Table 1. Open in new tab Simulation...

Table 1.

Simulation scenarios considered

	$f (d, z)$	$σ_{y}$	$z_{1}$	$z_{2}$	$d_{opt}$	$f_{opt}$	ses	monotone
Simulation Study	1) $- 1 {z_{1} = 0} \times g_{1} (d) -$	2.015	0		$(1, 1)$	$-$ 1.592	0.79	Yes
	$1 {z_{1} = 1} \times g_{1} (d)$		1		$(1, 1)$	$-$ 1.592	0.79	Yes
	2) $- 1 {z_{1} = 0} \times g_{2} (d) -$	0.319	0		$(0.25, 0.75)$	$-$ 1.203	3.77	No
	$1 {z_{1} = 1} \times g_{3} (d)$		1		$(0.75, 0.25)$	$-$ 1.203	3.77	No
	3) $- 1 {(z_{1} = 0, z_{2} = 0)} \times 0 -$	1	0	0	None	None	0	No
	$1 {(z_{1} = 0, z_{2} = 1)} \times 0.831 g_{2} (d) -$		0	1	$(0.25, 0.75)$	$-$ 1	1	No
	$1 {(z_{1} = 1, z_{2} = 0)} \times 3.134 g_{3} (d) -$		1	0	$(0.75, 0.25)$	$-$ 3.77	3.77	No
	$1 {(z_{1} = 1, z_{2} = 1)} \times 0.496 g_{1} (d)$		1	1	$(1, 1)$	$-$ 0.79	0.79	Yes
Implant	1) $- 1 (z_{1} = 0) \times 2.49 g_{2} (d) -$	5	0		$(0.25, 0.75)$	$-$ 5	1	No
	$1 (z_{1} = 1) \times 6.65 g_{3} (d) - 2$		1		$(0.75, 0.25)$	$-$ 10	2	No

	$f (d, z)$	$σ_{y}$	$z_{1}$	$z_{2}$	$d_{opt}$	$f_{opt}$	ses	monotone
Simulation Study	1) $- 1 {z_{1} = 0} \times g_{1} (d) -$	2.015	0		$(1, 1)$	$-$ 1.592	0.79	Yes
	$1 {z_{1} = 1} \times g_{1} (d)$		1		$(1, 1)$	$-$ 1.592	0.79	Yes
	2) $- 1 {z_{1} = 0} \times g_{2} (d) -$	0.319	0		$(0.25, 0.75)$	$-$ 1.203	3.77	No
	$1 {z_{1} = 1} \times g_{3} (d)$		1		$(0.75, 0.25)$	$-$ 1.203	3.77	No
	3) $- 1 {(z_{1} = 0, z_{2} = 0)} \times 0 -$	1	0	0	None	None	0	No
	$1 {(z_{1} = 0, z_{2} = 1)} \times 0.831 g_{2} (d) -$		0	1	$(0.25, 0.75)$	$-$ 1	1	No
	$1 {(z_{1} = 1, z_{2} = 0)} \times 3.134 g_{3} (d) -$		1	0	$(0.75, 0.25)$	$-$ 3.77	3.77	No
	$1 {(z_{1} = 1, z_{2} = 1)} \times 0.496 g_{1} (d)$		1	1	$(1, 1)$	$-$ 0.79	0.79	Yes
Implant	1) $- 1 (z_{1} = 0) \times 2.49 g_{2} (d) -$	5	0		$(0.25, 0.75)$	$-$ 5	1	No
	$1 (z_{1} = 1) \times 6.65 g_{3} (d) - 2$		1		$(0.75, 0.25)$	$-$ 10	2	No

Note. The data-generating mechanism for each scenario is $y = f (d, z) + ϵ$ where $ϵ \sim N (0, σ_{y}^{2})$ ⁠. The table columns contain the location of the optimal dose combination (⁠ $d_{opt}$ ⁠), the optimal value of the efficacy function (⁠ $f_{opt}$ ⁠), the standardized effect size (ses), and whether or not the dose-efficacy surface is monotonically increasing with respect to each dosing dimension (monotone). Note that $g_{i} (d)$ for $1 = 1, 2, 3$ represents the value of a bivariate normal density function with specific mean vector and covariance matrix evaluated at $d$ as defined in the text. The subtraction of 2 in $f (d, z)$ under the Implant scenario corresponds to a base level of drug response outside the regions of optimality.

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