c-Optimal Designs for the Bivariate Emax Model

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.

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

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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

$$\biggl( \frac{32(s+1)^4}{16s^2+(s+1)^4} \biggr) ^2 \frac{sx^2}{4} \biggl( \frac{1}{(x+1)^4}+\frac{s^2}{(x+s)^4} \biggr) \leq \frac{16s(s+1)^4}{16s^2+(s+1)^4},\quad\forall x \in\chi, $$

which is equivalent to

$$ \frac{16(s+1)^4}{16s^2+(s+1)^4} x^2 \biggl( \frac{1}{(x+1)^4}+ \frac{s^2}{(x+s)^4} \biggr) \leq1,\quad\forall x \in \chi. $$

(8)

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→∞. □