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The increase of exposure to toxic materials and hazardous chemicals is a major concern due to the adverse effect on human health. Among the major concerns of toxicologists is to determine acceptable levels of exposure to hazardous substances. Current approaches often evaluate each endpoint and stressor individually. We propose a novel method to simultaneously determine the Benchmark Dose Tolerable Region (BMDTR) for multiple endpoint and multiple stressor studies by adopting a Bayesian approach. A main concern while assessing the combined toxicological effect of a chemical mixture is the anticipated type of the combined action (i.e. synergistic or antagonistic); thus it was essential to account for interaction effects to handle this situation, imposing more challenges due to the non-linearity of the tolerable region. The proposed method will be evaluated using two approaches, the first one using the estimated value of the posterior median and the second approach using all MCMC samples from the posterior distribution. Furthermore, we propose a new method to determine the endpoint probabilities for each endpoint, which reflects the importance of each endpoint in helping determining the boundaries of the benchmark dose tolerable region (BMDTR). [ABSTRACT FROM AUTHOR]

Bayesian inclusion probabilities have become a popular tool for variable assessment. From a frequentist perspective, it is often difficult to evaluate these probabilities as typically no Type I error rates are considered, neither are any explorations of power of the methods given. This paper considers how a frequentist may evaluate Bayesian inclusion probabilities for screening predictors. This evaluation looks at both unrestricted and restricted model spaces and develops a framework which a frequentist can utilize inclusion probabilities that preserve Type I error rates. Furthermore, this framework is applied to an analysis of the Arabidopsis thaliana with respect to determining quantitative trait loci associated with cotelydon opening angle. [ABSTRACT FROM AUTHOR]