Logistic Regression Models with Unspecified Low Dose-Response Relationships and Experimental Designs for Hormesis Studies.

Hormesis refers to a nonmonotonic (biphasic) dose-response relationship in toxicology, environmental science, and related fields. In the presence of hormesis, a low dose of a toxic agent may have a lower risk than the risk at the control dose, and the risk may increase at high doses. When the sample size is small due to practical, logistic, and ethical considerations, a parametric model may provide an efficient approach to hypothesis testing at the cost of adopting a strong assumption, which is not guaranteed to be true. In this article, we first consider alternative parameterizations based on the traditional three-parameter logistic regression. The new parameterizations attempt to provide robustness to model misspecification by allowing an unspecified dose-response relationship between the control dose and the first nonzero experimental dose. We then consider experimental designs including the uniform design (the same sample size per dose group) and the c -optimal design (minimizing the standard error of an estimator for a parameter of interest). Our simulation studies showed that (1) the c -optimal design under the traditional three-parameter logistic regression does not help reducing an inflated Type I error rate due to model misspecification, (2) it is helpful under the new parameterization with three parameters (Type I error rate is close to a fixed significance level), and (3) the new parameterization with four parameters and the c -optimal design does not reduce statistical power much while preserving the Type I error rate at a fixed significance level.