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StatsToDo: Sample Size for Receiver Operator Characteristics (ROCs)

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Explanations and References

This page provides two sample size algorithms for Receiver Operator Characteristics (ROC), with Javascript programs and sample size tables.

Sample Size and Standard Error of a Single ROC

This is based on the calculation of the Standard Error (SE) from the value of the ROX (θ), as described by Hanley and McNeil (see references). The sample size is calculated by iterative approximation until the required Standard Error is obtained. The confidence interval is θ±z(SE)

The sample size table for a single ROC in the next panel is based upon the following arguments

The Standard Error (SE) from a nominated θ is calculated so that its one tail 95% confidence interval reaches but does not overlap the null value of 0.5.
As the one tail z value for 95% confidence interval is 1.65, θ-1.65SE=0.5, so SE = (θ-0.5) / 1.65
Sample size is then calculated using θ and SE

This sample size is the number of cases in each of the two groups (Outcome Positive (O+) and Outcome Negative (O-), so the total sample size is twice that presented.

This sample size is the minimum required to obtain a significant θ. In most cases, where a precise θ is required, or comparison between multiple θ is intended, a narrower SE and therefore a larger sample size is used.

Sample size and power calculations for comparing two ROCs

This is also based on the formula described by Hanley and McNeil (see references). It assumes that θ is a population measurement and approximately normally distributed, so that the difference is Diff=θ₁-θ₂, and the Standard Error of the difference is the root sum square of the individual Standard Errors SE_diff = sqrt(θ₁²+θ₂²)

The sample size calculated is for the one tail model. Those requiring a two tail model can use the same calculations but the Type I Error value (α) is halved.

The sample size is for the total number of cases (number of Outcome Positive + number of Outcome Negative) in each of the two ROCs.

Instead of using the z=diff/SE_Diff or the 95% confidence interval of the difference to assess statistical significance, the power calculation can also be used to see if the data contains sufficient power to satisfy the user's requirement. Commonly a power in the data exceeding 0.8 for a nominated α of 0.05 is taken as a statistically significant difference.

The table in the next panels provides sample size requirements for 3 levels of power (0.8, 0.9, 0.95), 3 levels of α (0.1, 0.05, 0.01), and difference between two θ values from 0.5 upwards at 0.02 intervals.

References

Hanley JA, McNeil BJ (1982) The meaning and use of the Area under a receiver operating characteristic (ROC) curve. Radiology 143:29-36

Sample Size Tables

Sample Size to Establish 1 ROC

Table on this page provides minimal sample size required, for each of the two groups (Outcome Positive and Outcome Negative), to obtain a statistically significant Receiver Operator Characteristics. Calculation is based on θ and its Standard Error (SE), the formula is as follows

θ = value of the ROC
Standard Error = one tail 95% confidence interval to the null value of 0.5, SE = (θ-0.5)/1.65

Please Note :

This calculation assumes that the researcher will onbtain equal sample size for Outcome Positive and Outcome Negative groups
The sample size is for each group, so the total sample size is twice that in the table
The sample size calculated is for the minimum number of cases required to achieve a statistically significant value, that the one tail 95% confidence does not overlap the null (0.5) value. In most research projects, a more precise (narrower) Standard Error is desirable, so the sample size will need to be bigger.
Only θ > 0.5 is calculated. A θ=0.5 has no predictive value at all, and θ < 0.5 predicts in the wrong direction.

θ	SSiz	θ	SSiz	θ	SSiz	θ	SSiz	θ	SSiz	θ	SSiz	θ	SSiz	θ	SSiz	θ	SSiz	θ	SSiz
0.51	1000	0.52	1000	0.53	503	0.54	282	0.55	180	0.56	125	0.57	91	0.58	70	0.59	55	0.6	44
0.61	36	0.62	30	0.63	26	0.64	22	0.65	19	0.66	17	0.67	15	0.68	13	0.69	11	0.7	10
0.71	9	0.72	8	0.73	7	0.74	7	0.75	6	0.76	6	0.77	5	0.78	5	0.79	4	0.8	4
0.81	4	0.82	3	0.83	3	0.84	3	0.85	3	0.86	3	0.87	3	0.88	3	0.89	3	0.9	3
0.91	3	0.92	3	0.93	3	0.94	3	0.95	3	0.96	3	0.97	3	0.98	3	0.99	3

