Bioequivalance

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StatsToDo: Bioequivalance

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Explanations and References

Bioequivalence is used mostly in the pharmaceutical industry, particularly by regulatory bodies such as the FDA. The definition provided by the FDA is as follows.

"Bioequivalence means the absence of a significant difference in the rate and extent to which the active ingredient or active moiety in pharmaceutical equivalents or pharmaceutical alternatives become available at the site of drug action when administered at the same molar dose under similar conditions in an appropriately designed study". When Bioequivalence is used in this sense, the "appropriately designed studies" required by the FDA will need to be adhered to. The formal descriptions of "appropriateness" are contained in FDA's formal guideline (see reference, but be aware that the latest edition of the guide line should be examined in case of changes). This page will merely provide a sketchy summary of the statistical aspects. These are

Depending on the problems and situations confronting the investigator, two research models can be used.
- The preferred model is the cross over trial, where each test subject is administered both the reference and test agents (pharmaceutical alternatives). The variability to be used for sample size estimation and analysis is the within-subject variability. Usually, this means a known and accepted reference Standard Deviation for repeated sampling in the same individuals, or the pooled within subject Standard Deviation from a previous or current trial.
- The parallel trial, where the reference and test agents (pharmaceutical alternatives) are randomly allocated into two equal size groups of subjects. The variability to be used for sample size estimation and analysis is the total variability, including within-subject and between-subject variability. Usually, this means a known and accepted reference population Standard Deviation or the pooled Standard Deviation of the reference and test groups combined.
The appropriate sample size is calculated and adhered to.
A higher or lower limits of difference to define equivalence is determined prior to data collection. The common limit set by the FDA is ±20% (0.2) of the reference mean value. Statistically this is sometimes called the Critical Difference (CD) or Tolerance Limit (TL)
A clearly defined protocol of allocation to groups or order of administration, control of dosage and environment, and collection and recording of data are adhered to throughout.
When the data from the planned sample size are collected, the mean values of the responses to the reference and test agents are compared. If the difference is within the the tolerance limit, Bioequivalence is accepted. The commonly accepted forms of results for Bioequivalence studies are
- Ratio of means r = mean_test / mean_reference. Bioequivalence is accepted if r is between 0.8 (80%) and 1.2 (120%)
- More commonly the logarithmic transformation of r is used, so that log(r) = log(mean_test / mean_reference) = log(mean_test) - log(mean_reference). if log(r) is between -0.22 (log(0.8)) and 0.18 (log(1.2)), Bioequivalence is accepted.

Example

Please note that the numbers of this example are computer generated to demonstrate the statistical procedure, and are not based on any observations.

We know, from repeated studies in the past that 2 hours after a standard Intramuscular dose of penicillin, the blood level has a mean value of 50 miliunits per ml (mU/ml), with a Standard Deviation of 7.5 mU/ml.

Now that the patent for penicillin has expired, a company has produced a new preparation of penicillin, and seek FDA's licence for commercial development. To obtain this the company needs to show that its preparation (test) is Bioequivalent to the current accepted preparation (reference). We are asked to help establishing this.

Sample Size estimation

We established from published data that the standard response blood level (reference mean) is 50 mU/ml, and Standard Deviation is 7.5mU/ml. From this the Coefficient of Variance CV = 7.5 / 50 = 0.15 (15%). We adopted the commonly used α=0.05, and power=0.9, and the FDA recommended Ω=0.2 (20%). Using the short table from sample size table panel, we decided that the sample size should be 15 cases per group, or 15 pairs in a cross over study.

Data and analysis

We decided to use the cross over design. We recruited 15 subjects, and administered to each subject both the standard (reference) and our new (test) penicillin, using the same dosage and administration, but two weeks apart in order to avoid residual levels. Our subjects receive the two preparation in random order, using a random number table. Blood is taken 2 hours after administration and the concentration of penicillin measured.

The mean values of blood concentration was 49.5mU/ml following the administration of a reference preparation, and 48.2mU/ml following the test preparation. The difference 48.2-49.5 = -1.3mU/ml. This results can be expressed as

Ω = 1.3 / 49.5 = 0.026 (2.6%), well short of the critical value of 0.2 (20%)
Ratio (test/reference) = 48.2 / 49.5 = 0.97, within the Bioequivalent range of 0.8 to 1.2
Log(ratio) = log(0.97) = -0.03, within the Bioequivalent range of -0.22 to +0.18

We can therefore conclude that Bioequivalence is demonstrated between the test and reference preparations.

The total variance in the data was partitioned into those within and between subjects. The within subject Standard Deviation was 8.0 mU/ml. From this, the Coefficient of variation is calculated as 8.0 / 49.5 = 0.162 (16.2%). Using the CV and Omega thus obtained, the one tail power is calculated to be 0.96, well over the usually accepted level of 0.8

We can therefore be confident that the sample size for the data was indeed adequate for the study.

References

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. 102-105, p. 126-130

FDA draft document on bioavailibility and bioequivalance, a pdf file

Calculations

data entry and results

This panel provides explanations for data entry in the Javascript calculations panel

Sample Size Calculations

The parameters to be used are as follows

Probability of Type I Error (α), and the most commonly used one is 0.05
Power (1-β), where β is the Probability of Type II Error. The most commonly used values are 0.8 or 0.9
The reference or expected mean and Standard Deviation
The Critical Difference (CD) or Tolerance Limit. The usual value used by the FDA is 20% (0.2) of the mean

The program firstly calculates two parameters

The Coefficient of Variance CV = Standard Deviation / Mean. Commonly this is multiplied by 100 and expressed as percent however, the ratio is used for calculations in this page
The ratio of Critical Difference and mean, represented by Omega Ω = CD / mean. The FDA recommends 0.2 (20%), so this is usually the value used for this parameter.

Sample size (per group in 2 group parallel study or number of cases in a cross over study) is then calculated using α, power, CV, and Ω. Although the program calculates sample size for both the one and two tail models, the one tail sample size is almost universally used, as the major concern is that the test agent is non-inferior to the reference.

Analysis of Power of results

The parameters to be used are obtained from the collected data, and are as follows

The sample size used, per group in a parallel study, and number of cases in a cross over study.
The mean and Standard Deviation of the reference group in a parallel study, or the reference arm in the cross over study
The difference in mean value between test and reference groups (Diff = mean_test - mean_reference)
Probability of Type I Error (α), and the most commonly used one is 0.05

The program calculates two intermediary parameters, both for comparison against that defined during planning, and also to be used in power analysis

The Coefficient of Variance CV = Standard Deviation / Mean.
The ratio of the Difference and the reference mean, Ω = Diff / mean of reference group

Under the current FDA guidelines, a difference between the test and reference mean that is less than 20% of the reference mean is accepted as Bioequivalence. This value can be represented in 3 expressions.

