Table 6.

Optimal allocation matrices for cross‐sectional designs with D = 4. Several optimal allocation matrices in the case $\mathfrak{I}=\{8\}$, $\mathfrak{C}=\{{\mathfrak{C}}_6\}=\{8\}$, ${\mathfrak{M}}_{6,8}=\{8\}$, σ ² = 1, ρ = 0.05, α = 0.05 with the Bonferroni correction, w = 0, and β = 0.12 for the individual power when δ=(1.5σ,0.75σ,0.75σ)^⊤ are shown. No restrictions are placed on $\mathfrak{X}$ other than the identifiability of Equation (1). Each allocation matrix was identified via our stochastic search method. The proposed design is also shown for comparison

	Design
Factor	Proposed	D‐optimal	A‐optimal	E‐optimal
X	$\left(\begin{array}{cccccccc}0& 0& 0& 1& 1& 2& 2& 3\\ 0& 0& 0& 1& 1& 2& 2& 3\\ 0& 0& 1& 1& 2& 2& 3& 3\\ 0& 0& 1& 1& 2& 2& 3& 3\\ 0& 1& 1& 2& 2& 3& 3& 3\\ 0& 1& 1& 2& 2& 3& 3& 3\end{array}\right)$	$\left(\begin{array}{cccccccc}0& 0& 0& 0& 0& 0& 1& 1\\ 0& 0& 0& 0& 1& 1& 2& 3\\ 0& 0& 1& 2& 2& 3& 3& 3\\ 0& 1& 1& 1& 1& 2& 2& 2\\ 1& 2& 2& 2& 3& 3& 3& 3\\ 2& 2& 3& 3& 3& 3& 3& 3\end{array}\right)$	$\left(\begin{array}{cccccccc}0& 0& 0& 0& 0& 0& 1& 1\\ 0& 0& 0& 0& 1& 1& 3& 3\\ 0& 0& 1& 1& 1& 2& 2& 2\\ 1& 1& 1& 1& 2& 2& 2& 2\\ 1& 1& 2& 2& 2& 3& 3& 3\\ 2& 2& 2& 3& 3& 3& 3& 3\end{array}\right)$	$\left(\begin{array}{cccccccc}0& 0& 0& 0& 0& 0& 1& 3\\ 0& 0& 0& 0& 1& 1& 2& 2\\ 0& 0& 0& 1& 1& 3& 3& 3\\ 1& 1& 1& 1& 2& 2& 2& 2\\ 1& 1& 2& 2& 3& 3& 3& 3\\ 2& 2& 3& 3& 3& 3& 3& 3\end{array}\right)$
$\mathbb{P}(\text{Reject}{H}_{01}|{\delta }_1)$	1.000	1.000 (±0%)	1.000 (±0%)	1.000 (±0%)
$\mathbb{P}(\text{Reject}{H}_{02}|{\delta }_2)$	0.852	0.992 (+11.6%)	0.996 (+11.7%)	0.989 (+11.6%)
$\mathbb{P}(\text{Reject}{H}_{03}|{\delta }_3)$	0.852	0.990 (+11.6%)	0.984 (+11.6%)	0.989 (+11.6%)
$\det ({\mathrm{\Lambda }}_{𝒟})$	1.559 × 10−4	1.985 × 10−5 (−87.3%)	2.108 × 10−5 (−86.5%)	2.090 × 10−5 (−86.6%)
${(D-1)}^{-1}\text{tr}({\mathrm{\Lambda }}_{𝒟})$	5.590 × 10−2	2.873 × 10−2 (−48.6%)	2.806 × 10−2 (−49.8%)	2.886 × 10−2 (−48.4%)
$\max \mathrm{Diag}({\mathrm{\Lambda }}_{𝒟})$	5.590 × 10−2	3.024 × 10−2 (−45.9%)	3.085 × 10−2 (−44.8%)	2.893 × 10−2 (−48.2%)