Dose-time-response cumulative multinomial generalized linear model

In toxicological and pharmaceutical experiments, a type of quantal bioassay experiment is designed in which a response, such as mortality, in a group of animals is recorded over time points under different dose levels in the course of the experiment. The application of the typical logit and probit analyses is no longer valid in this situation because it neglects the dependency on time and also the possible interaction of time and dose concentration on the response in the experiment. In this paper, a dose-time-response model is proposed for this type of experiment and a cumulative multinomial generalized linear model that incorporates time and the other experimental conditions as covariates is developed by the theory of maximum likelihood estimation. Both the point estimator and confidence bands for ED50(t), the concentration of a toxicant that will kill 50% of the animals by a specific time, t; as well as LT50 (d), the time to 50% mortalities for a specific concentration, d, is then formulated in closed form from the newly proposed dose-time-response model. Finally, the newly proposed model is considered for a real data set to demonstrate the application.