A ratio between the difference between means and reference mean that is less than 20%, (mean_test - mean_reference) / mean_reference between -0.2 and +0.2
A ratio of the two means, mean_test) / mean_reference between 0.8 and 1.2
Log(ratio of the two means), log(mean_test) / mean_reference) between -0.22 and +0.18

A power estimation is also carried out, to make sure that the data has sufficient power (power>=0.8) to draw confident conclusions. Power calculation is based on Probability of Type I Error α (usually 0.05), and the two parameters CV and Ω calculated from the data. In nearly all cases, the one tail power is used.

Javascript Program

Data Input for Sample Size is a table of 5 columns
    Each row contains data from a separate study
    Col 1 = probability of Type I error (α)
    Col 2 = power (1-β)
    Col 3 = reference mean or expected average of the two groups
    Col 4 = reference Standard Deviation (SD) or expected background SD
    Col 5 = Critical Difference (tolerable level) for equivalence

Data Input for Bioequivalence Analysis is a table of 5 columns
    Each row contains data from a separate study
    Col 1 = sample size (per group or number of pairs)
    Col 2 = reference mean or mean of reference group
    Col 3 = Standard Deviation
    Col 4 = Difference (mean of test group - reference mean)
    Col 5 = probability of Type I error to be used for power analysis

Sample Size :
Power Analysis :

Sample Size Tables

Full Tables

General samplesize table for bioequivalence
Power=(1-β), β = Probability of Type II Error
α=Probability of Type I Error
CV = Coefficient of Variance = Reference Standard Deviation / Reference Mean
Ω = Difference (Test Mean - Reference Mean) / Reference Mean
1 tail and 2 tail are sample size (per group or pairs) for one tail or two tail model
Cells contains sample size per group (or per arm in cross over model). Sample size <4 or >10000 are blanked out