Sample Size to Compare 2 ROCs

α=Probability of Type I Error, β=Probability of Type II Error
θ₁ and θ₂ are values of the two ROCs being compared
Sample Size is total of both groups (outcome Positive + Outcome Negative) for each of the two ROCs, for θ₁ and the same number of cases for θ₂
Sample size is for a One Tail model. For two tail model, use the same calculation, but halve the &alpha value

		Power(1-β)=0.8			Power(1-β)=0.9			Power(1-β)=0.95					Power(1-β)=0.8			Power(1-β)=0.9			Power(1-β)=0.95
		α=0.1	α=0.05	α=0.01	α=0.1	α=0.05	α=0.01	α=0.1	α=0.05	α=0.01			α=0.1	α=0.05	α=0.01	α=0.1	α=0.05	α=0.01	α=0.1	α=0.05	α=0.01
θ₁	θ₂	SSiz	SSiz	SSiz	SSiz	SSiz	SSiz	SSiz	SSiz	SSiz	θ₁	θ₂	SSiz	SSiz	SSiz	SSiz	SSiz	SSiz	SSiz	SSiz	SSiz
0.50	0.52	3756	5151	8362	5472	7134	10844	7133	9015	13137	0.50	0.54	938	1287	2089	1366	1782	2709	1781	2251	3281
0.50	0.56	416	571	928	606	790	1202	790	998	1455	0.50	0.58	234	321	521	340	443	675	443	560	817
0.50	0.60	149	205	333	217	283	431	282	357	521	0.50	0.62	103	142	231	150	196	298	195	247	360
0.50	0.64	76	104	169	110	143	218	142	180	263	0.50	0.66	58	79	129	83	109	166	108	137	201
0.50	0.68	45	62	102	65	86	131	85	107	157	0.50	0.70	37	50	82	53	69	105	68	86	127
0.50	0.72	30	41	67	43	56	86	56	71	104	0.50	0.74	25	35	56	36	47	72	46	59	87
0.50	0.76	21	29	48	30	40	61	39	50	73	0.50	0.78	18	25	41	26	34	52	33	42	62
0.50	0.80	16	22	36	22	29	45	28	36	54	0.50	0.82	14	19	31	19	25	39	25	31	47
0.50	0.84	12	17	27	17	22	34	21	27	41	0.50	0.86	11	15	24	15	20	30	19	24	36
0.50	0.88	10	13	22	13	17	27	17	21	32	0.50	0.90	9	12	19	12	15	24	15	19	28
0.50	0.92	8	11	17	10	14	22	13	17	25	0.50	0.94	7	10	16	9	12	19	12	15	23
0.50	0.96	6	9	14	8	11	18	10	14	20	0.50	0.98	6	8	13	8	10	16	9	12	18
0.52	0.54	3745	5137	8341	5456	7114	10815	7111	8988	13101	0.52	0.56	935	1283	2083	1361	1775	2700	1773	2242	3269
0.52	0.58	415	569	924	603	787	1197	785	993	1449	0.52	0.60	233	319	519	338	441	672	440	556	812
0.52	0.62	148	204	331	215	281	428	280	354	518	0.52	0.64	103	141	229	149	194	296	193	245	358
0.52	0.66	75	103	168	109	142	217	141	179	261	0.52	0.68	57	79	128	83	108	165	107	136	199
0.52	0.70	45	62	101	65	85	130	84	106	156	0.52	0.72	36	50	81	52	68	104	67	85	125
0.52	0.74	30	41	67	43	56	86	55	70	103	0.52	0.76	25	34	56	35	46	71	46	58	86
0.52	0.78	21	29	47	30	39	60	38	49	72	0.52	0.80	18	25	41	25	33	52	33	42	62
0.52	0.82	16	21	35	22	29	44	28	36	53	0.52	0.84	14	19	31	19	25	39	24	31	46
