Added risk and inverse estimation for count responses in reproductive aquatic toxicology studies

One of the experimental designs used to evaluate the toxicity of certain chemicals in aquatic organisms focuses on reproductive output. Toxic effects are manifested through a reduced level of reproduction in exposed organisms. Historically, evaluating risks in this context has focused on changes in the mean reproduction in a population of organisms. In this paper, we focus on the toxic effects at the level of the individual organism. This new method for count responses involves added risk, the probability of the production of young being suppressed below certain specified levels in individuals exposed to a particular concentration level relative to the probability of that level of suppression in control organisms. This probability serves as the basis of the individual-based risk estimation procedures. In particular, inverse estimation of the concentration associated with a specified added risk and estimates of the added risk associated with a particular concentration are discussed in the context of a negative binomial regression model. Confidence intervals are constructed for both of these quantities using the delta method. These methods are illustrated with a study of an aquatic organism, Ceriodaphnia dubia, exposed to the herbicide nitrofen.