General discussions on Phase II clinical trials are presented in PhaseII.php and not repeated here

This page provides calculations and tables for sample size required for a Phase II study, using Fleming's procedure. This is an adaptation of a quality control procedure commonly used in industry. The user defines two levels of successes, the existing one, and the expected improved level the new treatment should achieve. The procedure then calculate the sample size required to test the new treatment. After the required number of cases are studied, the proportion of success is compared with that required, to determine whether the expected improvement has been met, thus whether to accept the new treatment for further study and trial (phase III).

Fleming's procedure therefore has advantages over Gehan's procedure and Simon's procedure, as both concept and procedures are simpler, and intuitively easier to understand and undertake. The down side is that it requires a fixed sample size, so that decisions cannot be made until the full sample size is used. It is therefore more expensive and takes longer to conclude.

Example

We are treating the advanced stage of a particularly aggressive type of cancer, and after the current standard surgery and radiotherapy, only 20% (p0=0.2) of the patients survive 12 or more months. We are offered to trial a newly developed chemotherapy, and we decided that a formal large scale Phase III trial would be worth while if the Phase II trial find the 12 months survival rate is doubled to 40% (pn = 0.4).

We use the statistical parameters α=0.05, power=0.8, p0=0.2, pn=0.4, and determined that the sample size is 29 cases.

We administered the additional chemotherapy to 29 patients that fits the description, and 14 (0.48 or 48%) survived at the end of 12 months. As we define success as survival for more than 12 months, We concluded that the results are sufficiently encouraging for a formal phase III trial to be conducted.

Had the 12 month survival rate be less than 12 out of our 29 cases (<0.4 or 40%), then we would have abandoned this new treatment and concluded that further study of it is unlikely to be productive.

Reference

Machin D, Campbell M, Fayers, P, Pinol A (1997) Sample Size Tables for Clinical Studies. Second Ed. Blackwell Science IBSN 0-86542-870-0 p. 254-255