		Power(1-β)=0.8			Power(1-β)=0.9			Power(1-β)=0.95					Power(1-β)=0.8			Power(1-β)=0.9			Power(1-β)=0.95
		α=0.1	α=0.05	α=0.01	α=0.1	α=0.05	α=0.01	α=0.1	α=0.05	α=0.01			α=0.1	α=0.05	α=0.01	α=0.1	α=0.05	α=0.01	α=0.1	α=0.05	α=0.01
θ₁	θ₂	SSiz	SSiz	SSiz	SSiz	SSiz	SSiz	SSiz	SSiz	SSiz	θ₁	θ₂	SSiz	SSiz	SSiz	SSiz	SSiz	SSiz	SSiz	SSiz	SSiz
0.52	0.86	12	16	27	17	22	34	21	27	40	0.52	0.88	11	15	24	15	19	30	18	24	35
0.52	0.90	9	13	21	13	17	27	16	21	31	0.52	0.92	8	12	19	11	15	24	14	19	28
0.52	0.94	8	10	17	10	14	21	13	16	25	0.52	0.96	7	9	16	9	12	19	11	15	22
0.52	0.98	6	9	14	8	11	17	10	13	20	0.54	0.56	3720	5104	8288	5419	7066	10744	7062	8927	13013
0.54	0.58	928	1273	2068	1350	1762	2680	1759	2224	3244	0.54	0.60	411	565	917	598	780	1188	778	984	1437
0.54	0.62	231	317	515	335	437	666	435	551	805	0.54	0.64	147	202	329	213	278	424	277	351	513
0.54	0.66	102	140	227	147	192	293	191	242	354	0.54	0.68	74	102	166	107	140	214	139	176	258
0.54	0.70	57	78	127	81	107	163	105	134	196	0.54	0.72	44	61	100	64	84	128	82	105	154
0.54	0.74	36	49	80	51	67	103	66	84	124	0.54	0.76	29	40	66	42	55	84	54	69	101
0.54	0.78	24	34	55	35	46	70	45	57	84	0.54	0.80	21	29	47	29	39	59	37	48	71
0.54	0.82	18	24	40	25	33	51	32	41	60	0.54	0.84	15	21	35	21	28	44	27	35	52
0.54	0.86	13	18	30	19	25	38	24	30	45	0.54	0.88	12	16	27	16	21	33	20	26	39
0.54	0.90	10	14	24	14	19	29	18	23	35	0.54	0.92	9	13	21	13	17	26	16	20	31
0.54	0.94	8	11	19	11	15	23	14	18	27	0.54	0.96	7	10	17	10	13	21	12	16	24
0.54	0.98	7	9	15	9	12	19	11	14	22	0.56	0.58	3681	5050	8202	5360	6990	10631	6984	8830	12875
0.56	0.60	917	1259	2046	1334	1741	2650	1738	2198	3207	0.56	0.62	406	558	907	590	770	1173	768	972	1419
0.56	0.64	228	313	509	330	431	657	429	543	794	0.56	0.66	145	199	324	210	274	419	272	345	505
0.56	0.68	100	138	224	145	189	289	188	238	349	0.56	0.70	73	101	164	105	138	211	136	173	254
0.56	0.72	56	77	125	80	105	161	103	131	193	0.56	0.74	44	60	98	63	82	126	81	103	151
0.56	0.76	35	48	79	50	66	101	65	82	121	0.56	0.78	29	40	65	41	54	83	53	67	99
		Power(1-β)=0.8			Power(1-β)=0.9			Power(1-β)=0.95					Power(1-β)=0.8			Power(1-β)=0.9			Power(1-β)=0.95
		α=0.1	α=0.05	α=0.01	α=0.1	α=0.05	α=0.01	α=0.1	α=0.05	α=0.01			α=0.1	α=0.05	α=0.01	α=0.1	α=0.05	α=0.01	α=0.1	α=0.05	α=0.01
θ₁	θ₂	SSiz	SSiz	SSiz	SSiz	SSiz	SSiz	SSiz	SSiz	SSiz	θ₁	θ₂	SSiz	SSiz	SSiz	SSiz	SSiz	SSiz	SSiz	SSiz	SSiz
0.56	0.80	24	33	54	34	45	69	44	56	82	0.56	0.82	20	28	46	29	38	58	37	47	69