		Power=0.8						Power=0.9						Power=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
CV	Ω	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail
0.02	0.02	15	19	19	24	29	33	19	24	24	29	35	39	24	29	29	34	40	45
0.02	0.04	5	7	7	8	10	11	7	8	8	9	11	13	8	9	9	11	13	14
0.02	0.06	4			5	6	7	4	5	5	6	7	8	5	6	6	6	7	8
0.02	0.08					5	5		4	4		5	6	4	4	4	5	6	6
0.02	0.10						5						5	4	4	4	4	5	5
0.02	0.12																4		5
0.02	0.14																4
0.04	0.02	54	71	71	87	107	123	71	89	89	107	129	146	87	107	107	126	150	168
0.04	0.04	15	19	19	24	29	33	19	24	24	29	35	39	24	29	29	34	40	45
0.04	0.06	8	10	10	12	15	17	10	12	12	14	17	19	12	14	14	17	20	22
0.04	0.08	5	7	7	8	10	11	7	8	8	9	11	13	8	9	9	11	13	14
0.04	0.10	4	5	5	6	7		5	6	6	7	8	9	6	7	7	8	9	10
		Power=0.8						Power=0.9						Power=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
CV	Ω	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail
0.04	0.12	4			5	6	7	4	5	5	6	7	8	5	6	6	6	7	8
0.04	0.14		4	4		5	6	4	4	4	5	6	6	4	5	5	6	6	7
0.04	0.16					5	5		4	4		5	6	4	4	4	5	6	6
0.04	0.18						5		4	4		5	5	4	4	4	5	5	6
0.04	0.20						5						5	4	4	4	4	5	5
0.04	0.22												5		4	4	4		5
0.04	0.24																4		5
0.04	0.26																4
0.04	0.28																4
0.06	0.02	120	156	156	192	237	271	156	197	197	237	287	324	192	237	237	280	334	374
0.06	0.04	31	41	41	50	62	70	41	51	51	61	74	84	50	61	61	72	86	96
0.06	0.06	15	19	19	24	29	33	19	24	24	29	35	39	24	29	29	34	40	45
0.06	0.08	9	12	12	14	18	20	12	15	15	17	21	24	14	17	17	20	24	27
0.06	0.10	7	8	8	10	12	14	8	10	10	12	15	16	10	12	12	14	17	19
0.06	0.12	5	7	7	8	10	11	7	8	8	9	11	13	8	9	9	11	13	14
0.06	0.14	4	5	5	7	8	9	5	6	6	8	9	10	7	8	8	9	10	11
0.06	0.16	4	5	5	6	7		5	6	6	6	8	9	6	6	6	7	9	10
0.06	0.18	4			5	6	7	4	5	5	6	7	8	5	6	6	6	7	8
0.06	0.20		4	4	5	6	6	4	4	4	5	6	7	5	5	5	6	7	7
		Power=0.8						Power=0.9						Power=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
CV	Ω	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail
0.06	0.22		4	4		5	6	4	4	4	5	6	6	4	5	5	5	6	7
0.06	0.24					5	5		4	4		5	6	4	4	4	5	6	6
0.06	0.26					5	5		4	4		5	5	4	4	4	5	5	6
0.06	0.28						5		4	4		5	5	4	4	4	4	5	6
0.06	0.30						5						5	4	4	4	4	5	5
0.08	0.02	212	276	276	339	420	480	276	349	349	418	508	574	339	418	418	495	591	662
0.08	0.04	54	71	71	87	107	123	71	89	89	107	129	146	87	107	107	126	150	168
0.08	0.06	25	33	33	40	49	56	33	41	41	49	59	67	40	49	49	58	69	77
0.08	0.08	15	19	19	24	29	33	19	24	24	29	35	39	24	29	29	34	40	45
0.08	0.10	10	13	13	16	20	23	13	16	16	19	23	26	16	19	19	23	27	30
0.08	0.12	8	10	10	12	15	17	10	12	12	14	17	19	12	14	14	17	20	22
0.08	0.14	6	8	8	9	12	13	8	10	10	11	14	15	9	11	11	13	15	17
0.08	0.16	5	7	7	8	10	11	7	8	8	9	11	13	8	9	9	11	13	14
0.08	0.18	5	6	6	7	8	9	6	7	7	8	9	11	7	8	8	9	11	12
0.08	0.20	4	5	5	6	7		5	6	6	7	8	9	6	7	7	8	9	10
0.08	0.22	4	5	5	5	7		5	5	5	6	7	8	5	6	6	7	8	9
0.08	0.24	4			5	6	7	4	5	5	6	7	8	5	6	6	6	7	8
0.08	0.26		4	4	5	6	6	4	5	5	5	6	7	5	5	5	6	7	8
0.08	0.28		4	4		5	6	4	4	4	5	6	6	4	5	5	6	6	7
0.08	0.30		4	4		5	6	4	4	4	5	5	6	4	5	5	5	6	7
		Power=0.8						Power=0.9						Power=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
CV	Ω	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail
0.10	0.02	330	430	430	528	654	747	430	543	543	652	792	894	528	652	652	771	922	1032
0.10	0.04	84	109	109	134	166	189	109	138	138	165	200	226	134	165	165	195	233	261
0.10	0.06	38	50	50	61	75	86	50	62	62	75	91	103	61	75	75	88	105	118
0.10	0.08	22	29	29	35	44	50	29	36	36	43	52	59	35	43	43	51	61	68
0.10	0.10	15	19	19	24	29	33	19	24	24	29	35	39	24	29	29	34	40	45
0.10	0.12	11	14	14	17	21	24	14	17	17	21	25	28	17	21	21	24	29	32
0.10	0.14	9	11	11	13	16	19	11	13	13	16	19	22	13	16	16	19	22	25
0.10	0.16	7	9	9	11	13	15	9	11	11	13	16	18	11	13	13	15	18	20
0.10	0.18	6	8	8	9	11	13	8	9	9	11	13	15	9	11	11	12	15	16
0.10	0.20	5	7	7	8	10	11	7	8	8	9	11	13	8	9	9	11	13	14
0.10	0.22	5	6	6	7	9	10	6	7	7	8	10	11	7	8	8	9	11	12
0.10	0.24	4	5	5	6	8	9	5	6	6	7	9	10	6	7	7	8	10	11
0.10	0.26	4	5	5	6	7		5	6	6	7	8	9	6	7	7	8	9	10
0.10	0.28	4	5	5	5	6		5	5	5	6	7	8	5	6	6	7	8	9
0.10	0.30	4			5	6	7	4	5	5	6	7	8	5	6	6	6	7	8
0.12	0.02	475	619	619	759	940	1075	619	782	782	938	1139	1286	759	938	938	1109	1326	1485
0.12	0.04	120	156	156	192	237	271	156	197	197	237	287	324	192	237	237	280	334	374
0.12	0.06	54	71	71	87	107	123	71	89	89	107	129	146	87	107	107	126	150	168
0.12	0.08	31	41	41	50	62	70	41	51	51	61	74	84	50	61	61	72	86	96
0.12	0.10	21	27	27	33	41	46	27	34	34	40	49	55	33	40	40	47	56	63
		Power=0.8						Power=0.9						Power=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
CV	Ω	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail
0.12	0.12	15	19	19	24	29	33	19	24	24	29	35	39	24	29	29	34	40	45
0.12	0.14	12	15	15	18	22	25	15	18	18	22	26	30	18	22	22	26	30	34
0.12	0.16	9	12	12	14	18	20	12	15	15	17	21	24	14	17	17	20	24	27
0.12	0.18	8	10	10	12	15	17	10	12	12	14	17	19	12	14	14	17	20	22
0.12	0.20	7	8	8	10	12	14	8	10	10	12	15	16	10	12	12	14	17	19
0.12	0.22	6	7	7	9	11	12	7	9	9	10	13	14	9	10	10	12	14	16
0.12	0.24	5	7	7	8	10	11	7	8	8	9	11	13	8	9	9	11	13	14
0.12	0.26	5	6	6	7	9	10	6	7	7	8	10	11	7	8	8	10	11	12
0.12	0.28	4	5	5	7	8	9	5	6	6	8	9	10	7	8	8	9	10	11
0.12	0.30	4	5	5	6	7		5	6	6	7	8	9	6	7	7	8	9	10
0.14	0.02	646	841	841	1032	1279	1462	841	1063	1063	1276	1549	1749	1032	1276	1276	1509	1804	2020
0.14	0.04	163	212	212	260	322	368	212	268	268	321	390	440	260	321	321	379	453	508
0.14	0.06	73	95	95	117	145	166	95	120	120	144	175	198	117	144	144	170	203	228
0.14	0.08	42	55	55	67	83	95	55	69	69	82	100	113	67	82	82	97	116	130
0.14	0.10	28	36	36	44	54	62	36	45	45	54	65	73	44	54	54	63	75	84
0.14	0.12	20	25	25	31	39	44	25	32	32	38	46	52	31	38	38	45	53	60
0.14	0.14	15	19	19	24	29	33	19	24	24	29	35	39	24	29	29	34	40	45
