Statistical approaches to detect and estimate hormesis

Hormesis is described as a dose response relationship that is stimulatory at low doses, but inhibitory at higher doses in dose-response experiments. In order to be able to incorporate hormesis in quantitative risk assessment and regulatory decision-making activities, hormesis must be quantitatively defined and statistically tested. A non-parameric rank test and a model-based approach are proposed to detect hormesis. In non-parametric rank test, the Mark-Wolfe's procedure is applied to test the null hypothesis H₀: μ ₁ = … = μ_k vs. the alternative hormesis hypothesis (or umbrella hypothesis) H_A2: there exists an l, 1 < l < k, such that μ1 ≤ … ≤ μ_l ≥ … ≥ μ_k. In model-based approach, we use the full model to describe the dose-response relationship incorporating the hormetic effect and the reduced model to describe the dose-response relationship without the hormetic effect. The full model has an extra parameter which could measure the amount of increase in response at low doses. A test of significance of this extra parameter is essentially a test for hormesis. Under the model-based approach, the ratio of the area under the hormetic zone (AUC_H) and the area under the best-fitted curve from zero to zero equivalent point (AUC_ZEP) is proposed to estimate the magnitude of the hormetic effect. Finally, to incorporate the hormetic effect into the estimation of no-observed-effect-concentration (NOEC), a Stepwise Practical Equivalence method is proposed. The stepwise practical equivalence test is more powerful and strongly controls the probability of falsely identifying a toxic concentration as the NOEC. It is robust to the hormetic and interrupted dose-response relationships.