Table 3:

Comparison between the proposed optimal two-stage and Belin’s design under the exponential distribution with nominal type I error 5% and power of 80%, a constant accrual rate r = 10, a restricted follow-up x = 1 and x₀ = 1. Overall empirical type I error ($\widehat{\alpha }$) and power ($1-\widehat{\beta }$) for the two-stage designs were estimated from 10,000 simulated trials.

(S0, S1)	t 1	c 1	c	ES	ETSL	n 1	n	PS	$\widehat{\alpha }$	$1-\widehat{\beta }$
Belin’s two-stage design under exponential model
(.1, .3)	1.11	0.068	1.615	14.8	2.9	12	19	0.53	.033	.727
(.1, .25)	1.67	0.097	1.673	22.9	4.0	17	30	0.54	.035	.722
(.2, .4)	1.72	0.082	1.621	23.2	4.0	18	30	0.53	.036	.748
(.2, .35)	2.84	0.150	1.623	37.9	6.0	29	50	0.56	.037	.750
(.3, .5)	2.16	0.107	1.621	29.1	4.8	22	38	0.54	.037	.764
(.3, .45)	3.68	0.358	1.627	49.0	7.4	37	64	0.55	.039	.765
(.4, .6)	2.50	0.167	1.622	32.4	5.2	25	42	0.57	.039	.780
(.4, .55)	4.11	0.137	1.626	55.3	8.3	42	73	0.55	.040	.777
(.5, .7)	2.44	0.114	1.621	32.9	5.3	25	43	0.55	.038	.797
(.5, .65)	4.27	0.149	1.626	56.9	8.5	43	75	0.56	.039	.789
(.6, .8)	2.28	0.106	1.620	30.7	5.0	23	40	0.54	.037	.814
(.6, .75)	3.93	0.122	1.624	53.6	8.1	40	71	0.55	.039	.802
(.7, .85)	3.42	0.191	1.618	45.6	7.1	35	61	0.58	.040	.827
Proposed two-stage design under exponential model
(.1, .3)	1.48	−0.106	1.638	18.7	3.2	15	22	0.46	.032	.786
(.1, .25)	2.21	0.054	1.634	28.3	4.5	23	35	0.52	.038	.810
(.2, .4)	2.12	0.052	1.633	26.8	4.3	22	33	0.52	.036	.784
(.2, .35)	3.33	0.110	1.632	43.2	6.5	34	55	0.54	.038	.801
(.3, .5)	2.48	0.082	1.631	31.9	5.0	26	40	0.53	.038	.800
(.3, .45)	4.12	0.130	1.632	53.2	7.8	42	68	0.55	.039	.798
(.4, .6)	2.56	0.129	1.623	33.9	5.4	26	44	0.55	.039	.797
(.4, .55)	4.46	0.185	1.627	58.0	8.6	45	76	0.57	.040	.797
(.5, .7)	2.54	0.184	1.620	32.9	5.3	26	43	0.57	.038	.798
(.5, .65)	4.45	0.202	1.625	57.7	8.6	45	76	0.58	.039	.797
(.6, .8)	2.14	0.121	1.616	28.9	4.8	22	38	0.55	.038	.795
(.6, .75)	3.87	0.135	1.617	52.2	7.9	39	69	0.55	.039	.793
(.7, .85)	2.98	0.126	1.682	41.6	6.6	30	56	0.55	.040	.791

S₀ and S₁ are the survival probabilities at x₀ under the null hypothesis and alternative, respectively; t₁ is the calendar time for the first-stage interim analysis; c₁ and c are the boundaries for the first-stage and final analysis; n₁ and n are the sample sizes for the first-stage and final analysis; PS is the stopping probability of first stage under the null hypothesis; ES is the expected sample size under the null hypothesis; ETSL is expected total study duration.