0.14	0.16	12	15	15	19	23	26	15	19	19	23	27	31	19	23	23	26	32	35
0.14	0.18	10	13	13	15	19	22	13	16	16	18	22	25	15	18	18	22	26	29
0.14	0.20	8	11	11	13	16	18	11	13	13	15	19	21	13	15	15	18	21	24
		Power=0.8						Power=0.9						Power=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
CV	Ω	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail
0.14	0.22	7	9	9	11	14	16	9	11	11	13	16	18	11	13	13	15	18	20
0.14	0.24	6	8	8	10	12	14	8	10	10	12	14	16	10	12	12	13	16	18
0.14	0.26	6	7	7	9	11	12	7	9	9	10	12	14	9	10	10	12	14	16
0.14	0.28	5	7	7	8	10	11	7	8	8	9	11	13	8	9	9	11	13	14
0.14	0.30	5	6	6	7	9	10	6	7	7	8	10	11	7	8	8	10	11	13
0.16	0.02	843	1098	1098	1347	1669	1908	1098	1388	1388	1666	2022	2284	1347	1666	1666	1970	2355	2637
0.16	0.04	212	276	276	339	420	480	276	349	349	418	508	574	339	418	418	495	591	662
0.16	0.06	95	124	124	152	188	215	124	156	156	187	227	257	152	187	187	221	265	296
0.16	0.08	54	71	71	87	107	123	71	89	89	107	129	146	87	107	107	126	150	168
0.16	0.10	35	46	46	56	70	80	46	58	58	69	84	95	56	69	69	82	97	109
0.16	0.12	25	33	33	40	49	56	33	41	41	49	59	67	40	49	49	58	69	77
0.16	0.14	19	25	25	30	37	42	25	31	31	37	44	50	30	37	37	43	51	57
0.16	0.16	15	19	19	24	29	33	19	24	24	29	35	39	24	29	29	34	40	45
0.16	0.18	12	16	16	19	24	27	16	20	20	23	28	32	19	23	23	27	32	36
0.16	0.20	10	13	13	16	20	23	13	16	16	19	23	26	16	19	19	23	27	30
0.16	0.22	9	11	11	14	17	19	11	14	14	16	20	22	14	16	16	19	23	25
0.16	0.24	8	10	10	12	15	17	10	12	12	14	17	19	12	14	14	17	20	22
0.16	0.26	7	9	9	11	13	15	9	11	11	13	15	17	11	13	13	15	17	19
0.16	0.28	6	8	8	9	12	13	8	10	10	11	14	15	9	11	11	13	15	17
0.16	0.30	6	7	7	9	11	12	7	9	9	10	12	14	9	10	10	12	14	15
		Power=0.8						Power=0.9						Power=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
CV	Ω	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail
0.18	0.02	1066	1389	1389	1705	2112	2414	1389	1756	1756	2108	2558	2889	1705	2108	2108	2492	2980	3337
0.18	0.04	268	349	349	428	530	606	349	441	441	529	642	725	428	529	529	625	747	837
0.18	0.06	120	156	156	192	237	271	156	197	197	237	287	324	192	237	237	280	334	374
0.18	0.08	68	89	89	109	135	154	89	112	112	134	163	184	109	134	134	159	189	212
0.18	0.10	44	58	58	71	87	100	58	72	72	87	105	119	71	87	87	103	122	137
0.18	0.12	31	41	41	50	62	70	41	51	51	61	74	84	50	61	61	72	86	96
0.18	0.14	24	30	30	37	46	53	30	38	38	46	55	62	37	46	46	54	64	72
0.18	0.16	19	24	24	29	36	41	24	30	30	36	43	49	29	36	36	42	50	56
0.18	0.18	15	19	19	24	29	33	19	24	24	29	35	39	24	29	29	34	40	45
0.18	0.20	13	16	16	20	24	28	16	20	20	24	29	32	20	24	24	28	33	37
0.18	0.22	11	14	14	17	21	23	14	17	17	20	24	27	17	20	20	24	28	31
0.18	0.24	9	12	12	14	18	20	12	15	15	17	21	24	14	17	17	20	24	27
0.18	0.26	8	10	10	13	16	18	10	13	13	15	18	21	13	15	15	18	21	23
0.18	0.28	7	9	9	11	14	16	9	11	11	13	16	18	11	13	13	16	19	21
0.18	0.30	7	8	8	10	12	14	8	10	10	12	15	16	10	12	12	14	17	19
0.20	0.02	1316	1715	1715	2104	2606	2979	1715	2167	2167	2602	3157	3566	2104	2602	2602	3076	3678	4118
0.20	0.04	330	430	430	528	654	747	430	543	543	652	792	894	528	652	652	771	922	1032
0.20	0.06	148	192	192	236	292	334	192	243	243	291	354	399	236	291	291	344	412	461
0.20	0.08	84	109	109	134	166	189	109	138	138	165	200	226	134	165	165	195	233	261
0.20	0.10	54	71	71	87	107	123	71	89	89	107	129	146	87	107	107	126	150	168
		Power=0.8						Power=0.9						Power=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
CV	Ω	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail
0.20	0.12	38	50	50	61	75	86	50	62	62	75	91	103	61	75	75	88	105	118
0.20	0.14	29	37	37	45	56	64	37	47	47	56	68	76	45	56	56	66	78	88
0.20	0.16	22	29	29	35	44	50	29	36	36	43	52	59	35	43	43	51	61	68
0.20	0.18	18	23	23	28	35	40	23	29	29	35	42	48	28	35	35	41	49	55
0.20	0.20	15	19	19	24	29	33	19	24	24	29	35	39	24	29	29	34	40	45
0.20	0.22	13	16	16	20	25	28	16	20	20	24	29	33	20	24	24	28	34	38
0.20	0.24	11	14	14	17	21	24	14	17	17	21	25	28	17	21	21	24	29	32
0.20	0.26	10	12	12	15	19	21	12	15	15	18	22	25	15	18	18	21	25	28
0.20	0.28	9	11	11	13	16	19	11	13	13	16	19	22	13	16	16	19	22	25
0.20	0.30	8	10	10	12	15	17	10	12	12	14	17	19	12	14	14	17	20	22
0.22	0.02	1592	2075	2075	2545	3153	3604	2075	2621	2621	3147	3820	4315	2545	3147	3147	3721	4449	4983
0.22	0.04	399	520	520	638	791	904	520	657	657	789	957	1081	638	789	789	933	1115	1248
0.22	0.06	178	232	232	285	353	404	232	293	293	352	427	483	285	352	352	416	497	557
0.22	0.08	101	132	132	161	200	229	132	166	166	199	242	273	161	199	199	235	281	315
0.22	0.10	65	85	85	104	129	148	85	107	107	128	156	176	104	128	128	152	181	203
0.22	0.12	46	60	60	73	91	104	60	75	75	90	109	123	73	90	90	106	127	142
0.22	0.14	34	44	44	54	67	77	44	56	56	67	81	92	54	67	67	79	94	105
0.22	0.16	27	35	35	42	52	60	35	43	43	52	63	71	42	52	52	61	73	82
0.22	0.18	21	28	28	34	42	48	28	35	35	42	50	57	34	42	42	49	58	65
0.22	0.20	18	23	23	28	35	40	23	29	29	34	41	47	28	34	34	40	48	54
		Power=0.8						Power=0.9						Power=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
CV	Ω	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail
0.22	0.22	15	19	19	24	29	33	19	24	24	29	35	39	24	29	29	34	40	45
0.22	0.24	13	17	17	20	25	29	17	21	21	25	30	34	20	25	25	29	34	38
0.22	0.26	11	14	14	18	22	25	14	18	18	21	26	29	18	21	21	25	30	33
0.22	0.28	10	13	13	16	19	22	13	16	16	19	23	26	16	19	19	22	26	29
0.22	0.30	9	11	11	14	17	20	11	14	14	17	20	23	14	17	17	19	23	26
0.24	0.02	1894	2469	2469	3029	3752	4289	2469	3119	3119	3745	4545	5134	3029	3745	3745	4428	5295	5929
0.24	0.04	475	619	619	759	940	1075	619	782	782	938	1139	1286	759	938	938	1109	1326	1485
0.24	0.06	212	276	276	339	420	480	276	349	349	418	508	574	339	418	418	495	591	662
0.24	0.08	120	156	156	192	237	271	156	197	197	237	287	324	192	237	237	280	334	374
0.24	0.10	78	101	101	124	153	175	101	127	127	152	185	209	124	152	152	180	215	241
