Adaptive sequential design for phase II single-arm oncology trials: an expansion of Simon’s design

Figures & data

Figure 1. Interim analysis using DACT.

Figure 2. Necessity of continuity correction in a two-stage design.

Figure 3. Necessity of continuity correction in a three-stage design.

Table 1. Notations.

Parameters	Notations
	Simon’s design	The hybrid design		The midpoint design	The three-stage design
Design	$\mathit{S}\left(p_0,\mathit{p},\mathit{\alpha },\mathit{\beta }\right)$
$r_1$	$r_1^{Si\mathrm{mon}:opt}\left(p_0,\mathit{p}\right)$	$r_1^{hy\mathrm{brid}}$		$r_1^{mi\mathrm{dpoint}}$	$r_1^{3-st\mathrm{age}}$
$r_{1,\mathit{f}}$		$r_{1,f}^{hy\mathrm{brid}}$		$r_{1,f}^{mi\mathrm{dpoint}}$	$r_{1,f}^{3-st\mathrm{age}}$
$r_{1,\mathit{e}}$		$r_{1,e}^{hy\mathrm{brid}}$		$r_{1,e}^{mi\mathrm{dpoint}}$	$r_{1,e}^{3-st\mathrm{age}}$
$n_1$	$n_1^{Si\mathrm{mon}:opt}\left(p_0,\mathit{p}\right)$	$n_1^{hy\mathrm{brid}}$		$n_1^{mi\mathrm{dpoint}}$	$n_1^{3-st\mathrm{age}}$
$r_2$	$r_2^{Si\mathrm{mon}:opt}1\left(p_0,\mathit{p}\right)$	$r_2^{hy\mathrm{brid}}$		$r_2^{mi\mathrm{dpoint}}$	$r_2^{3-st\mathrm{age}}$
$r_{2,\mathit{f}}$		$r_{2,f}^{hy\mathrm{brid}}$		$r_{2,f}^{mi\mathrm{dpoint}}$	$r_{2,f}^{3-st\mathrm{age}}$
$r_{2,\mathit{e}}$		$r_{2,e}^{hy\mathrm{brid}}$		$r_{2,e}^{mi\mathrm{dpoint}}$	$r_{2,e}^{3-st\mathrm{age}}$
$n_2$	$n_2^{Si\mathrm{mon}:opt}\left(p_0,\mathit{p}\right)$	$n_2^{hy\mathrm{brid}}$		$n_2^{mi\mathrm{dpoint}}$	$n_2^{3-st\mathrm{age}}$
$\mathit{PET}\left(\mathit{p}\right)$	$\mathit{PE}{T}^{\mathit{Simon}}\left(p_0,\mathit{p}\right)$	$\mathit{PE}{T}^{\mathit{hybrid}}\left(p_0\right)$		$\mathit{PE}{T}^{\mathit{midpoint}}\left(p_0\right)$	$\mathit{PE}{T}^{3-\mathit{stage}}\left(p_0\right)$
$\mathit{E}\left(p_0\right)$	$\mathit{E}{N}^{\mathit{Simon}}\left(p_0,\mathit{p}\right)$	$\mathit{E}{N}^{\mathit{hybrid}}\left(p_0\right)$		$\mathit{E}{N}^{\mathit{midpoint}}\left(p_0\right)$	$\mathit{E}{N}^{3-\mathit{stage}}\left(p_0\right)$
$n_1$: stage 1 sample size $r_1$: stage 1 futility criterion $r_{1,\mathit{f}}$: stage 1 futility criterion $r_{1,\mathit{e}}$: stage 1 efficacy criterion $n_2$: Sample size at end of stage 2 $r_2$: stage 2 efficacy criterion in a two-stage design			$r_{2,\mathit{e}}$: stage 2 efficacy criterion $n_3$: Sample size at end of stage 3$r_{2,\mathit{f}}$: stage 2 futility criterion $r_3$: stage 3 efficacy criterion $r_1^{Si\mathrm{mon}:opt}\left(p_0,\mathit{p}\right),\mathit{or}{r}_1^{Si\mathrm{mon}:mi\mathrm{nimax}}\left(p_0,\mathit{p}\right)$ means the $r_1$ from the optimal or minimax design

Table 2. Comparing the average design and Jung’s admissible design.

