## Optimal designs for dose response curves with common parameters.(English)Zbl 1421.62149

Consider the dose-response parametric regression models $f(d,\theta_0,\theta_1,\theta_2^{(i)})=\theta_0+\frac{\theta_1 d}{\theta_2^{(i)}+d}, \quad i=1,2.$ Here, $$d \in [0, d_{\max}^{(i)}]$$ is the dose of the drug ($$d_{\max}^{(i)}\gg 1$$), $$\theta_0$$, $$\theta_1$$, $$\theta_2^{(i)}$$ are unknown parameters and $$f(d,\theta_0,\theta_1,\theta_2^{(i)})$$ is the clinical effect of the drug in group $$i$$. It is assumed that both groups ($$i=1,2$$) differ only in the administration frequencies but not in the sort of the drug, e.g., monthly and weekly administrations are considered. Hence, the parameter $$\theta_0$$, which describes the placebo effect at $$d=0$$, and the parameter $$\theta_1$$, which describes the biological maximum attainable effect $$\theta_0+\theta_1$$ at $$d=d_{\max}$$, are assumed to be common in both models, while $$\theta_2^{(1)}$$, $$\theta_2^{(2)}$$ are considered to be different.
A design is a vector $$(\xi_1,\xi_2,\mu)$$, where $$\xi_{i}$$ is a probability measure with masses $$\xi_{ij}$$ at the experimental conditions $$d_j^{(i)}\in [0,d_{\max}^{(i)}]$$, $$i=1,2$$, and $$\mu$$ is a probability measure on $$i=1,2$$ describing the choice of models. This allows one to introduce the information matrix and to define the $$D$$-optimal design.
In the article, dose-response models with common parameters and optimal designs in a general setting are considered. Upper bounds on the number of support points of admissible designs are derived. For $$D$$-optimal designs minimally supported designs are determined and sufficient conditions for their optimality are derived. The results are illustrated by examples.

### MSC:

 62P10 Applications of statistics to biology and medical sciences; meta analysis 62K05 Optimal statistical designs 62F03 Parametric hypothesis testing 62J02 General nonlinear regression 62C15 Admissibility in statistical decision theory
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