| Hormesis is that decreased risk of adverse effects would occur with increased dose levels when subjects are exposed to toxin at low levels. This implies the existence of a threshold level, as hormesis, if exists, would exist below this level. Below the threshold dose level, no adverse events above the response at the control level which is called the background response occur, then background response also occurs at the threshold level. In order to describe this phenomenon, we characterize the overall dose-response relationship with a piecewise function that consists of a quadratic curve which is assumed to be U-shape curve and another quadratic curve at high dose levels. Here we use finite mixture normal models because the experimental subjects may have different responses like "more susceptible, less susceptible " after a toxicant experiment. We first use the EM algorithm to obtain the estimates of the parameters in the mixture normal models, and then discuss the model selection problems using Bayes factor method and Bootstrap confidence interval method, we adopt the rough but simple percentile method for Bootstrap confidence interval, we study the following three model selection problems: first, the model we choose is the mixture models or the single-component model; second, the overall dose-response function we choose is a piecewise function or not; last, if we choose the piecewise dose-response function, how to determine the cutpoint which is also called the threshold level. The results of the simulations will indicate that the mixture models are significantly better than the single-component population for the developmental toxicity studies, the piecewise dose-response function is significantly better than the current dose-response function which is naively assumed strictly increasing over the range of toxic doses for the hormesis, the suitable threshold level can improve the dose-response model's ability to fit the hormetic data.
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