0.24	0.12	54	71	71	87	107	123	71	89	89	107	129	146	87	107	107	126	150	168
0.24	0.14	40	52	52	64	80	91	52	66	66	79	96	108	64	79	79	93	111	125
0.24	0.16	31	41	41	50	62	70	41	51	51	61	74	84	50	61	61	72	86	96
0.24	0.18	25	33	33	40	49	56	33	41	41	49	59	67	40	49	49	58	69	77
0.24	0.20	21	27	27	33	41	46	27	34	34	40	49	55	33	40	40	47	56	63
0.24	0.22	18	23	23	28	34	39	23	28	28	34	41	46	28	34	34	40	47	53
0.24	0.24	15	19	19	24	29	33	19	24	24	29	35	39	24	29	29	34	40	45
0.24	0.26	13	17	17	20	25	29	17	21	21	25	30	34	20	25	25	29	35	39
0.24	0.28	12	15	15	18	22	25	15	18	18	22	26	30	18	22	22	26	30	34
0.24	0.30	10	13	13	16	20	23	13	16	16	19	23	26	16	19	19	23	27	30
		Power=0.8						Power=0.9						Power=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
CV	Ω	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail
0.26	0.02	2222	2897	2897	3554	4403	5033	2897	3660	3660	4395	5334	6025	3554	4395	4395	5197	6213	6958
0.26	0.04	557	726	726	890	1103	1261	726	917	917	1101	1336	1509	890	1101	1101	1301	1556	1742
0.26	0.06	249	324	324	397	492	562	324	409	409	491	595	673	397	491	491	580	693	776
0.26	0.08	141	183	183	224	278	318	183	231	231	277	336	380	224	277	277	328	391	438
0.26	0.10	91	118	118	145	179	205	118	149	149	178	216	244	145	178	178	211	252	282
0.26	0.12	64	83	83	101	125	143	83	104	104	125	151	171	101	125	125	147	176	197
0.26	0.14	47	61	61	75	93	106	61	77	77	92	112	126	75	92	92	109	130	146
0.26	0.16	37	47	47	58	72	82	47	60	60	71	86	98	58	71	71	84	100	112
0.26	0.18	29	38	38	46	57	66	38	48	48	57	69	78	46	57	57	67	80	90
0.26	0.20	24	31	31	38	47	54	31	39	39	47	56	64	38	47	47	55	65	73
0.26	0.22	20	26	26	32	39	45	26	33	33	39	47	53	32	39	39	46	55	61
0.26	0.24	17	22	22	27	34	38	22	28	28	33	40	45	27	33	33	39	46	52
0.26	0.26	15	19	19	24	29	33	19	24	24	29	35	39	24	29	29	34	40	45
0.26	0.28	13	17	17	21	26	29	17	21	21	25	30	34	21	25	25	29	35	39
0.26	0.30	12	15	15	18	23	26	15	19	19	22	27	30	18	22	22	26	31	35
0.28	0.02	2577	3359	3359	4121	5106	5836	3359	4245	4245	5097	6185	6987	4121	5097	5097	6026	7205	8068
0.28	0.04	646	841	841	1032	1279	1462	841	1063	1063	1276	1549	1749	1032	1276	1276	1509	1804	2020
0.28	0.06	288	375	375	460	570	652	375	474	474	569	690	779	460	569	569	672	804	900
0.28	0.08	163	212	212	260	322	368	212	268	268	321	390	440	260	321	321	379	453	508
0.28	0.10	105	136	136	167	207	237	136	172	172	206	250	283	167	206	206	244	291	326
		Power=0.8						Power=0.9						Power=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
CV	Ω	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail
0.28	0.12	73	95	95	117	145	166	95	120	120	144	175	198	117	144	144	170	203	228
0.28	0.14	54	71	71	87	107	123	71	89	89	107	129	146	87	107	107	126	150	168
0.28	0.16	42	55	55	67	83	95	55	69	69	82	100	113	67	82	82	97	116	130
0.28	0.18	34	44	44	53	66	76	44	55	55	66	79	90	53	66	66	77	92	103
0.28	0.20	28	36	36	44	54	62	36	45	45	54	65	73	44	54	54	63	75	84
0.28	0.22	23	30	30	37	45	52	30	37	37	45	54	61	37	45	45	53	63	70
0.28	0.24	20	25	25	31	39	44	25	32	32	38	46	52	31	38	38	45	53	60
0.28	0.26	17	22	22	27	33	38	22	27	27	33	40	45	27	33	33	39	46	51
0.28	0.28	15	19	19	24	29	33	19	24	24	29	35	39	24	29	29	34	40	45
0.28	0.30	13	17	17	21	26	29	17	21	21	25	31	35	21	25	25	30	35	40
0.30	0.02	2958	3856	3856	4731	5861	6699	3856	4872	4872	5850	7100	8020	4731	5850	5850	6918	8271	9262
0.30	0.04	741	966	966	1185	1467	1677	966	1220	1220	1465	1777	2008	1185	1465	1465	1732	2070	2318
0.30	0.06	330	430	430	528	654	747	430	543	543	652	792	894	528	652	652	771	922	1032
0.30	0.08	187	243	243	298	369	422	243	307	307	368	447	505	298	368	368	435	520	582
0.30	0.10	120	156	156	192	237	271	156	197	197	237	287	324	192	237	237	280	334	374
0.30	0.12	84	109	109	134	166	189	109	138	138	165	200	226	134	165	165	195	233	261
0.30	0.14	62	81	81	99	123	140	81	102	102	122	148	167	99	122	122	144	172	193
0.30	0.16	48	62	62	76	95	108	62	78	78	94	114	129	76	94	94	111	133	148
0.30	0.18	38	50	50	61	75	86	50	62	62	75	91	103	61	75	75	88	105	118
0.30	0.20	31	41	41	50	62	70	41	51	51	61	74	84	50	61	61	72	86	96
		Power=0.8						Power=0.9						Power=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
CV	Ω	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail
0.30	0.22	26	34	34	42	51	59	34	43	43	51	62	70	42	51	51	60	72	80
0.30	0.24	22	29	29	35	44	50	29	36	36	43	52	59	35	43	43	51	61	68
0.30	0.26	19	25	25	31	38	43	25	31	31	37	45	51	31	37	37	44	52	59
0.30	0.28	17	22	22	27	33	38	22	27	27	33	39	44	27	33	33	38	46	51
0.30	0.30	15	19	19	24	29	33	19	24	24	29	35	39	24	29	29	34	40	45
0.32	0.02	3365	4387	4387	5382	6668	7622	4387	5543	5543	6656	8078	9124	5382	6656	6656	7870	9410
0.32	0.04	843	1098	1098	1347	1669	1908	1098	1388	1388	1666	2022	2284	1347	1666	1666	1970	2355	2637
0.32	0.06	376	489	489	600	744	850	489	618	618	742	900	1017	600	742	742	877	1049	1174
0.32	0.08	212	276	276	339	420	480	276	349	349	418	508	574	339	418	418	495	591	662
0.32	0.10	136	178	178	218	270	308	178	224	224	269	326	368	218	269	269	318	380	425
0.32	0.12	95	124	124	152	188	215	124	156	156	187	227	257	152	187	187	221	265	296
0.32	0.14	70	92	92	112	139	159	92	115	115	138	168	190	112	138	138	163	195	219
0.32	0.16	54	71	71	87	107	123	71	89	89	107	129	146	87	107	107	126	150	168
0.32	0.18	43	56	56	69	85	98	56	71	71	85	103	116	69	85	85	100	119	134
0.32	0.20	35	46	46	56	70	80	46	58	58	69	84	95	56	69	69	82	97	109
0.32	0.22	30	38	38	47	58	66	38	48	48	58	70	79	47	58	58	68	81	91
0.32	0.24	25	33	33	40	49	56	33	41	41	49	59	67	40	49	49	58	69	77
0.32	0.26	22	28	28	34	42	49	28	35	35	42	51	58	34	42	42	49	59	66
0.32	0.28	19	25	25	30	37	42	25	31	31	37	44	50	30	37	37	43	51	57
0.32	0.30	17	22	22	26	33	37	22	27	27	32	39	44	26	32	32	38	45	51
		Power=0.8						Power=0.9						Power=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
