Estimating Potency for Hierarchical Dichotomous Responses in an Aquatic Toxicology Study

Toxicants in aquatic organisms may impact a variety of responses including hatching, survival, growth, deformity, and reproduction. These responses have a hierarchical relationship since survival is only possible for hatched organisms and deformity is only possible if organisms survive. Linear models and generalized linear models have been widely used to analyze individual responses; however, an analysis that simultaneously considers the structure of all responses would allow a more comprehensive assessment of toxicity. Ignoring the hierarchy of these responses and applying separate modeling approaches could be misleading. Motivated by a study of hatching–survival–deformities in fish, we developed a method to model these responses simultaneously. In the present study, we propose a Bayesian multinomial regression model to analyze the hierarchical toxicity responses simultaneously and estimate the concentrations associated with a specified level of reproductive inhibition (RI _p ) on hatching success, post hatching survival and deformity, respectively. Simulation studies are conducted to compare the estimates from the hierarchical model with those from the response-specific separate models. The proposed model outperforms the separate model by producing more precise point estimates and interval estimates with better coverage probabilities. The proposed method is illustrated with data from a study of hatching–survival–deformities in fish. The proposed model captures the hierarchy in the multiple responses from animals in the toxicity test and hence improves the understanding of concentration–response relationships for these responses. It provides a useful template of modeling hierarchical dichotomous responses in the aquatic toxicity studies.