Table II.

Optimal designs for the linear regression and E_max model for various optimality criteria. The quantity Δ* is defined by Δ* = ϑ₁ϑ₂ (d̄ − ḏ)/{2(ḏ + ϑ₂)(ḏ + ϑ₂)}, while the quantity w(ϑ₂) is given by $w({\vartheta }_2)=1/4-(\overline{d}-\underset{¯}{d}){\vartheta }_2{\vartheta }_1/\{8[(\overline{d}-\underset{¯}{d}){\vartheta }_2{\vartheta }_1+\mathrm{\Delta }{\vartheta }_2(\underset{¯}{d}+\overline{d})+\mathrm{\Delta }({\vartheta }_2^2+\overline{d}\underset{¯}{d})]\}$.

	linear	variance	Emax	variance
D	$\left(\begin{array}{cc}\underset{¯}{d}& \overline{d}\\ \frac{1}{2}& \frac{1}{2}\end{array}\right)$	n.a.	$\left(\begin{array}{ccc}\underset{¯}{d}& \frac{(\overline{d}+{\vartheta }_2)\underset{¯}{d}+(\underset{¯}{d}+{\vartheta }_2)\overline{d}}{2{\vartheta }_2+\underset{¯}{d}+\overline{d}}& \overline{d}\\ \frac{1}{3}& \frac{1}{3}& \frac{1}{3}\end{array}\right)$	n.a.
EDp	any	0	$\left(\begin{array}{ccc}\underset{¯}{d}& \frac{(\overline{d}+{\vartheta }_2)\underset{¯}{d}+(\underset{¯}{d}+{\vartheta }_2)\overline{d}}{2{\vartheta }_2+\underset{¯}{d}+\overline{d}}& \overline{d}\\ \frac{1}{4}& \frac{1}{2}& \frac{1}{4}\end{array}\right)$	$\frac{{\sigma }^2}{n}{\left(\frac{8p(1-p){({\vartheta }_2+\underset{¯}{d})}^2{({\vartheta }_2+\overline{d})}^2}{{\vartheta }_1{\vartheta }_2{({\vartheta }_2+p\underset{¯}{d}+(1-p)\overline{d})}^2}\right)}^2$
MED Δ < Δ*	$\left(\begin{array}{cc}\underset{¯}{d}& \overline{d}\\ \frac{1}{2}& \frac{1}{2}\end{array}\right)$	$\frac{2{\sigma }^2{\mathrm{\Delta }}^2}{n{\vartheta }_1^4({\underset{¯}{d}}^2+{\overline{d}}^2)}$	$\left(\begin{array}{ccc}\underset{¯}{d}& \frac{(\overline{d}+{\vartheta }_2)\underset{¯}{d}+(\underset{¯}{d}+{\vartheta }_2)\overline{d}}{2{\vartheta }_2+\underset{¯}{d}+\overline{d}}& \overline{d}\\ w(\vartheta )& \frac{1}{2}& \frac{1}{2}-w(\vartheta )\end{array}\right)$	$\frac{{\sigma }^2}{n}\frac{4{\vartheta }_1^2{\vartheta }_2^2{(\underset{¯}{d}+{\vartheta }_2)}^4}{{({\vartheta }_1{\vartheta }_2-\mathrm{\Delta }(\underset{¯}{d}+{\vartheta }_2))}^4}$
MED Δ > Δ*	$\left(\begin{array}{cc}\underset{¯}{d}& \overline{d}\\ \frac{1}{2}& \frac{1}{2}\end{array}\right)$	$\frac{2{\sigma }^2{\mathrm{\Delta }}^2}{n{\vartheta }_1^4({\underset{¯}{d}}^2+{\overline{d}}^2)}$	$\left(\begin{array}{cc}\underset{¯}{d}& \frac{{\vartheta }_2(\mathrm{\Delta }{\vartheta }_2+(\mathrm{\Delta }+{\vartheta }_1)\underset{¯}{d}}{{\vartheta }_2{\vartheta }_1-\mathrm{\Delta }(\underset{¯}{d}+{\vartheta }_2)}\\ \frac{1}{2}& \frac{1}{2}\end{array}\right)$	$\frac{8{\sigma }^2}{n}\frac{{\mathrm{\Delta }}^2{(\underset{¯}{d}+{\vartheta }_2)}^6{(\overline{d}+{\vartheta }_2)}^2{\{(\underset{¯}{d}-\overline{d}){\vartheta }_1{\vartheta }_2+\mathrm{\Delta }({\vartheta }_2+\underset{¯}{d})({\vartheta }_2+\overline{d})\}}^2}{{\vartheta }_2^2{\vartheta }_2^4{(\overline{d}-\underset{¯}{d})}^4{(\mathrm{\Delta }(\underset{¯}{d}+{\vartheta }_2)-{\vartheta }_1{\vartheta }_2)}^2}$

n.a. = not available