CV	Ω	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail
0.34	0.02	3799	4952	4952	6076	7527	8604	4952	6258	6258	7514	9118		6076	7514	7514	8884
0.34	0.04	951	1240	1240	1521	1884	2154	1240	1566	1566	1880	2282	2578	1521	1880	1880	2223	2658	2977
0.34	0.06	424	552	552	677	839	959	552	697	697	837	1016	1148	677	837	837	990	1183	1325
0.34	0.08	239	311	311	382	473	541	311	393	393	472	573	647	382	472	472	558	667	747
0.34	0.10	154	200	200	245	304	348	200	253	253	303	368	415	245	303	303	358	428	479
0.34	0.12	107	140	140	171	212	242	140	176	176	211	256	290	171	211	211	250	298	334
0.34	0.14	79	103	103	126	157	179	103	130	130	156	189	214	126	156	156	184	220	246
0.34	0.16	61	79	79	97	121	138	79	100	100	120	146	164	97	120	120	142	169	190
0.34	0.18	49	63	63	77	96	110	63	80	80	95	116	131	77	95	95	113	134	151
0.34	0.20	40	52	52	63	78	90	52	65	65	78	94	107	63	78	78	92	110	123
0.34	0.22	33	43	43	53	65	75	43	54	54	65	78	89	53	65	65	76	91	102
0.34	0.24	28	37	37	45	55	63	37	46	46	55	66	75	45	55	55	65	77	86
0.34	0.26	24	31	31	38	48	54	31	39	39	47	57	64	38	47	47	55	66	74
0.34	0.28	21	27	27	34	41	47	27	34	34	41	50	56	34	41	41	48	58	64
0.34	0.30	19	24	24	30	37	42	24	30	30	36	44	49	30	36	36	42	51	57
0.36	0.02	4259	5551	5551	6811	8438		5551	7015	7015	8423			6811	8423	8423	9960
0.36	0.04	1066	1389	1389	1705	2112	2414	1389	1756	1756	2108	2558	2889	1705	2108	2108	2492	2980	3337
0.36	0.06	475	619	619	759	940	1075	619	782	782	938	1139	1286	759	938	938	1109	1326	1485
0.36	0.08	268	349	349	428	530	606	349	441	441	529	642	725	428	529	529	625	747	837
0.36	0.10	172	224	224	275	340	389	224	283	283	339	412	465	275	339	339	401	480	537
		Power=0.8						Power=0.9						Power=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
CV	Ω	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail
0.36	0.12	120	156	156	192	237	271	156	197	197	237	287	324	192	237	237	280	334	374
0.36	0.14	89	115	115	141	175	200	115	145	145	175	212	239	141	175	175	206	246	276
0.36	0.16	68	89	89	109	135	154	89	112	112	134	163	184	109	134	134	159	189	212
0.36	0.18	54	71	71	87	107	123	71	89	89	107	129	146	87	107	107	126	150	168
0.36	0.20	44	58	58	71	87	100	58	72	72	87	105	119	71	87	87	103	122	137
0.36	0.22	37	48	48	59	73	83	48	60	60	72	88	99	59	72	72	85	102	114
0.36	0.24	31	41	41	50	62	70	41	51	51	61	74	84	50	61	61	72	86	96
0.36	0.26	27	35	35	43	53	61	35	44	44	53	64	72	43	53	53	62	74	83
0.36	0.28	24	30	30	37	46	53	30	38	38	46	55	62	37	46	46	54	64	72
0.36	0.30	21	27	27	33	41	46	27	34	34	40	49	55	33	40	40	47	56	63
0.38	0.02	4745	6185	6185	7589	9401		6185	7816	7816	9385			7589	9385	9385
0.38	0.04	1188	1548	1548	1899	2353	2689	1548	1956	1956	2348	2850	3219	1899	2348	2348	2776	3320	3717
0.38	0.06	529	689	689	845	1047	1197	689	871	871	1045	1268	1433	845	1045	1045	1236	1477	1654
0.38	0.08	298	389	389	477	590	675	389	491	491	589	715	807	477	589	589	696	832	932
0.38	0.10	192	249	249	306	379	433	249	315	315	378	459	518	306	378	378	447	534	598
0.38	0.12	134	174	174	213	264	302	174	219	219	263	319	361	213	263	263	311	372	416
0.38	0.14	99	128	128	157	195	223	128	162	162	194	236	266	157	194	194	229	274	307
0.38	0.16	76	99	99	121	150	171	99	124	124	149	181	205	121	149	149	176	211	236
0.38	0.18	60	78	78	96	119	136	78	99	99	118	144	162	96	118	118	140	167	187
0.38	0.20	49	64	64	78	97	111	64	81	81	96	117	132	78	96	96	114	136	152
		Power=0.8						Power=0.9						Power=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
CV	Ω	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail
0.38	0.22	41	53	53	65	81	92	53	67	67	80	97	110	65	80	80	95	113	126
0.38	0.24	35	45	45	55	68	78	45	57	57	68	82	93	55	68	68	80	95	107
0.38	0.26	30	39	39	47	59	67	39	49	49	58	71	80	47	58	58	69	82	92
0.38	0.28	26	34	34	41	51	58	34	42	42	51	61	69	41	51	51	60	71	80
0.38	0.30	23	30	30	36	45	51	30	37	37	44	54	61	36	44	44	52	62	70
0.40	0.02	5257	6853	6853	8408			6853	8660	8660				8408
0.40	0.04	1316	1715	1715	2104	2606	2979	1715	2167	2167	2602	3157	3566	2104	2602	2602	3076	3678	4118
0.40	0.06	586	763	763	936	1160	1326	763	964	964	1158	1405	1587	936	1158	1158	1369	1636	1832
0.40	0.08	330	430	430	528	654	747	430	543	543	652	792	894	528	652	652	771	922	1032
0.40	0.10	212	276	276	339	420	480	276	349	349	418	508	574	339	418	418	495	591	662
0.40	0.12	148	192	192	236	292	334	192	243	243	291	354	399	236	291	291	344	412	461
0.40	0.14	109	142	142	174	216	246	142	179	179	215	261	294	174	215	215	254	303	340
0.40	0.16	84	109	109	134	166	189	109	138	138	165	200	226	134	165	165	195	233	261
0.40	0.18	67	87	87	106	132	150	87	109	109	131	159	179	106	131	131	155	185	207
0.40	0.20	54	71	71	87	107	123	71	89	89	107	129	146	87	107	107	126	150	168
0.40	0.22	45	59	59	72	89	102	59	74	74	89	107	121	72	89	89	105	125	140
0.40	0.24	38	50	50	61	75	86	50	62	62	75	91	103	61	75	75	88	105	118
0.40	0.26	33	43	43	52	65	74	43	54	54	64	78	88	52	64	64	76	90	101
0.40	0.28	29	37	37	45	56	64	37	47	47	56	68	76	45	56	56	66	78	88
0.40	0.30	25	33	33	40	49	56	33	41	41	49	59	67	40	49	49	58	69	77
		Power=0.8						Power=0.9						Power=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
CV	Ω	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail
0.42	0.02	5796	7555	7555	9270			7555	9548	9548				9270
0.42	0.04	1450	1890	1890	2319	2873	3284	1890	2389	2389	2868	3481	3932	2319	2868	2868	3391	4054	4540
0.42	0.06	646	841	841	1032	1279	1462	841	1063	1063	1276	1549	1749	1032	1276	1276	1509	1804	2020
0.42	0.08	364	474	474	582	721	824	474	599	599	719	872	986	582	719	719	850	1016	1138
0.42	0.10	234	304	304	373	462	528	304	384	384	461	560	632	373	461	461	545	652	730
0.42	0.12	163	212	212	260	322	368	212	268	268	321	390	440	260	321	321	379	453	508
0.42	0.14	120	156	156	192	237	271	156	197	197	237	287	324	192	237	237	280	334	374
