Single-arm phase II three-outcome designs with handling of over-running/under-running

Figures & data

Table 1. Differences between Simon's two-stage design and the new design.

	Simon's design	New design
Setup of threshold	Some uninteresting level, $p_0$. Some interesting level, $p_1$	The minimal effective response rate $p_c$ or an interval $[p_l,p_u]$
Hypothesis setup	$H_0:p\le p_0$ vs. $H_1:p\ge p_1$	$H_0:p=p_c\mathrm{v}\mathrm{s}.H_1:p\ne p_c$ or $H_0:p\in [p_l,p_u]\mathrm{v}\mathrm{s}.H_1:p\notin [p_l,p_u]$
Sample size at interim	Fixed or limited	Flexible
Trial conclusions	Inefficacy (no-go, $p<p_1$)	Inefficacy (no-go, $p<p_c$ or $p<p_l$ ) Efficacy (go, $p>p_c$ or $p>p_u$ )

Table 2. Two-stage designs for Case 1: the minimal effective response rate is 0.40. The postulate response rate is at least 0.55. The type I errors to conclude the compound are ineffective (conclude p<0.4) when p = 0.4 and effective (conclude p>0.4) when p = 0.4 are 0.3, 0.1, respectively. The power to conclude the compound is effective (conclude p>0.4) when p = 0.55 is at least 0.80.

(a) No early stopping for efficacy.
${\alpha }_{11}$	${\alpha }_{12}$	${\alpha }_{22}$	β	$r_1$	$r_2$	$s_2$	$n_1$	$n_2$	$E\left(n\right)$
0.090	0.272	0.097	0.198	3	17	24	15	35	46.832
0.065	0.259	0.098	0.197	3	17	24	16	34	47.785
0.126	0.286	0.097	0.198	4	17	24	17	33	45.842
0.094	0.269	0.098	0.197	4	17	24	18	32	46.987
0.070	0.257	0.098	0.197	4	17	24	19	31	47.842
0.126	0.281	0.098	0.197	5	17	24	20	30	46.232
0.096	0.266	0.098	0.197	5	17	24	21	29	47.224
0.158	0.294	0.098	0.197	6	17	24	22	28	45.564
0.124	0.275	0.098	0.197	6	17	24	23	27	46.653
0.096	0.262	0.098	0.197	6	17	24	24	26	47.505
0.154	0.285	0.098	0.197	7	17	24	25	25	46.161
0.122	0.269	0.098	0.197	7	17	24	26	24	47.083
0.184	0.297	0.098	0.197	8	17	24	27	23	45.769
0.148	0.278	0.098	0.197	8	17	24	28	22	46.733
0.119	0.263	0.098	0.197	8	17	24	29	21	47.507
0.176	0.287	0.098	0.197	9	17	24	30	20	46.474
(b) With early stopping for efficacy.
${\alpha }_{11}$	${\alpha }_{12}$	${\alpha }_{21}$	${\alpha }_{22}$	β	$r_1$	$r_2$	$s_1$	$s_2$	$n_1$	$n_2$	$E\left(n\right)$
0.090	0.272	0.002	0.098	0.198	3	17	11	24	15	35	46.765
0.065	0.259	0.005	0.099	0.196	3	17	11	24	16	34	47.619
0.126	0.286	0.000	0.097	0.198	4	17	13	24	17	33	45.827
0.094	0.269	0.006	0.099	0.196	4	17	12	24	18	32	46.803
0.070	0.257	0.003	0.098	0.196	4	17	13	24	19	31	47.747
0.126	0.281	0.006	0.099	0.196	5	17	13	24	20	30	46.038
0.096	0.266	0.004	0.098	0.196	5	17	14	24	21	29	47.120
0.158	0.294	0.007	0.099	0.196	6	17	14	24	22	28	45.366
0.124	0.275	0.004	0.098	0.196	6	17	15	24	23	27	46.546
0.096	0.262	0.007	0.099	0.196	6	17	15	24	24	26	47.310
0.154	0.285	0.004	0.098	0.197	7	17	16	24	25	25	46.053
0.122	0.269	0.008	0.099	0.196	7	17	16	24	26	24	46.894
0.184	0.297	0.005	0.098	0.197	8	17	17	24	27	23	45.663
0.148	0.278	0.008	0.099	0.196	8	17	17	24	28	22	46.555
0.119	0.263	0.013	0.100	0.195	8	17	17	24	29	21	47.224
0.176	0.287	0.008	0.099	0.196	9	17	18	24	30	20	46.308

Table 3. Two-stage designs for Case 1 with over-running and under-running at stage 2 (no stopping for efficacy). The planned design has $n_1=22,n_2=28$. The sample size at stage 2 is changed from 28 to 25–27 (under-running) and 29–31 (over-running).

${\alpha }_{11}$	${\alpha }_{12}$	${\alpha }_{22}$	β	$r_1$	$r_2^{\ast }$	$s_2^{\ast }$	$n_1$	$n_2^{\ast }$
Under-running at stage 2
0.158	0.238	0.082	0.245	6	15	23	22	25
0.158	0.275	0.060	0.290	6	16	24	22	26
0.158	0.254	0.077	0.241	6	16	24	22	27
Planned design
0.158	0.294	0.098	0.197	6	17	24	22	28
Over-running at stage 2
0.158	0.271	0.073	0.236	6	17	25	22	29
0.158	0.250	0.092	0.195	6	17	25	22	30
0.158	0.288	0.070	0.232	6	18	26	22	31

Table 4. Two-stage designs for Case 2: the minimal effective response rate is between 0.40 and 0.45. The expected response rate is at least 0.60. The type I errors to conclude the compound is ineffective (conclude p<0.4) when p = 0.4 and effective (conclude p>0.45) when p = 0.45 are 0.3, 0.1, respectively. The power to conclude the compound is effective (conclude $p>0.4$) when p = 0.60 is at least 0.80.

${\alpha }_{11}$	${\alpha }_{12}$	${\alpha }_{22}$	β	$r_1$	$r_2$	$s_2$	$n_1$	$n_2$	$E\left(n\right)$
0.090	0.263	0.100	0.177	3	18	28	15	38	49.561
0.065	0.250	0.100	0.177	3	18	28	16	37	50.590
0.126	0.279	0.100	0.177	4	18	28	17	36	48.464
0.094	0.261	0.100	0.177	4	18	28	18	35	49.704
0.070	0.248	0.100	0.177	4	18	28	19	34	50.633
0.126	0.274	0.100	0.177	5	18	28	20	33	48.855
0.096	0.258	0.100	0.177	5	18	28	21	32	49.936
0.158	0.288	0.100	0.177	6	18	28	22	31	48.088
0.124	0.268	0.100	0.177	6	18	28	23	30	49.281
0.096	0.254	0.100	0.177	6	18	28	24	29	50.217
0.154	0.280	0.100	0.177	7	18	28	25	28	48.701
0.122	0.263	0.100	0.177	7	18	28	26	27	49.718
0.184	0.292	0.100	0.177	8	18	28	27	26	48.217
0.148	0.272	0.100	0.177	8	18	28	28	25	49.288
0.143	0.264	0.100	0.177	9	18	28	31	22	49.846
0.205	0.293	0.100	0.177	10	18	28	32	21	48.704