Historical perspective

This page provides only a quick summary to provide context for discussions regarding equivalence.

In the 19th Century, Fisher developed the idea of Type I Error, based on the Normal Distribution, thus allows a probability estimate for whether the null hypothesis can be rejected. This allows decisions to be made in science and industry on whether a new product or process is better or worse than the current ones available. However, if the null hypothesis cannot be rejected, the researcher cannot draw any statistical conclusion, as a failure to reject null is not the same as an ability to accept null.

A generation later, Pearson added the idea of Type II Error and the statistical significance, so that both the ability to reject and accept the null hypothesis can be made. Although this method was widely used in the twentieth century, it was increasingly criticised because results of research are often nor reproducible because of difficulties in determining the population Standard Deviation.

To provide robustness to statistical conclusions, researchers increasingly used the 95% confidence interval of the difference, which is an intuitively easier to understand expression of the Type I Error, and to carry out power analysis, which makes no assumption about population parameters. The combination of these two approaches allows researchers to draw confident conclusions whether two sets of observations can be considered significantly different. However, the problem remains that a failure to demonstrate significant difference is not the same as to demonstrate similarity, and the ability to robustly demonstrate similarity is increasingly required, particularly in biomedical research.

An example is in cancer treatment. The current treatment may have severe side effects, and a new treatment may have much more acceptable side effects, but the researcher needs to know whether the effectiveness in controlling the cancer is the same, or at least not inferior to the current treatment.

The concept of equivalence

Given the random variations inherent in any set of observations, it is very unlikely to demonstrate two groups to have the same mean values. The term equivalence is therefore used to represent similarity. This is dependent on a pre-determined and arbitrarily assigned Critical Difference (CD) or Tolerance Limit (TL), a difference that can be considered as trivial in the practical sense. Using the 95% confidence interval of the difference to illustrate, the various conclusions that can be drawn are as shown in the diagram (Forest Plot) to the right. Assuming that the difference is that between group 1 and 2 (diff = mean₁ - mean₂)

• If the 95% CI does not intercept the null value, as shown in the 1^st and the 5^th lines in the diagram, then a significant difference exists. This is true whether the 95% CI is calculated using the one or two tail models.
• If the 95% CI is on the left (diff<null), intercepts the null, but does not intercept the positive Critical Difference (CD) value, as shown in the 2^nd line in the diagram, then group 1 is significantly not greater than group 2. This is true whether the 95% CI is calculated using the one or two tail models.
• If the 95% CI is on the right (diff>null), intercepts the null, but does not intercept the negative Critical Difference (CD) value, as shown in the 4^th line in the diagram, then group 1 is significantly not less than group 2. This is true whether the 95% CI is calculated using the one or two tail models.
• If the 95% CI is near the center, intercepts the null, but does not intercept either the negative positive Critical Difference (CD) value, as shown in the 3^rd line in the diagram, then the two groups is significantly equivalent. This is only true if the 95% CI is calculated using the two tail models.
• If the 95% CI intercepts both negative and positive Critical Difference values, as shown in the 6^th line in the diagram, then the data lack sufficient power to drawn statistical conclusions, usually because the variance is too large or the sample size not big enough. This is true whether the 95% CI is calculated using the one or two tail models.

One tail or two tail In most text books and published papers, statistics related to equivalence uses the one tail model. This is because most equivalence related research are concerned with non-inferiority, so that not significantly greater or not significantly less are the hypotheses to be tested. As the one tail model allows these conclusions and requires smaller sample size, this is the model to use. StatsToDo however provides calculations for both one and two tails in case any user requires them.

Sample size and power calculation

Sample size and power calculations for equivalence differ from those of significant differences in two ways.

Firstly, the decision is based not only on the relationship between the 95% confidence interval and the null value, but in addition with the positive and negative critical values.

Secondly, robustness is required for the conclusion of significant equivalence and not on significant difference. The Probability of Type I Error (α) is therefore relaxed, and the common values of 0.1 or 0.2 are used instead of 0.05 or 0.01. The Probability of Type II Error, in terms of power, is made more strict, so the power value of 0.9, 0.95, or 0.99 are used instead of 0.8

Statistical Decisions

Users should remember that good statistical practice requires that the hypothesis to be tested is defined at the planning stage, and statistical procedures are used to reject or support that hypothesis. A common malpractice of doing the statistical calculations first, then cherry pick the hypothesis according to how the numbers come together should be avoided. A study to test significant difference or equivalence, the direction of non-inferiority, whether the model should be one or two tail, must be determined at the planning stage before data collection and analysis.

