Table 4.

Optimal allocation matrices for cross‐sectional designs with D = 3. Several optimal allocation matrices in the case $\mathfrak{I}=\{2,\text{…},6\}$, $\mathfrak{C}=\{{\mathfrak{C}}_2,\text{…},{\mathfrak{C}}_6\}$, ${\mathfrak{C}}_T=\{2,\text{…},6\}$, ${\mathfrak{M}}_{C,T}=\{2,\text{…},\lfloor 48/T\rfloor \}$, σ ² = 1, ρ = 0.05, α = 0.05 with the Bonferroni correction, and β = 0.12 for the individual power when δ=(1.5σ,0.75σ)^⊤ are shown. Specifically, the optimal design for the optimality criteria is given for w ∈ {0,0.5}. Restrictions are placed on $\mathfrak{X}$ such that Equation (1) is identifiable and that each cluster must receive each of the interventions. Each allocation matrix was identified via our exhaustive search method. The utilized design is also shown for comparison

	Design
Factor	Proposed	D‐Optimal: w  = 0	D‐Optimal: w  = 0.5	A/E‐Optimal: w  = 0	A/E‐Optimal: w  = 0.5
C	6	6	6	6	6
T	6	6	6	6	6
m	8	8	5	8	5
X	$\left(\begin{array}{cccccc}0& 0& 0& 1& 1& 2\\ 0& 0& 0& 1& 1& 2\\ 0& 0& 1& 1& 2& 2\\ 0& 0& 1& 1& 2& 2\\ 0& 1& 1& 2& 2& 2\\ 0& 1& 1& 2& 2& 2\end{array}\right)$	$\left(\begin{array}{cccccc}0& 0& 0& 0& 1& 2\\ 0& 0& 0& 0& 1& 2\\ 0& 0& 0& 1& 2& 2\\ 0& 0& 1& 2& 2& 2\\ 0& 1& 2& 2& 2& 2\\ 0& 1& 2& 2& 2& 2\end{array}\right)$	$\left(\begin{array}{cccccc}0& 0& 0& 0& 1& 2\\ 0& 0& 0& 0& 1& 2\\ 0& 0& 1& 1& 2& 2\\ 0& 1& 1& 2& 2& 2\\ 0& 1& 2& 2& 2& 2\\ 0& 1& 2& 2& 2& 2\end{array}\right)$	$\left(\begin{array}{cccccc}0& 0& 0& 0& 1& 2\\ 0& 0& 0& 0& 1& 2\\ 0& 0& 1& 1& 2& 2\\ 0& 0& 1& 2& 2& 2\\ 0& 1& 2& 2& 2& 2\\ 0& 1& 2& 2& 2& 2\end{array}\right)$	$\left(\begin{array}{cccccc}0& 0& 0& 0& 1& 2\\ 0& 0& 0& 0& 1& 2\\ 0& 0& 1& 1& 2& 2\\ 0& 0& 1& 2& 2& 2\\ 0& 1& 2& 2& 2& 2\\ 0& 1& 2& 2& 2& 2\end{array}\right)$
$\mathbb{P}(\text{Reject}{H}_{01}|{\delta }_1)$	1.0000	1.0000 (±0%)	1.0000 (±0%)	1.0000 (±0%)	1.0000 (±0%)
$\mathbb{P}(\text{Reject}{H}_{02}|{\delta }_2)$	0.8815	0.9528 (+8.1%)	0.8507 (−3.5%)	0.9570 (+8.6%)	0.8440 (−4.3%)
$f(𝒟)$	288	288 (±0%)	180 (−37.5%)	288 (±0%)	180 (−37.5%)
$\det ({\mathrm{\Lambda }}_{𝒟})$	3.090 × 10−3	1.670 × 10−3 (−46.0%)	3.881 × 10−3 (+25.6%)	1.712 × 10−3 (−44.6%)	3.973 × 10−3 (+25.6%)
${(D-1)}^{-1}\text{tr}({\mathrm{\Lambda }}_{𝒟})$	5.696 × 10−2	4.264 × 10−2 (−25.1%)	6.392 × 10−2 (+12.2%)	4.160 × 10−2 (−27.0%)	6.373 × 10−2 (+11.9%)
$\max \mathrm{Diag}({\mathrm{\Lambda }}_{𝒟})$	5.696 × 10−2	4.264 × 10−2 (−25.1%)	6.531 × 10−2 (+14.7%)	4.160 × 10−2 (−27.0%)	6.373 × 10−2 (+11.9%)