		Power=0.8			Power=0.9			Power=0.95					Power=0.8			Power=0.9			Power=0.95					Power=0.8			Power=0.9			Power=0.95
α		0.1	0.05	0.01	0.1	0.05	0.01	0.1	0.05	0.01	α		0.1	0.05	0.01	0.1	0.05	0.01	0.1	0.05	0.01	α		0.1	0.05	0.01	0.1	0.05	0.01	0.1	0.05	0.01
p0	pn										p0	pn										p0	pn
0.05	0.1	114	150	231	177	221	318	239	291	401	0.05	0.15	34	44	66	55	67	94	76	90	120	0.05	0.2	17	22	32	28	34	47	40	46	61
0.05	0.25	11	14	19	18	21	29	25	29	38	0.05	0.3	8	9	13	13	15	20	18	20	26	0.05	0.35	6	7	10	9	11	14	13	15	19
0.05	0.4	4	5	7	7	8	11	10	12	15	0.05	0.45	4	4	6	6	7	9	8	9	11	0.05	0.5	3	3	5	5	5	7	6	7	9
0.05	0.55	2	3	4	4	4	6	5	6	8	0.05	0.6	2	2	3	3	4	5	4	5	6	0.05	0.65	2	2	3	3	3	4	4	4	5
0.05	0.7	2	2	2	2	3	3	3	3	4	0.05	0.75	1	2	2	2	2	3	3	3	4	0.05	0.8	1	1	2	2	2	2	2	2	3
0.05	0.85	1	1	2	1	2	2	2	2	2	0.05	0.9	1	1	1	1	1	2	1	2	2	0.05	0.95	1	1	1	1	1	1	1	1	1
0.1	0.15	188	253	399	284	362	535	378	468	661	0.1	0.2	53	69	108	81	102	147	109	133	184	0.1	0.25	25	33	51	40	49	70	54	65	89
0.1	0.3	15	20	30	24	30	42	33	39	53	0.1	0.35	10	13	20	16	20	28	22	27	36	0.1	0.4	8	10	14	12	14	20	16	19	26
0.1	0.45	6	7	11	9	11	15	12	15	19	0.1	0.5	5	6	8	7	9	12	10	11	15	0.1	0.55	4	5	7	6	7	9	8	9	12
0.1	0.6	3	4	5	5	6	8	6	7	10	0.1	0.65	3	3	4	4	5	6	5	6	8	0.1	0.7	2	3	4	3	4	5	4	5	6
0.1	0.75	2	2	3	3	3	4	3	4	5	0.1	0.8	2	2	3	2	3	3	3	3	4	0.1	0.85	1	2	2	2	2	3	2	3	3
0.1	0.9	1	1	2	1	2	2	2	2	3	0.1	0.95	1	1	2	1	1	2	1	2	2	0.15	0.2	253	342	546	377	484	722	498	621	887
0.15	0.25	68	91	143	103	131	192	137	169	239	0.15	0.3	32	43	66	49	62	90	66	80	112	0.15	0.35	19	25	38	29	36	52	39	48	66
0.15	0.4	13	16	25	19	24	35	26	32	43	0.15	0.45	9	12	18	14	17	24	19	22	31	0.15	0.5	7	9	13	10	13	18	14	17	23
0.15	0.55	5	7	10	8	10	14	11	13	17	0.15	0.6	4	5	8	6	8	11	8	10	14	0.15	0.65	3	4	7	5	6	9	7	8	11
0.15	0.7	3	4	5	4	5	7	5	6	9	0.15	0.75	2	3	4	3	4	6	4	5	7	0.15	0.8	2	3	4	3	3	5	3	4	6
0.15	0.85	2	2	3	2	3	4	3	3	5	0.15	0.9	1	2	3	2	2	3	2	3	4	0.15	0.95	1	1	2	1	2	2	2	2	3
0.2	0.25	308	419	671	456	589	883	601	751	1080	0.2	0.3	81	109	174	121	156	231	161	200	284	0.2	0.35	38	50	79	57	72	106	75	93	131
0.2	0.4	22	29	46	33	42	61	44	54	76	0.2	0.45	14	19	30	22	27	40	29	35	49	0.2	0.5	10	13	21	15	19	28	20	25	35
0.2	0.55	8	10	15	11	14	21	15	18	25	0.2	0.6	6	8	12	9	11	16	11	14	19	0.2	0.65	5	6	9	7	8	12	9	11	15
0.2	0.7	4	5	7	5	7	10	7	8	12	0.2	0.75	3	4	6	4	5	8	5	7	9	0.2	0.8	3	3	5	3	4	6	4	5	8
0.2	0.85	2	3	4	3	3	5	3	4	6	0.2	0.9	2	2	3	2	3	4	3	3	5	0.2	0.95	1	2	3	2	2	3	2	2	3
0.25	0.3	354	483	777	522	676	1018	686	860	1241	0.25	0.35	92	125	199	136	176	262	180	225	322	0.25	0.4	42	57	90	63	80	119	83	103	147
0.25	0.45	24	32	51	36	46	68	48	59	84	0.25	0.5	16	21	33	23	30	44	31	38	54	0.25	0.55	11	15	23	16	21	31	21	27	38
0.25	0.6	8	11	17	12	15	22	16	19	27	0.25	0.65	6	8	13	9	11	17	12	15	21	0.25	0.7	5	6	10	7	9	13	9	11	16
0.25	0.75	4	5	8	5	7	10	7	9	12	0.25	0.8	3	4	6	4	5	8	5	7	10	0.25	0.85	3	3	5	3	4	6	4	5	8
0.25	0.9	2	3	4	3	3	5	3	4	6	0.25	0.95	2	2	3	2	3	4	2	3	4	0.3	0.35	392	534	862	575	746	1126	753	947	1370
		Power=0.8			Power=0.9			Power=0.95					Power=0.8			Power=0.9			Power=0.95					Power=0.8			Power=0.9			Power=0.95
α		0.1	0.05	0.01	0.1	0.05	0.01	0.1	0.05	0.01	α		0.1	0.05	0.01	0.1	0.05	0.01	0.1	0.05	0.01	α		0.1	0.05	0.01	0.1	0.05	0.01	0.1	0.05	0.01
p0	pn										p0	pn										p0	pn
0.3	0.4	100	136	219	148	191	287	195	244	351	0.3	0.45	45	62	98	67	87	129	88	110	158	0.3	0.5	26	35	56	38	49	73	50	63	90
0.3	0.55	17	22	36	25	31	47	32	40	57	0.3	0.6	12	16	25	17	22	32	22	28	39	0.3	0.65	8	11	18	12	16	23	16	20	28
0.3	0.7	6	9	14	9	12	18	12	15	21	0.3	0.75	5	7	11	7	9	13	9	11	16	0.3	0.8	4	5	8	5	7	10	7	8	12
0.3	0.85	3	4	7	4	5	8	5	6	10	0.3	0.9	2	3	5	3	4	6	4	5	7	0.3	0.95	2	3	4	2	3	5	3	3	5
0.35	0.4	420	573	927	615	798	1208	804	1012	1468	0.35	0.45	107	145	234	156	203	306	205	257	372	0.35	0.5	48	65	105	70	91	137	92	115	166