Equivalence Between 2 Means

For equivalence to be validly established, the correct sample size must be estimated and used, and this requires 5 parameters.

• Probability of Type I Error (α). For equivalence, the value of 0.05 most commonly used, to be compatible with the 95% confidence interval. α of 0.1 and 0.2 are occasionally used in association with 90% and 80% confidence intervals of the difference.
• Power (1 - β). For equivalence, the value of 0.95 or 0.99 are often used
• Expected Within Group or expected population Standard Deviation
• Critical Difference, the tolerated limit of difference, a difference which the researcher considers to be within measurement error or small enough to be trivial
• Ratio of sample sizes (φ = n_group2 / n_group1) to be used. In most clinical research the groups are of equal size (φ=1), but availability or clinical priority may require groups of unequal size. Please note the φ is calculated as n_group2 / n_group1).

From these parameters, the program calculates the sample size for each of the two groups, in the one and two tail models. For most non-inferiority studies, the sample size for the one tail model is used.

The two tail model is only required if the purpose of the study is to establish true equivalence, a confidence interval that is both significantly not less than the negative Critical Difference value, and significantly not greater than the positive Critical Difference value.

Analysis of data collected Once the data is collected, analysis requires the following information

• The sample size, mean, and Standard Deviation from the two groups
• The Probability of Type I Error (α) for power estimation
• Critical Difference, the tolerated limit of difference, a difference which the researcher considers to be within measurement error or small enough to be trivial. This is also to be used for power estimation

The program summarizes the data

• The pooled Standard Deviation from the two groups, representing the population or within group Standard Deviation
• The difference between the means Diff = mean_{group 1} - mean_{group 2}
• The Standard Error of the mean

The program then produce two analysis

• The confidence interval of the difference. 80% CI is presented if α=0.2, 90% CI if α=0.1, 95% CI if α=0.1, 99% CI if α=0.01
  • If the hypothesis is that group 1 is not less than group 2, then the first confidence interval, difference - t x SE→+∞ should be used
  • If the hypothesis is that group 1 is not more than group 2, then the second confidence interval, -∞→difference + t x SE should be used
  • If the hypothesis is true equivalence, that group 1 in neither less nor more than group 2, then the two tail confidence interval should be used
• The power for equivalence estimation from the data
  • If the hypothesis is that group 1 is not less than group 2, then the first one tail power calculation, (Power1!<) should be used
  • If the hypothesis is that group 1 is not more than group 2, then the second one tail power calculation, (Power1!>) should be used
  • If the hypothesis is true equivalence, that group 1 in neither less nor more than group 2, then the two tail power calculation (Power2) should be used

Please note that the two calculations produces similar results when the sample size is large, but differs if the sample size is small. In most cases the two calculations lead to the same conclusions, but with data on the margin of statistical significance, confidence interval and power analysis may lead to different conclusions. The reasons are as follow.

The confidence interval is the same as that calculated for comparing the differences between two means, assuming the data to be a sample, and calculations based on the t distribution. This is offered as it is the most common parameter for making statistical decisions, particularly that associated with the 95% confidence interval (α=0.05). Such a decision allows the interpretation for significant difference, significant non inferiority, and equivalence.

The power calculation assumes the data to represent population values, so calculations are based on the z distribution. It estimates the probability of detecting the non-inferiority or equivalence at the level of α, if it is truly present.

Most clinicians would be content to use the 95% confidence interval to decide whether equivalence exists or not. Statisticians however may wish for more nuance to determine the confidence of the decision, and use the power analysis also.

Example

The data for the example is artificially created to demonstrate the statistical method, and not based on any real observations.

We are offered two nutritional supplements for muscle building, supplement 1 is expensive, and supplement 2 is cheap. We would wish to use supplement 2 if it is not inferior (not building less muscle than supplement 1). We use weight gain over a period of time as the outcome measurement.

From experience, we have concluded that the Standard Deviation of weight gain over the trial period with supplements is around 1.5Kg, and decided that difference in weight gain between the two supplement groups is only of interest if it is more than 1Kg. In other words, a difference of less than 1Kg is trivial and has no clinical relevance (CD=1Kg). As we have no difficulty recruiting for either group, we decided to use equal size groups (φ=1)

Sample size

We decided to use Probability of Type I Error α=0.05 and power=0.90. We anticipated standard deviation=1.5, Critical Difference =1.0, ratio of sample sizes φ=1, and the one tail model. The ratio CD/SD = 1.0 / 1.5 = 0.67.

Looking up the table for sample size comparing two means, α=0.05, Power=0.90, CD/SD=0.67, the sample size required is 39 cases per group.