0.42	0.16	92	120	120	147	182	209	120	152	152	182	220	249	147	182	182	215	257	287
0.42	0.18	73	95	95	117	145	166	95	120	120	144	175	198	117	144	144	170	203	228
0.42	0.20	60	78	78	95	118	135	78	98	98	117	142	161	95	117	117	138	165	185
0.42	0.22	50	65	65	79	98	112	65	81	81	97	118	133	79	97	97	115	137	154
0.42	0.24	42	55	55	67	83	95	55	69	69	82	100	113	67	82	82	97	116	130
0.42	0.26	36	47	47	57	71	81	47	59	59	70	85	97	57	70	70	83	99	111
0.42	0.28	31	41	41	50	62	70	41	51	51	61	74	84	50	61	61	72	86	96
0.42	0.30	28	36	36	44	54	62	36	45	45	54	65	73	44	54	54	63	75	84
0.44	0.02	6361	8292	8292				8292
0.44	0.04	1592	2075	2075	2545	3153	3604	2075	2621	2621	3147	3820	4315	2545	3147	3147	3721	4449	4983
0.44	0.06	708	923	923	1133	1403	1604	923	1166	1166	1400	1699	1920	1133	1400	1400	1656	1979	2216
0.44	0.08	399	520	520	638	791	904	520	657	657	789	957	1081	638	789	789	933	1115	1248
0.44	0.10	256	334	334	409	507	580	334	421	421	506	614	693	409	506	506	598	715	800
		Power=0.8						Power=0.9						Power=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
CV	Ω	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail
0.44	0.12	178	232	232	285	353	404	232	293	293	352	427	483	285	352	352	416	497	557
0.44	0.14	132	171	171	210	260	297	171	216	216	259	315	355	210	259	259	306	366	410
0.44	0.16	101	132	132	161	200	229	132	166	166	199	242	273	161	199	199	235	281	315
0.44	0.18	80	104	104	128	159	181	104	132	132	158	192	216	128	158	158	187	223	250
0.44	0.20	65	85	85	104	129	148	85	107	107	128	156	176	104	128	128	152	181	203
0.44	0.22	54	71	71	87	107	123	71	89	89	107	129	146	87	107	107	126	150	168
0.44	0.24	46	60	60	73	91	104	60	75	75	90	109	123	73	90	90	106	127	142
0.44	0.26	39	51	51	63	78	89	51	64	64	77	93	106	63	77	77	91	109	122
0.44	0.28	34	44	44	54	67	77	44	56	56	67	81	92	54	67	67	79	94	105
0.44	0.30	30	39	39	48	59	68	39	49	49	59	71	80	48	59	59	69	82	92
0.46	0.02	6952	9063	9063				9063
0.46	0.04	1739	2267	2267	2782	3446	3939	2267	2865	2865	3440	4174	4715	2782	3440	3440	4067	4863	5445
0.46	0.06	774	1009	1009	1238	1533	1753	1009	1275	1275	1530	1857	2098	1238	1530	1530	1809	2163	2422
0.46	0.08	436	568	568	697	864	987	568	718	718	862	1046	1182	697	862	862	1019	1218	1364
0.46	0.10	280	365	365	447	554	633	365	460	460	553	671	757	447	553	553	653	781	874
0.46	0.12	195	254	254	311	386	441	254	320	320	385	467	527	311	385	385	455	543	608
0.46	0.14	144	187	187	229	284	325	187	236	236	283	344	388	229	283	283	335	400	448
0.46	0.16	110	144	144	176	218	249	144	181	181	217	264	298	176	217	217	257	307	344
0.46	0.18	88	114	114	140	173	198	114	144	144	172	209	236	140	172	172	204	243	272
0.46	0.20	71	93	93	114	141	161	93	117	117	140	170	192	114	140	140	165	198	221
		Power=0.8						Power=0.9						Power=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
CV	Ω	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail
0.46	0.22	59	77	77	94	117	134	77	97	97	116	141	159	94	116	116	137	164	184
0.46	0.24	50	65	65	80	99	113	65	82	82	98	119	134	80	98	98	116	138	155
0.46	0.26	43	56	56	68	85	97	56	70	70	84	102	115	68	84	84	99	118	133
0.46	0.28	37	48	48	59	73	84	48	61	61	73	88	100	59	73	73	86	103	115
0.46	0.30	33	42	42	52	64	73	42	53	53	64	77	87	52	64	64	75	90	100
0.48	0.02	7570	9868	9868				9868
0.48	0.04	1894	2469	2469	3029	3752	4289	2469	3119	3119	3745	4545	5134	3029	3745	3745	4428	5295	5929
0.48	0.06	843	1098	1098	1347	1669	1908	1098	1388	1388	1666	2022	2284	1347	1666	1666	1970	2355	2637
0.48	0.08	475	619	619	759	940	1075	619	782	782	938	1139	1286	759	938	938	1109	1326	1485
0.48	0.10	305	397	397	487	603	689	397	501	501	601	730	824	487	601	601	711	850	952
0.48	0.12	212	276	276	339	420	480	276	349	349	418	508	574	339	418	418	495	591	662
0.48	0.14	156	203	203	250	309	353	203	257	257	308	374	422	250	308	308	364	435	487
0.48	0.16	120	156	156	192	237	271	156	197	197	237	287	324	192	237	237	280	334	374
0.48	0.18	95	124	124	152	188	215	124	156	156	187	227	257	152	187	187	221	265	296
0.48	0.20	78	101	101	124	153	175	101	127	127	152	185	209	124	152	152	180	215	241
0.48	0.22	64	84	84	103	127	145	84	105	105	126	153	173	103	126	126	149	178	200
0.48	0.24	54	71	71	87	107	123	71	89	89	107	129	146	87	107	107	126	150	168
0.48	0.26	47	61	61	74	92	105	61	76	76	91	111	125	74	91	91	108	129	144
0.48	0.28	40	52	52	64	80	91	52	66	66	79	96	108	64	79	79	93	111	125
0.48	0.30	35	46	46	56	70	80	46	58	58	69	84	95	56	69	69	82	97	109
		Power=0.8						Power=0.9						Power=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
CV	Ω	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail
0.50	0.02	8214
0.50	0.04	2055	2678	2678	3286	4071	4653	2678	3384	3384	4064	4931	5571	3286	4064	4064	4805	5745	6433
0.50	0.06	914	1192	1192	1462	1811	2070	1192	1505	1505	1807	2193	2478	1462	1807	1807	2137	2555	2861
0.50	0.08	515	671	671	823	1020	1166	671	848	848	1018	1235	1395	823	1018	1018	1203	1439	1611
0.50	0.10	330	430	430	528	654	747	430	543	543	652	792	894	528	652	652	771	922	1032
0.50	0.12	230	299	299	367	455	520	299	378	378	454	551	622	367	454	454	536	641	718
0.50	0.14	169	221	221	271	335	383	221	278	278	334	405	458	271	334	334	395	472	529
0.50	0.16	130	169	169	208	257	294	169	214	214	256	311	351	208	256	256	303	362	406
0.50	0.18	103	134	134	165	204	233	134	169	169	203	247	278	165	203	203	240	287	321
0.50	0.20	84	109	109	134	166	189	109	138	138	165	200	226	134	165	165	195	233	261
0.50	0.22	70	91	91	111	138	157	91	114	114	137	166	188	111	137	137	162	193	216
0.50	0.24	59	76	76	94	116	133	76	96	96	115	140	158	94	115	115	136	163	182
0.50	0.26	50	65	65	80	99	114	65	82	82	99	120	135	80	99	99	117	139	156
0.50	0.28	44	57	57	70	86	98	57	71	71	86	104	117	70	86	86	101	121	135
0.50	0.30	38	50	50	61	75	86	50	62	62	75	91	103	61	75	75	88	105	118