0.56	0.84	17	24	39	24	32	50	31	40	59	0.56	0.86	15	21	34	21	28	43	26	34	51
0.56	0.88	13	18	30	18	24	37	23	29	44	0.56	0.90	11	16	26	16	21	33	20	26	38
0.56	0.92	10	14	23	14	18	29	17	22	34	0.56	0.94	9	12	21	12	16	25	15	20	30
0.56	0.96	8	11	18	11	14	23	13	17	26	0.56	0.98	7	10	17	10	13	20	12	16	24
0.58	0.60	3627	4977	8084	5281	6887	10476	6880	8699	12686	0.58	0.62	903	1240	2015	1313	1714	2609	1710	2163	3157
0.58	0.64	400	549	893	580	758	1155	755	955	1396	0.58	0.66	224	307	500	324	424	646	421	534	781
0.58	0.68	142	196	319	206	269	411	267	339	496	0.58	0.70	98	135	221	142	186	284	184	233	342
0.58	0.72	72	99	161	103	135	207	133	170	249	0.58	0.74	55	75	123	78	103	157	101	128	189
0.58	0.76	43	59	97	61	80	123	79	100	148	0.58	0.78	34	47	78	49	64	99	63	80	119
0.58	0.80	28	39	64	40	53	81	51	65	97	0.58	0.82	23	32	53	33	44	67	42	54	80
0.58	0.84	20	27	45	28	37	57	35	45	68	0.58	0.86	17	23	39	24	31	48	30	39	58
0.58	0.88	15	20	33	20	27	42	26	33	49	0.58	0.90	13	18	29	17	23	36	22	28	43
0.58	0.92	11	15	26	15	20	32	19	25	37	0.58	0.94	10	14	23	13	18	28	17	22	33
0.58	0.96	9	12	20	12	16	25	15	19	29	0.58	0.98	8	11	18	10	14	22	13	17	26
0.60	0.62	3559	4884	7934	5180	6757	10280	6748	8533	12446	0.60	0.64	885	1216	1977	1287	1680	2558	1675	2120	3095
0.60	0.66	391	538	875	568	742	1131	738	935	1367	0.60	0.68	219	301	490	317	414	633	411	522	764
0.60	0.70	139	192	312	201	263	402	261	331	485	0.60	0.72	96	132	216	138	181	277	179	227	334
0.60	0.74	70	97	158	100	132	202	130	165	243	0.60	0.76	53	73	120	76	100	154	98	125	184
0.60	0.78	42	58	94	59	78	120	76	97	144	0.60	0.80	33	46	76	47	63	96	61	78	115
		Power(1-β)=0.8			Power(1-β)=0.9			Power(1-β)=0.95					Power(1-β)=0.8			Power(1-β)=0.9			Power(1-β)=0.95
		α=0.1	α=0.05	α=0.01	α=0.1	α=0.05	α=0.01	α=0.1	α=0.05	α=0.01			α=0.1	α=0.05	α=0.01	α=0.1	α=0.05	α=0.01	α=0.1	α=0.05	α=0.01
θ₁	θ₂	SSiz	SSiz	SSiz	SSiz	SSiz	SSiz	SSiz	SSiz	SSiz	θ₁	θ₂	SSiz	SSiz	SSiz	SSiz	SSiz	SSiz	SSiz	SSiz	SSiz
0.60	0.82	27	38	62	39	51	79	50	63	94	0.60	0.84	23	32	52	32	42	66	41	52	78
0.60	0.86	19	27	44	27	36	55	34	44	66	0.60	0.88	16	23	38	23	30	47	29	37	56
0.60	0.90	14	20	32	20	26	40	25	32	48	0.60	0.92	12	17	28	17	22	35	21	27	41
0.60	0.94	11	15	25	15	20	31	18	24	36	0.60	0.96	9	13	22	13	17	27	16	21	32
0.60	0.98	8	12	19	11	15	24	14	18	28	0.62	0.64	3476	4772	7753	5059	6600	10043	6589	8334	12158
0.62	0.66	864	1187	1931	1255	1639	2497	1633	2068	3020	0.62	0.68	382	525	854	553	723	1103	719	911	1333