$\left(p_0,p_1,\mathit{\alpha },\mathit{\beta }\right)=\left(0.1,\mathit{\hspace{0.17em}}0.3,\mathit{\hspace{0.17em}}0.05,\mathit{\hspace{0.17em}}0.15\right)$
	$r_1$	$n_1$	$r_2$	$n_2$	$\mathit{err}$	$\mathit{EN}\left(p_0\right)$	$\mathit{PW}\left(p_1\right)$
minimax	2	18	5	27	0.0444	20.4	0.851
optimal	1	11	6	35	0.0422	18.3	0.851
Average	2	15	6	36	0.0433	18.9	0.852
Jung’s admissible	1	13	5	28	0.0498	18.7	0.858
$\left(p_0,p_1,\mathit{\alpha },\mathit{\beta }\right)=\left(0.05,\mathit{\hspace{0.17em}}0.25,\mathit{\hspace{0.17em}}0.05,\mathit{\hspace{0.17em}}0.1\right)$
	$r_1$	$n_1$	$r_2$	$n_2$	$\mathit{err}$	$\mathit{EN}\left(p_0\right)$	$\mathit{PW}\left(p_1\right)$
minimax	0	15	3	25	0.0336	20.4	0.901
optimal	0	9	3	30	0.0489	16.8	0.902
Average	0	12	3	26	0.0365	18.4	0.905
Jung’s admissible 1	0	12	3	26	0.0365	18.4	0.905
Jung’s admissible 2	0	10	3	28	0.0426	17.2	0.906

Figure 4. Conducting final analysis using DACT.

Table 3. Simon’s two-stage design with augmentation (fixed sample).

$\mathit{S}\left(p_0,p_{\mathit{low}},\mathit{\alpha },\mathit{\beta }\right)$
	$r_1$	$n_1$	$r_2$	$n_2$	$\mathit{err}$	$\mathit{EN}\left(p_0\right)$	$\mathit{PET}\left(p_{\mathit{low}}\right)$	$\mathit{PW}\left(p_1\right)$	$\mathit{PW}\left(p_{\mathit{low}}\right)$
minimax	25	100	32	119	0.0248	101.7	0.053	0.900	0.900
optimal	11	50	36	137	0.0244	75.2	0.063	0.902	0.902
Average	18	75	34	128	0.0238	83.3	0.060	0.900	0.900
$\mathit{S}\left(p_0,p_1,\mathit{\alpha },\mathit{\beta }\right)$
minimax	9	37	16	53	0.0248	40.1	0.172	0.900	0.596
optimal	5	22	19	66	0.0242	33.8	0.216	0.904	0.632
Average	7	30	17	57	0.0236	36.5	0.176	0.900	0.607

Table 4. The expanded designs (fixed sample).

	$r_{1,\mathit{f}}$		$n_1$	$r_{1,\mathit{e}}$	$r_2$	$n_2$	$\mathit{err}$	$\mathit{EN}\left(p_0\right)$	$\mathit{PET}\left(p_{\mathit{low}}\right)$		$\mathit{PW}\left(p_1\right)$	$\mathit{PW}\left(p_{\mathit{low}}\right)$
Two-stage: The hybrid design: $\mathit{min}\text{(}\mathit{\Delta }{n}_{12}\text{)}=5$, $\left({\alpha }_1,{\alpha }_2\right)=\left(0.001,0.024\right)$
Stage 1 minimax	25		100	33	29	105	0.0223	100.4	0.053		0.994	0.858
Stage 1 optimal	11		50	NA	17	56	0.0219	51.7	0.063		0.911	0.604
Stage 1 Average	18		75	26	23	80	0.0217	75.8	0.060		0.976	0.752
Two-stage: The midpoint design: $\mathit{\rho }=0.5$. $\mathit{min}\left(\mathrm{\Delta }{n}_{12}\right)=5$, $\left({\alpha }_1,{\alpha }_2\right)=\left(0.001,\mathrm{.0.024}\right)$
Stage 1 minimax	17		69	25	22	74	0.0159	69.7	0.086		0.956	0.678
Stage 1 optimal	7		36	NA	17	56	0.0218	44.7	0.056		0.910	0.602
Stage 1 Average	12		53	NA	18	59	0.0185	54.5	0.069		0.914	0.600
	$r_{1,\mathit{f}}$	$n_1$	$r_2$	$r_{2,\mathit{e}}$	$n_2$	$r_3$	$n_3$	$\mathit{err}$	$\mathit{EN}\left(p_0\right)$	$\mathit{PET}\left(p_{\mathit{low}}\right)$	$\mathit{PW}\left(p_1\right)$	$\mathit{PW}\left(p_{\mathit{low}}\right)$
The three-stage design. $\mathit{min}\left(\mathrm{\Delta }{n}_{23}\right)=5$ , $\left({\alpha }_1,{\alpha }_2,{\alpha }_3\right)=\left(0,0.001,\mathrm{.0.024}\right)$
Stage 1 minimax	8	37	22	33	100	29	105	0.019	57.5	0.098	0.981	0.810
Stage 1 optimal	3	22	11	NA	50	17	56	0.022	42.3	0.084	0.908	0.600
Stage 1 Average	6	30	16	26	75	23	80	0.020	48.5	0.097	0.963	0.724

Table 5. Simulation on SSR with Simon’s two-stage design.