Short Tables

Table specifically for Ratio between two means for FDA approval studies
Power=(1-β), β = Probability of Type II Error
α=Probability of Type I Error
CV = Coefficient of Variance = Reference Standard Deviation / Reference Mean
Table uses Ω = 0.2 as stipulated by FDA, ans cells contains sample size (per group or pairs) for the one tail model
Cells contain sample size (per group or per arm in cross over studies). Sample size of <4 are blanked out

	Power=0.8		Power=0.9			Power=0.8		Power=0.9			Power=0.8		Power=0.9
CV	α=0.05	α=0.01	α=0.05	α=0.01	CV	α=0.05	α=0.01	α=0.05	α=0.01	CV	α=0.05	α=0.01	α=0.05	α=0.01
0.01					0.02					0.03
0.04					0.05		5	4	5	0.06	4	6	4	6
0.07		6	5	7	0.08	5	7	6	8	0.09	6	8	7	10
0.10	7	10	8	11	0.11	7	11	9	13	0.12	8	12	10	15
0.13	9	14	12	17	0.14	11	16	13	19	0.15	12	18	15	21
0.16	13	20	16	23	0.17	15	22	18	26	0.18	16	24	20	29
0.19	18	27	22	32	0.20	19	29	24	35	0.21	21	32	26	38
0.22	23	35	29	41	0.23	25	37	31	45	0.24	27	41	34	49
0.25	29	44	36	52	0.26	31	47	39	56	0.27	33	51	42	61
0.28	36	54	45	65	0.29	38	58	48	69	0.30	41	62	51	74
0.31	43	66	54	79	0.32	46	70	58	84	0.33	49	74	61	89
0.34	52	78	65	94	0.35	55	83	69	100	0.36	58	87	72	105
0.37	61	92	76	111	0.38	64	97	81	117	0.39	67	102	85	123
0.40	71	107	89	129	0.41	74	112	93	136	0.42	78	118	98	142
0.43	81	123	102	149	0.44	85	129	107	156	0.45	89	135	112	163
0.46	93	141	117	170	0.47	97	147	122	177	0.48	101	153	127	185
0.49	105	159	132	192	0.50	109	166	138	200