0.62	0.70	213	293	478	308	404	616	400	508	744	0.62	0.72	135	187	305	195	256	392	253	322	472
0.62	0.74	93	129	210	134	176	270	174	221	325	0.62	0.76	68	94	154	97	128	197	126	160	236
0.62	0.78	52	71	117	74	97	149	95	121	179	0.62	0.80	40	56	92	57	76	117	74	94	140
0.62	0.82	32	45	74	46	60	93	59	75	112	0.62	0.84	26	37	60	37	49	76	48	61	91
0.62	0.86	22	31	50	31	41	63	39	50	75	0.62	0.88	19	26	43	26	34	53	33	42	63
0.62	0.90	16	22	36	22	29	45	28	36	54	0.62	0.92	14	19	31	19	25	39	24	31	46
0.62	0.94	12	16	27	16	22	34	20	26	40	0.62	0.96	10	14	24	14	19	30	18	23	35
0.62	0.98	9	13	21	12	16	26	15	20	30	0.64	0.66	3380	4640	7541	4918	6417	9767	6405	8102	11822
0.64	0.68	839	1153	1877	1219	1592	2427	1585	2008	2934	0.64	0.70	370	509	830	537	702	1071	697	884	1293
0.64	0.72	207	285	464	299	391	598	387	492	721	0.64	0.74	131	181	295	189	248	380	245	311	457
0.64	0.76	90	125	204	130	170	261	167	213	314	0.64	0.78	66	91	149	94	124	190	121	155	228
0.64	0.80	50	69	113	71	93	144	91	117	172	0.64	0.82	39	54	89	55	73	113	71	91	135
0.64	0.84	31	43	71	44	58	90	56	72	107	0.64	0.86	26	35	58	36	47	74	46	59	87
0.64	0.88	21	29	49	30	39	61	38	48	72	0.64	0.90	18	25	41	25	33	51	31	40	61
		Power(1-β)=0.8			Power(1-β)=0.9			Power(1-β)=0.95					Power(1-β)=0.8			Power(1-β)=0.9			Power(1-β)=0.95
		α=0.1	α=0.05	α=0.01	α=0.1	α=0.05	α=0.01	α=0.1	α=0.05	α=0.01			α=0.1	α=0.05	α=0.01	α=0.1	α=0.05	α=0.01	α=0.1	α=0.05	α=0.01
θ₁	θ₂	SSiz	SSiz	SSiz	SSiz	SSiz	SSiz	SSiz	SSiz	SSiz	θ₁	θ₂	SSiz	SSiz	SSiz	SSiz	SSiz	SSiz	SSiz	SSiz	SSiz
0.64	0.92	15	21	35	21	28	44	26	34	51	0.64	0.94	13	18	30	18	24	37	22	29	44
0.64	0.96	11	16	26	15	21	32	19	25	38	0.64	0.98	10	14	23	13	18	28	17	22	33
0.66	0.68	3271	4491	7299	4757	6209	9451	6195	7837	11438	0.66	0.70	811	1115	1815	1178	1539	2346	1531	1940	2836
0.66	0.72	358	492	802	518	677	1035	672	852	1249	0.66	0.74	199	275	448	288	377	577	373	474	695
0.66	0.76	126	174	285	182	239	366	235	299	440	0.66	0.78	87	120	197	125	164	252	161	205	302
0.66	0.80	63	87	143	90	119	183	116	148	219	0.66	0.82	48	66	109	68	90	138	87	112	166
0.66	0.84	37	52	85	53	70	108	68	87	129	0.66	0.86	30	42	69	42	56	86	54	69	103
0.66	0.88	24	34	56	34	45	71	43	56	84	0.66	0.90	20	28	47	28	37	58	36	46	69
0.66	0.92	17	24	39	23	31	49	30	38	58	0.66	0.94	14	20	34	20	27	42	25	32	49
0.66	0.96	12	17	29	17	23	36	21	28	42	0.66	0.98	11	15	25	15	20	31	18	24	36
0.68	0.70	3148	4323	7028	4578	5975	9098	5960	7541	11009	0.68	0.72	780	1073	1746	1131	1479	2256	1471	1864	2726
