Multinomial responses frequently occur in dose level experiments. For example, in a study of the influence of gamma radiation on the emergence of house flies (Musca domestica L., 1758), three disjoint outcomes occurred: death before the pupae opened, death during emergence, and life after emergence. Although the flies are easy to breed, this sort of bioassay is, in general, very expensive since it requires the use of a gamma radiation source. Experiments therefore need to be designed to involve the minimum number of different doses. Here the theory of optimum experimental design is applied to provide efficient experiments to estimate the parameters of those multinomial logistic models that are a special case of the multivariate logistic models of Glonek and McCullagh (1995, Journal of the Royal Statistical Society, Series B 57, 533-546). The purpose is to reduce the overall experimental cost. The general equivalence theorem (Fedorov, 1972, Theory of Optimal Experiments) is adapted to this class of models, providing an effective method of generating and checking the optimality of designs. One example on flies demonstrates the method, which can be easily implemented
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