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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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