0.68	0.74	343	473	771	497	650	994	645	818	1199	0.68	0.76	191	264	431	276	361	554	357	454	667
0.68	0.78	121	167	274	174	228	351	225	286	422	0.68	0.80	83	115	189	119	157	241	153	196	289
0.68	0.82	60	84	137	86	113	175	111	141	209	0.68	0.84	46	63	104	65	85	132	83	106	158
0.68	0.86	36	49	82	50	66	103	64	82	123	0.68	0.88	28	40	66	40	53	82	51	65	98
0.68	0.90	23	32	54	32	43	67	41	53	79	0.68	0.92	19	27	45	27	35	56	34	44	66
0.68	0.94	16	23	38	22	30	47	28	36	55	0.68	0.96	14	19	32	19	25	40	23	30	46
0.68	0.98	12	16	28	16	21	34	20	26	40	0.70	0.72	3013	4138	6728	4380	5718	8708	5700	7214	10535
0.70	0.74	746	1026	1671	1081	1413	2157	1404	1780	2605	0.70	0.76	328	452	737	474	620	949	614	780	1144
0.70	0.78	182	252	411	263	344	528	340	432	636	0.70	0.80	115	159	261	165	217	334	213	272	401
		Power(1-β)=0.8			Power(1-β)=0.9			Power(1-β)=0.95					Power(1-β)=0.8			Power(1-β)=0.9			Power(1-β)=0.95
		α=0.1	α=0.05	α=0.01	α=0.1	α=0.05	α=0.01	α=0.1	α=0.05	α=0.01			α=0.1	α=0.05	α=0.01	α=0.1	α=0.05	α=0.01	α=0.1	α=0.05	α=0.01
θ₁	θ₂	SSiz	SSiz	SSiz	SSiz	SSiz	SSiz	SSiz	SSiz	SSiz	θ₁	θ₂	SSiz	SSiz	SSiz	SSiz	SSiz	SSiz	SSiz	SSiz	SSiz
0.70	0.82	79	109	180	113	149	229	145	186	275	0.70	0.84	57	80	131	81	108	166	105	134	199
0.70	0.86	43	60	99	61	81	126	78	100	150	0.70	0.88	34	47	78	47	63	98	60	78	116
0.70	0.90	27	38	62	38	50	78	48	62	92	0.70	0.92	22	31	51	30	41	63	38	50	75
0.70	0.94	18	25	42	25	33	52	31	41	62	0.70	0.96	15	21	36	21	28	44	26	34	51
0.70	0.98	13	18	30	17	23	37	22	28	43	0.72	0.74	2865	3936	6401	4164	5437	8282	5418	6859	10018
0.72	0.76	708	974	1588	1026	1342	2049	1332	1690	2474	0.72	0.78	311	429	700	449	588	901	582	739	1085
0.72	0.80	173	238	390	248	326	501	321	409	602	0.72	0.82	109	151	248	156	205	316	201	257	379
0.72	0.84	75	104	170	106	140	217	137	175	259	0.72	0.86	54	75	124	77	101	157	98	126	187
0.72	0.88	41	57	94	57	76	118	73	94	141	0.72	0.90	32	44	73	44	59	92	56	73	109
0.72	0.92	25	35	59	35	47	73	44	58	87	0.72	0.94	21	29	48	28	38	60	36	46	70
0.72	0.96	17	24	40	23	31	49	29	38	58	0.72	0.98	14	20	33	19	26	41	24	31	48
0.74	0.76	2706	3718	6049	3931	5134	7823	5114	6475	9461	0.74	0.78	668	919	1499	967	1265	1933	1255	1592	2333
0.74	0.80	293	404	660	422	554	848	547	695	1021	0.74	0.82	162	224	368	233	306	471	301	384	566
0.74	0.84	102	142	233	146	193	297	188	240	356	0.74	0.86	70	97	160	99	131	203	128	163	243
