Optimal designs for dose-response models with restricted design spaces

In dose-response studies, the dose range is often restricted due to concerns over drug toxicity and/or efficacy. We derive optimal designs for estimating the underlying dose-response curve for a restricted or unrestricted dose range with respect to a broad class of optimality criteria. The underlying curve belongs to a diversified set of link functions suitable for the dose response studies and having a common canonical form. These include the fundamental binary response models -- the logit and the probit as well as the skewed versions of these models. Our methodology is based on a new geometric interpretation of optimal designs with respect to Kiefer's $\Phi_p$-criteria in regression models with two parameters, which is of independent interest. It provides an intuitive illustration of the number and locations of the support points of $\Phi_p$-optimal designs. Moreover, the geometric results generalize the classical characterization of $D$-optimal designs by the minimum covering ellipsoid [see Silvey (1972) or Sibson (1972)] to the class of Kiefer's $\Phi_p$-criteria. The results are illustrated through the re-design of a dose ranging trial