Confidence bounds for nonlinear dose-response relationships

An important aim of drug trials is to characterize the dose-response relationship of a new compound. Such a relationship can often be described by a parametric (nonlinear) function that is monotone in dose. If such a model is fitted, it is useful to know the uncertainty of the fitted curve. It is well known that Wald confidence intervals are based on linear approximations and are often unsatisfactory in nonlinear models. Apart from incorrect coverage rates, they can be unreasonable in the sense that the lower confidence limit of the difference to placebo can be negative, even when an overall test shows a significant positive effect. Bootstrap confidence intervals solve many of the problems of the Wald confidence intervals but are computationally intensive and prone to undercoverage for small sample sizes. In this work, we propose a profile likelihood approach to compute confidence intervals for the dose-response curve. These confidence bounds have better coverage than Wald intervals and are more precise and generally faster than bootstrap methods. Moreover, if monotonicity is assumed, the profile likelihood approach takes this automatically into account. The approach is illustrated using a public dataset and simulations based on the Emax and sigmoid Emax models.