0.74	0.88	51	70	116	71	95	147	91	117	175	0.74	0.90	38	53	88	53	71	111	68	88	131
0.74	0.92	30	41	69	41	55	86	52	68	102	0.74	0.94	24	33	55	33	43	68	41	53	80
0.74	0.96	19	27	45	26	35	55	33	43	65	0.74	0.98	16	22	37	21	29	46	27	35	53
0.76	0.78	2536	3485	5671	3682	4810	7332	4789	6065	8865	0.76	0.80	625	860	1404	903	1183	1809	1172	1488	2182
0.76	0.82	274	377	618	394	517	793	509	648	954	0.76	0.84	151	209	344	217	285	439	280	357	527
0.76	0.86	95	132	217	136	179	277	174	223	331	0.76	0.88	65	90	149	92	122	189	118	151	225
		Power(1-β)=0.8			Power(1-β)=0.9			Power(1-β)=0.95					Power(1-β)=0.8			Power(1-β)=0.9			Power(1-β)=0.95
		α=0.1	α=0.05	α=0.01	α=0.1	α=0.05	α=0.01	α=0.1	α=0.05	α=0.01			α=0.1	α=0.05	α=0.01	α=0.1	α=0.05	α=0.01	α=0.1	α=0.05	α=0.01
θ₁	θ₂	SSiz	SSiz	SSiz	SSiz	SSiz	SSiz	SSiz	SSiz	SSiz	θ₁	θ₂	SSiz	SSiz	SSiz	SSiz	SSiz	SSiz	SSiz	SSiz	SSiz
0.76	0.90	47	65	108	66	88	136	84	108	162	0.76	0.92	35	49	82	49	65	102	62	81	121
0.76	0.94	27	38	64	38	51	79	48	62	94	0.76	0.96	22	30	51	30	40	63	37	49	74
0.76	0.98	18	25	41	24	32	51	30	39	60	0.78	0.80	2355	3238	5270	3418	4466	6811	4444	5630	8233
0.78	0.82	579	798	1303	837	1096	1677	1085	1378	2022	0.78	0.84	253	349	572	364	478	734	470	599	882
0.78	0.86	140	194	318	200	263	406	257	329	486	0.78	0.88	88	122	201	125	165	255	160	205	305
0.78	0.90	60	83	138	84	112	174	108	138	207	0.78	0.92	43	60	100	60	80	125	77	99	148
0.78	0.94	32	45	75	45	60	94	57	73	111	0.78	0.96	25	35	59	34	46	73	43	56	85
0.78	0.98	20	28	47	27	36	57	34	44	67	0.80	0.82	2165	2977	4848	3140	4105	6262	4081	5172	7566
0.80	0.84	531	732	1197	766	1005	1539	993	1262	1854	0.80	0.86	231	320	525	332	437	672	429	547	806
0.80	0.88	128	177	291	182	240	371	234	299	443	0.80	0.90	80	111	184	113	150	232	145	186	277
0.80	0.92	54	76	126	76	101	158	97	125	187	0.80	0.94	39	55	91	54	72	113	69	89	134
0.80	0.96	29	41	68	40	54	85	51	66	100	0.80	0.98	22	32	53	31	41	65	38	50	76
0.82	0.84	1966	2704	4406	2849	3726	5688	3702	4693	6870	0.82	0.86	481	664	1086	693	909	1394	897	1141	1678
0.82	0.88	209	289	475	299	394	607	385	492	728	0.82	0.90	115	159	263	163	216	334	209	268	399
0.82	0.92	72	100	166	101	134	209	129	166	248	0.82	0.94	49	68	113	68	90	141	86	111	167
0.82	0.96	35	49	81	48	64	101	61	79	119	0.82	0.98	26	36	61	35	48	75	44	58	88
0.84	0.86	1759	2420	3946	2547	3332	5090	3307	4195	6145	0.84	0.88	429	592	970	617	810	1244	797	1015	1496
0.84	0.90	186	257	424	265	349	540	341	436	646	0.84	0.92	102	141	234	144	190	296	184	236	353
