Simultaneous confidence bands for log-logistic regression with applications in risk assessment

In risk assessment, it is often desired to make inferences on the low dose levels at which a specific benchmark risk is attained. Applications of simultaneous hyperbolic confidence bands for low-dose risk estimation with quantal data under different dose-response models (multistage, Abbott-adjusted Weibull, and Abbott-adjusted log-logistic models) have appeared in the literature. The use of simultaneous three-segment bands under the multistage model has also been proposed recently. In this article, we present explicit formulas for constructing asymptotic one-sided simultaneous hyperbolic and three-segment bands for the simple log-logistic regression model. We use the simultaneous construction to estimate upper hyperbolic and three-segment confidence bands on extra risk and to obtain lower limits on the benchmark dose by inverting the upper bands on risk under the Abbott-adjusted log-logistic model. Monte Carlo simulations evaluate the characteristics of the simultaneous limits. An example is given to illustrate the use of the proposed methods and to compare the two types of simultaneous limits at very low dose levels.