, and poor or the severeness of symptoms in categories like none, mild, moderate, marked. These examples show that there are at least two cases to be distinguished, namely, the case where response categories are mere labels that have no inherent ordering and the case where categories are ordered. In the first case, the response Y is measured on a nominal scale. Instead of using the numbers 1, …, k for the response categories, any set of k numbers would do. In the latter case, the response is measured on an ordinal scale, where the ordering of the categories and the corresponding numbers may be interpreted but not the distance or spacing between categories. Figures 8.1 and 8.2 illustrate different scalings of response categories. In the nominal case the response categories are given in an unsystematic way, while in the ordinal case the response categories are given on a straight line, thus illustrating the ordering of the categories.