0.84	0.94	63	88	147	89	118	184	113	145	219	0.84	0.96	43	60	100	59	79	124	75	97	147
0.84	0.98	30	43	72	42	56	89	52	68	104	0.86	0.88	1545	2127	3470	2234	2925	4472	2900	3680	5396
		Power(1-β)=0.8			Power(1-β)=0.9			Power(1-β)=0.95					Power(1-β)=0.8			Power(1-β)=0.9			Power(1-β)=0.95
		α=0.1	α=0.05	α=0.01	α=0.1	α=0.05	α=0.01	α=0.1	α=0.05	α=0.01			α=0.1	α=0.05	α=0.01	α=0.1	α=0.05	α=0.01	α=0.1	α=0.05	α=0.01
θ₁	θ₂	SSiz	SSiz	SSiz	SSiz	SSiz	SSiz	SSiz	SSiz	SSiz	θ₁	θ₂	SSiz	SSiz	SSiz	SSiz	SSiz	SSiz	SSiz	SSiz	SSiz
0.86	0.90	375	519	851	538	708	1089	695	886	1307	0.86	0.92	162	225	370	230	304	470	295	378	562
0.86	0.94	88	123	204	124	165	257	158	204	305	0.86	0.96	54	76	127	76	101	159	96	124	188
0.86	0.98	36	51	86	50	67	107	63	82	125	0.88	0.90	1325	1826	2981	1913	2507	3838	2481	3152	4626
0.88	0.92	320	443	728	457	603	929	589	753	1114	0.88	0.94	137	191	316	194	257	399	248	319	476
0.88	0.96	74	104	173	103	138	216	131	170	256	0.88	0.98	45	64	108	63	84	133	79	103	156
0.90	0.92	1100	1518	2482	1586	2080	3189	2053	2612	3840	0.90	0.94	263	366	603	375	495	767	482	617	917
0.90	0.96	112	156	260	157	209	327	200	258	387	0.90	0.98	60	84	141	83	111	175	104	136	206
0.92	0.94	872	1205	1974	1253	1647	2530	1620	2064	3041	0.92	0.96	206	287	475	291	387	601	373	480	716
0.92	0.98	86	121	203	120	160	253	151	197	298	0.94	0.96	641	889	1461	917	1209	1865	1182	1510	2235
0.94	0.98	148	208	347	207	277	434	263	341	514	0.96	0.98	410	572	947	580	770	1197	742	955	1426

Javascript Programs

Sample Size Estimating a Single ROC : The data is in 2 columns.
    - Each row a separate study
    - Col 1 = ROC to be estimated θ
    - Col 2 = Standard Error Required

Standard Error of a Single ROC : The data is in 3 columns.
    - Each row a separate study
    - Col 1 = ROC found θ
    - Col 2 = Number of cases with Outcome Negative
    - Col 3 = Number of cases with Outcome Positive

Sample Size Comparing Two ROCs : The data is in 4 columns.
    - Each row a separate study
    - Col 1 = Probability of Type I Error for Decision(α)
    - Col 2 = Power (1-β)
    - Col 3 & 4 = Anticipated Values of the Two ROCs

Sample Size Compare Two ROCs

Power Estimation Comparing Two ROCs : The data is in 4 columns.
    - Each row a separate study
    - Col 1 = Probability of Type I Error Found(α)
    - Col 2 = sample size per ROC used (Outcome pos + outcome neg)
    - Col 3 & Col 4 = Observed Values of the Two ROCs