Analysis of results

Grp	n	mean	SD
1	39	4.9Kg	1.4Kg
2	39	5.1Kg	1.6Kg

After randomly allocated trial subject into the two groups for the two treatment (group 1 received supplement 1 and group 2 supplement 2), the weight gain obtained are as in the table to the right.

These results, with α=0.05 and CD=1.0, are entered into analysis, and the results are as the next table to the right.

Data							Summary			Confidence Interval			Power
n1	mean1	SD1	n2	mean2	SD2	α	SDpooled	Diff	SEDiff	%CI	CI 1 tail	CI 2 tail	CD(tl)	Power1!<	Power1!>	Power2
39	4.9	1.6	39	5.1	1.4	0.05	1.50	-0.2	0.34	95%CI	>-0.77 or <0.37	-0.88 to 0.48	±1.0	0.76	0.97	0.66

Looking at the 95% confidence interval, if the conclusion that supplement 2 is non-inferior to supplement 1, the one tail 95% confidence interval of the difference (mean_{group 1} - mean_{group 2}) must be from -∞ up to +1Kg (positive Critical Difference). The results of the calculation show that the 95% CI is from -∞ to +0.37Kg, not intercepting the positive Critical Difference. We can therefore draw the conclusion that group 1 is significantly not greater than group 2, and that supplement 2 is non-inferior to supplement 1.

The data also allow us to draw the conclusion that supplement 1 is non inferior to supplement 2, if we had planned to test that hypothesis, as the one tail 95% confidence interval of the difference would then be -0.77Kg to +∞, not intercepting the negative Critical Difference value.

We cannot use these two one tail confidence intervals to conclude that the two supplements are truely equivalent (similar or same), as this will require the two tail confidence interval, which is very much wider. The two tail confidence interval for the difference is -0.88Kg to + 0.48Kg, which allows us to conclude that both supplements are truly equivalent.

Another approach to examining the data is to ask the question whether the data has sufficient power to detect equivalence at the α p<0.05 level

To test the hypothesis that supplement is non-inferior to supplement 1, we test the power of the difference (mean_{group 1} - mean_{group 2}) not greater than 1Kg., and the power is 0.97. We can therefore conclude with confidence that group 1 is significantly not greater than group 2, and therefore supplement 2 is non-inferior to supplement 1.

Power analysis also allows us to test whether supplement 1 is non-inferior to supplement 2, had we planned to test this. The power for this is 0.76, short of the 0.8 required to draw such a conclusion, so the question of whether supplement 1 is non-inferior to supplement 2 remaqins unanswered with the current data sign.

The power to test true equivalence requires the two tail model, and the two tail power is 0.66, very much short of the 0.8 required to conclude that the two means are equivalent.

References

Rogers JL, Howard KI, Vessey JT. (1993) Using significance tests to evaluate equivalence between two experimental groups. Psychological Bulletin 113:553-565.

Jones B, Jarvis P, Lewis JA, Ebbutt AF. (1996) Trials to assess equivalence: the importance of rigorous methods. British Medical Journal 313:36-39

Hwang IK, Morikawa T. (1999) Design issues in noninferiority/equivalence trials. Drug Information Journal 33:1205-1218

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. 100-104

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 = Expected Within Group Standard Deviation     Col 4 = Critical Difference, the tolerated limit of difference     Col 5 = Ratio of sample sizes (ngroup2 / ngroup1) Data Input for Equivalence Analysis is a table of 8 columns     Each row contains data from a separate study     Col 1, 2, and 3 = n, mean, and SD of group 1     Col 4, 5, and 6 = n, mean, and SD of group 2     Col 7 = Probability of Type I Error α     Col 8 = Critical Difference, the tolerated limit of difference

Sample Size :     
Power Analysis :     

Sample size Table for equivalence between two means
Power=(1-β), β = Probability of Type II Error
α=Probability of Type I Error
CD/SD = Ratio of Critical Difference (tolerance limit) / Standard Deviation
Cells contain sample size per group, assuming equal size groups

