Abstract
This paper explores c-optimal design problems for non-linear, bivariate response models. The focus is on bivariate dose response models, one response being a primary efficacy variable and the other a primary safety variable. The aim is to construct designs that are optimal for estimating the dose that gives the best possible combination of effects and side-effects.
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Acknowledgements
I am most grateful for all the support and encouragement from Prof. Hans Nyquist. I also want to express my gratitude to Prof. Ziad Taib and a referee for valuable comments, as well as to Carl-Fredrik Burman for help with formulating the problem.
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Appendix
Appendix
Lemma 1
(Pázman 1986)
Let ϕ(x,ξ) stand for the derivative of Ψ in the direction ξ x and let Ψ be a general criterion function to be minimized. A design, ξ, is locally optimal with respect to Ψ if and only if ϕ(x,ξ)≥0, ∀x∈χ. This further implies that ϕ(x,ξ)=0 for x∈{x 1,…,x n }.
Proof of Theorem 1
First note that the directional derivative can be written of the form \(\phi(x,\xi) = \mathrm{tr} ( \frac{\partial\varPsi}{\partial M(\xi)} ( M(\xi_{x}) - M(\xi)) )\) (Pázman 1986). We have
From the above we get
Now
□
Proof of Corollary 1
(i) Let s:=SD 50/ED 50>1, θ s =(1,s) and \(\xi_{s}=\{\sqrt {s};1\}\). Theorem 1 gives that ξ 2 is locally c-optimal iff \(f(x):=x^{2}{ (}\frac{1}{s(x+1)^{4}}+\frac{s}{(x+s)^{4}}{ )} \leq \frac{2}{(\sqrt{s}+1)^{4}},\:\forall x \in\chi\). Equality holds when \(x=\sqrt{s}\) and it is easy to show that this point is a local maximum given that \(s \in ]1 , \frac{7+3\sqrt{5}}{2} [\) (\(f'(\sqrt{s})=0 \) ∀s∈χ and \(f''(\sqrt{s}) < 0 \Leftrightarrow s \in ] \frac{7-3\sqrt{5}}{2} , \frac{7+3\sqrt{5}}{2} [\)). Now \(f'(x)=2x(x-\sqrt{s})(x+\sqrt {s})g(x)/((x+s)^{5}(x+1)^{5})\), where g(x) is a polynomial of degree 4. If \(s \in ]1,\frac{5+\sqrt{21}}{2} ]\) then all coefficients of g(x) are negative and hence \(x=\sqrt{s}\) is a global maximum. This means that ξ s is locally c-optimal for this interval and thus \(\xi=\{\mathit{ED}_{50}\sqrt{s};1 \}=\{\sqrt{\mathit{ED}_{50}\mathit{SD}_{50}};1 \}\) is locally c-optimal for the model with θ=ED 50 θ s =(ED 50,SD 50) given that \({\mathit{SD}_{50}}/{\mathit{ED}_{50}} \in ]1,\frac{5+\sqrt{21}}{2} ]\). (ii) Let s:=SD 50/ED 50, θ s =(1,s) and ξ s ={1,s;0.5,0.5}. Theorem 1 implies that ξ 2 is locally c-optimal iff
which is equivalent to
The left-hand side of (8) tends to 16x 2/(x+1)4 as s→∞. Finally, 16x 2/(x+1)4≤1,∀x∈χ. This gives that ξ s is locally c-optimal and thus ξ={ED 50,ED 50 s;0.5,0.5} ={ED 50,SD 50;0.5,0.5} is locally c-optimal for the model with θ= (ED 50,ED 50 s) =(ED 50,SD 50) as SD 50/ED 50→∞. □
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Magnusdottir, B.T. (2013). c-Optimal Designs for the Bivariate Emax Model. In: Ucinski, D., Atkinson, A., Patan, M. (eds) mODa 10 – Advances in Model-Oriented Design and Analysis. Contributions to Statistics. Springer, Heidelberg. https://doi.org/10.1007/978-3-319-00218-7_18
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