Design: The original Simon’s design $\mathit{S}\left(p_0,p_1,\mathit{\alpha },\mathit{\beta }\right)$ with $p_1=0.4$; Simulations: $\mathit{p}=p_{\mathit{low}}=0.33$
Fixed sample $\mathit{S}\left(p_0,p_1,\mathit{\alpha },\mathit{\beta }\right)$design					With SSR
	$r_1$	$n_1$	$r_2$	$n_2$	$\mathit{EN}\left(p_0\right)$	$\mathit{err}$	$\mathit{er}{r}_{\mathit{CC}}$	PET$\left(p_{\mathit{low}}\right)$	$\mathit{EN}\left(p_{\mathit{low}}\right)$	$\mathit{PW}\left(p_{\mathit{low}}\right)$	$\mathit{P}{W}_{\mathit{CC}}\left(p_{\mathit{low}}\right)$
minimax	9	37	16	53	50.7	0.0245^1	0.0201	0.171	66.5	0.727^1	0.691
optimal	5	22	19	66	42.7	0.0213^1	0.021	0.213	65.4	0.668^1	0.634
Average	7	30	17	57	48.1	0.0248^1	0.0211	0.177	66.5	0.682^1	0.662

^aContinuity corrected critical boundaries unnecessary; ²Continuity corrected critical boundaries necessary.

Table 6. Simulation SSR with the expanded two-stage designs.

Fixed sample hybrid design						With SSR
	$r_{1,\mathit{f}}$	$n_1$	$r_{1,\mathit{e}}$	$r_2$	$n_2$	$\mathit{err}$	$\mathit{er}{r}_{\mathit{CC}}$	$\mathit{EN}\left(p_0\right)$	PET$\left(p_{\mathit{low}}\right)$	$\mathit{EN}\left(p_{\mathit{low}}\right)$	$\mathit{PW}\left(p_{\mathit{low}}\right)$	$\mathit{P}{W}_{\mathit{CC}}\left(p_{\mathit{low}}\right)$
Two-stage: The hybrid design: $\mathit{min}\left(\mathrm{\Delta }{n}_{12}\right)=5$
Stage 1 minimax	25	100	33	29	105	0.0259	0.0235^2	102.3	0.054	107.3	0.90	0.881^2
Stage 1 optimal	11	50	NA	17	56	0.0273	0.0243^2	72.3	0.064	80.5	0.789	0.735^2
Stage 1 Average	18	75	26	23	80	0.0261	0.0224^2	83.1	0.062	91	0.862	0.816^2
Stage 1 minimax	17	69	25	22	74	0.0194^1	0.0173	76.7	0.086	87.7	0.807^1	0.763
Stage 1 optimal	7	36	NA	17	56	0.0234^1	0.0198	75	0.055	80.6	0.774^1	0.740
Stage 1 Average	12	53	NA	18	59	0.0223^1	0.0196	71.9	0.068	82.7	0.787^1	0.720

^aContinuity corrected critical boundaries unnecessary; ²Continuity corrected critical boundaries necessary.

Table 7. Simulation SSR with the expanded three-stage designs.

Fixed sample three-stage design								With SSR
	$r_{1,\mathit{f}}$	$n_1$	$r_2$	$r_{2,\mathit{e}}$	$n_2$	$r_3$	$n_3$	$\mathit{err}$	$\mathit{er}{r}_{\mathit{CC}}$	$\mathit{EN}\left(p_0\right)$	PET$\left(p_{\mathit{low}}\right)$	$\mathit{EN}\left(p_{\mathit{low}}\right)$	$\mathit{PW}\left(p_{\mathit{low}}\right)$	$\mathit{P}{W}_{\mathit{CC}}\left(p_{\mathit{low}}\right)$
The three-stage design. $\mathit{min}\left(\mathrm{\Delta }{n}_{23}\right)=5$
Stage 1 minimax	8	37	22	33	100	29	105	0.0223^1	0.0193	63.8	0.098	102	0.850^1	0.834
Stage 1 optimal	3	22	11	NA	50	17	56	0.0272	0.0233^2	63.3	0.084	80	0.786	0.736^2
Stage 1 Average	6	30	16	26	75	23	80	0.0247^1	0.0207	62.3	0.098	87.4	0.819^1	0.784

^aContinuity corrected critical boundaries unnecessary; ²Continuity corrected critical boundaries necessary.