	Power=0.9						Power=0.95						Power=0.99
	α=0.2		α=0.1		α=0.05		α=0.2		α=0.1		α=0.05		α=0.2		α=0.1		α=0.05
CD/SD	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
0.01	90157	131389	131389	171278	171278	210147	123652	171278	171278	216446	216446	259894	200720	260338	260338	315409	315409	367447
0.02	22540	32848	32848	42820	42820	52537	30913	42820	42820	54112	54112	64974	50180	65085	65085	78853	78853	91862
0.03	10018	14599	14599	19031	19031	23350	13740	19031	19031	24050	24050	28878	22303	28927	28927	35046	35046	40828
0.04	5635	8212	8212	10705	10705	13135	7729	10705	10705	13528	13528	16244	12545	16272	16272	19714	19714	22966
0.05	3607	5256	5256	6852	6852	8406	4947	6852	6852	8658	8658	10396	8029	10414	10414	12617	12617	14698
0.06	2505	3650	3650	4758	4758	5838	3435	4758	4758	6013	6013	7220	5576	7232	7232	8762	8762	10207
0.07	1840	2682	2682	3496	3496	4289	2524	3496	3496	4418	4418	5304	4097	5314	5314	6437	6437	7499
0.08	1409	2053	2053	2677	2677	3284	1933	2677	2677	3382	3382	4061	3137	4068	4068	4929	4929	5742
0.09	1114	1623	1623	2115	2115	2595	1527	2115	2115	2673	2673	3209	2479	3215	3215	3894	3894	4537
0.10	902	1314	1314	1713	1713	2102	1237	1713	1713	2165	2165	2599	2008	2604	2604	3155	3155	3675
0.11	746	1086	1086	1416	1416	1737	1022	1416	1416	1789	1789	2148	1659	2152	2152	2607	2607	3037
0.12	627	913	913	1190	1190	1460	859	1190	1190	1504	1504	1805	1394	1808	1808	2191	2191	2552
0.13	534	778	778	1014	1014	1244	732	1014	1014	1281	1281	1538	1188	1541	1541	1867	1867	2175
0.14	460	671	671	874	874	1073	631	874	874	1105	1105	1326	1025	1329	1329	1610	1610	1875
0.15	401	584	584	762	762	934	550	762	762	962	962	1156	893	1158	1158	1402	1402	1634
0.16	353	514	514	670	670	821	484	670	670	846	846	1016	785	1017	1017	1233	1233	1436
0.17	312	455	455	593	593	728	428	593	593	749	749	900	695	901	901	1092	1092	1272
0.18	279	406	406	529	529	649	382	529	529	669	669	803	620	804	804	974	974	1135
0.19	250	364	364	475	475	583	343	475	475	600	600	720	557	722	722	874	874	1018
0.20	226	329	329	429	429	526	310	429	429	542	542	650	502	651	651	789	789	919
	Power=0.9						Power=0.95						Power=0.99
	α=0.2		α=0.1		α=0.05		α=0.2		α=0.1		α=0.05		α=0.2		α=0.1		α=0.05
CD/SD	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
0.21	205	298	298	389	389	477	281	389	389	491	491	590	456	591	591	716	716	834
0.22	187	272	272	354	354	435	256	354	354	448	448	537	415	538	538	652	652	760
0.23	171	249	249	324	324	398	234	324	324	410	410	492	380	493	493	597	597	695
0.24	157	229	229	298	298	365	215	298	298	376	376	452	349	452	452	548	548	638
0.25	145	211	211	275	275	337	198	275	275	347	347	416	322	417	417	505	505	588
0.26	134	195	195	254	254	311	183	254	254	321	321	385	297	386	386	467	467	544
0.27	124	181	181	235	235	289	170	235	235	297	297	357	276	358	358	433	433	505
0.28	115	168	168	219	219	269	158	219	219	277	277	332	257	333	333	403	403	469
0.29	108	157	157	204	204	250	148	204	204	258	258	310	239	310	310	376	376	437
0.30	101	146	146	191	191	234	138	191	191	241	241	289	224	290	290	351	351	409
0.31	94	137	137	179	179	219	129	179	179	226	226	271	209	271	271	329	329	383
0.32	89	129	129	168	168	206	121	168	168	212	212	254	197	255	255	309	309	359
0.33	83	121	121	158	158	193	114	158	158	199	199	239	185	240	240	290	290	338
0.34	78	114	114	149	149	182	107	149	149	188	188	225	174	226	226	273	273	318
0.35	74	108	108	140	140	172	101	140	140	177	177	213	164	213	213	258	258	300