0.35	0.55	27	37	59	39	51	77	52	65	93	0.35	0.6	17	23	38	25	32	49	33	41	59	0.35	0.65	12	16	26	17	22	33	22	28	40
0.35	0.7	9	12	19	12	16	24	16	20	29	0.35	0.75	6	9	14	9	12	18	11	15	21	0.35	0.8	5	7	11	7	9	13	8	11	16
0.35	0.85	4	5	8	5	7	10	6	8	12	0.35	0.9	3	4	7	4	5	8	5	6	9	0.35	0.95	2	3	5	3	4	6	3	4	6
0.4	0.45	439	600	972	641	834	1264	837	1056	1534	0.4	0.5	110	151	244	161	210	317	211	266	385	0.4	0.55	49	67	108	72	93	141	93	118	171
0.4	0.6	28	38	61	40	52	79	52	65	95	0.4	0.65	17	24	38	25	33	50	32	41	60	0.4	0.7	12	16	26	17	22	34	22	28	40
0.4	0.75	9	12	19	12	16	24	15	19	28	0.4	0.8	6	9	14	9	11	18	11	14	21	0.4	0.85	5	7	11	6	8	13	8	10	15
0.4	0.9	4	5	8	5	6	10	6	7	11	0.4	0.95	3	4	6	3	4	7	4	5	8	0.45	0.5	449	615	997	654	852	1294	853	1077	1568
0.45	0.55	112	154	249	163	212	323	212	268	391	0.45	0.6	49	68	110	72	93	142	93	118	172	0.45	0.65	27	38	61	39	52	79	51	65	95
0.45	0.7	17	24	39	25	32	49	31	40	59	0.45	0.75	12	16	26	16	21	33	21	27	39	0.45	0.8	8	11	19	11	15	23	14	18	27
0.45	0.85	6	8	14	8	11	17	10	13	20	0.45	0.9	4	6	10	6	8	12	7	9	14	0.45	0.95	3	5	8	4	5	9	4	6	10
0.5	0.55	449	617	1001	654	853	1298	852	1077	1571	0.5	0.6	111	153	249	161	211	321	210	266	388	0.5	0.65	49	67	109	70	92	140	91	115	169
0.5	0.7	27	37	60	38	50	77	49	63	92	0.5	0.75	17	23	38	23	31	48	30	38	57	0.5	0.8	11	15	25	15	20	32	19	25	37
0.5	0.85	8	11	18	10	14	22	13	17	26	0.5	0.9	5	8	13	7	10	15	9	11	18	0.5	0.95	4	5	9	5	6	11	5	7	12
0.55	0.6	441	606	986	641	837	1275	834	1056	1542	0.55	0.65	108	149	243	156	205	313	203	257	378	0.55	0.7	47	65	106	67	88	136	87	110	163
0.55	0.75	26	35	58	36	48	74	46	59	88	0.55	0.8	16	22	36	22	29	45	27	35	53	0.55	0.85	10	14	24	14	19	29	17	22	34
0.55	0.9	7	10	17	9	12	20	11	15	23	0.55	0.95	5	7	12	6	8	13	7	9	15	0.6	0.65	424	583	950	615	804	1227	798	1012	1482
0.6	0.7	103	142	233	148	195	299	191	244	359	0.6	0.75	44	61	101	63	83	128	80	103	153	0.6	0.8	24	33	55	33	44	69	42	54	81
0.6	0.85	14	20	34	19	26	41	24	32	48	0.6	0.9	9	13	22	12	16	26	14	19	30	0.6	0.95	6	8	15	7	10	17	8	12	19
0.65	0.7	398	548	895	575	753	1152	746	947	1389	0.65	0.75	96	133	218	136	180	278	176	225	332	0.65	0.8	40	56	93	57	75	117	72	93	139
0.65	0.85	21	30	50	29	39	62	36	48	72	0.65	0.9	12	18	30	16	22	36	20	27	42	0.65	0.95	8	11	19	9	13	22	11	15	24
0.7	0.75	363	501	819	522	686	1052	676	860	1265	0.7	0.8	86	119	197	121	161	250	156	200	298	0.7	0.85	36	50	84	49	66	104	62	80	122
0.7	0.9	18	26	44	24	33	53	30	39	61	0.7	0.95	10	15	25	13	18	29	15	20	33	0.75	0.8	318	441	723	456	601	925	589	751	1110
0.75	0.85	74	103	172	103	137	215	131	169	255	0.75	0.9	29	42	71	40	54	87	49	65	101	0.75	0.95	14	21	36	18	25	42	21	29	47
0.8	0.85	265	368	607	377	498	771	484	621	922	0.8	0.9	59	83	140	81	109	173	102	133	203	0.8	0.95	22	32	56	28	40	66	34	46	74
0.85	0.9	202	283	470	284	378	591	362	468	702	0.85	0.95	42	60	103	55	76	124	67	90	142	0.9	0.95	129	184	311	177	239	382	221	291	447

0.05 0.8 0.1 0.3 0.05 0.8 0.2 0.4

The data is in 4 columns.   - Each row a separate study   - Col 1 = Probability of Type I error (α)   - Col 2 = Power (1-β)   - Col 3 = Lower proportion of success for rejection of treatment(p0)   - Col 4 = Higher proportion of success for acceptance of treatment(pn)

The R code for Fleming's procedure for sample size fpr Phase II studies is as follows

"                      Phase II Fleming
Machin D, Campbell M, Fayers, P, Pinol A (1997) Sample Size Tables for Clinical Studies. Second Ed. 
Blackwell Science IBSN 0-86542-870-0 p. 254-255 
"
# Data
errorI = 0.05   # alpha Type I Error
power = 0.8     # power 1-beta
pr = 0.2        # success rate below which treatment is rejected
pa = 0.4        # success rate above which treatment is accepted
# Calculations
za = -(qnorm(errorI))
zb = -(qnorm(1 - power))
top = za * sqrt(pr * (1 - pr)) + zb * sqrt(pa * (1 -pa))
bot = pa - pr;
ssiz = ceiling(top^2 / bot^2)
ssiz

The result is

> ssiz
[1] 29