0.36	70	102	102	133	133	163	96	133	133	168	168	201	155	201	201	244	244	284
0.37	66	96	96	126	126	154	91	126	126	159	159	190	147	191	191	231	231	269
0.38	63	91	91	119	119	146	86	119	119	150	150	180	140	181	181	219	219	255
0.39	60	87	87	113	113	139	82	113	113	143	143	171	132	172	172	208	208	242
0.40	57	83	83	108	108	132	78	108	108	136	136	163	126	163	163	198	198	230
	Power=0.9						Power=0.95						Power=0.99
	α=0.2		α=0.1		α=0.05		α=0.2		α=0.1		α=0.05		α=0.2		α=0.1		α=0.05
CD/SD	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
0.41	54	79	79	102	102	126	74	102	102	129	129	155	120	155	155	188	188	219
0.42	52	75	75	98	98	120	71	98	98	123	123	148	114	148	148	179	179	209
0.43	49	72	72	93	93	114	67	93	93	118	118	141	109	141	141	171	171	199
0.44	47	68	68	89	89	109	64	89	89	112	112	135	104	135	135	163	163	190
0.45	45	65	65	85	85	104	62	85	85	107	107	129	100	129	129	156	156	182
0.46	43	63	63	81	81	100	59	81	81	103	103	123	95	124	124	150	150	174
0.47	41	60	60	78	78	96	56	78	78	98	98	118	91	118	118	143	143	167
0.48	40	58	58	75	75	92	54	75	75	94	94	113	88	113	113	137	137	160
0.49	38	55	55	72	72	88	52	72	72	91	91	109	84	109	109	132	132	154
0.50	37	53	53	69	69	85	50	69	69	87	87	104	81	105	105	127	127	147
0.51	35	51	51	66	66	81	48	66	66	84	84	100	78	101	101	122	122	142
0.52	34	49	49	64	64	78	46	64	64	81	81	97	75	97	97	117	117	136
0.53	33	47	47	61	61	75	45	61	61	78	78	93	72	93	93	113	113	131
0.54	31	46	46	59	59	73	43	59	59	75	75	90	69	90	90	109	109	127
0.55	30	44	44	57	57	70	41	57	57	72	72	86	67	87	87	105	105	122
0.56	29	42	42	55	55	68	40	55	55	70	70	83	65	84	84	101	101	118
0.57	28	41	41	53	53	65	39	53	53	67	67	80	62	81	81	98	98	114
0.58	27	40	40	51	51	63	37	51	51	65	65	78	60	78	78	94	94	110
0.59	26	38	38	50	50	61	36	50	50	63	63	75	58	75	75	91	91	106
0.60	26	37	37	48	48	59	35	48	48	61	61	73	56	73	73	88	88	103
	Power=0.9						Power=0.95						Power=0.99
	α=0.2		α=0.1		α=0.05		α=0.2		α=0.1		α=0.05		α=0.2		α=0.1		α=0.05
CD/SD	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
0.61	25	36	36	47	47	57	34	47	47	59	59	70	54	70	70	85	85	99
0.62	24	35	35	45	45	55	33	45	45	57	57	68	53	68	68	83	83	96
0.63	23	34	34	44	44	53	32	44	44	55	55	66	51	66	66	80	80	93
0.64	23	33	33	42	42	52	31	42	42	53	53	64	50	64	64	78	78	90
0.65	22	32	32	41	41	50	30	41	41	52	52	62	48	62	62	75	75	87
0.66	21	31	31	40	40	49	29	40	40	50	50	60	47	60	60	73	73	85
0.67	21	30	30	39	39	47	28	39	39	49	49	58	45	58	58	71	71	82
0.68	20	29	29	38	38	46	27	38	38	47	47	57	44	57	57	69	69	80
0.69	19	28	28	36	36	45	26	36	36	46	46	55	43	55	55	67	67	78
0.70	19	27	27	35	35	43	26	35	35	45	45	54	41	54	54	65	65	75
0.71	18	27	27	34	34	42	25	34	34	43	43	52	40	52	52	63	63	73
0.72	18	26	26	34	34	41	24	34	34	42	42	51	39	51	51	61	61	71
0.73	17	25	25	33	33	40	24	33	33	41	41	49	38	49	49	60	60	69
0.74	17	24	24	32	32	39	23	32	32	40	40	48	37	48	48	58	58	68
0.75	17	24	24	31	31	38	22	31	31	39	39	47	36	47	47	57	57	66
0.76	16	23	23	30	30	37	22	30	30	38	38	45	35	46	46	55	55	64
0.77	16	23	23	29	29	36	21	29	29	37	37	44	34	44	44	54	54	62
0.78	15	22	22	29	29	35	21	29	29	36	36	43	33	43	43	52	52	61
0.79	15	22	22	28	28	34	20	28	28	35	35	42	33	42	42	51	51	59
0.80	15	21	21	27	27	33	20	27	27	34	34	41	32	41	41	50	50	58
	Power=0.9						Power=0.95						Power=0.99
	α=0.2		α=0.1		α=0.05		α=0.2		α=0.1		α=0.05		α=0.2		α=0.1		α=0.05
CD/SD	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
0.81	14	21	21	27	27	33	19	27	27	33	33	40	31	40	40	49	49	57
0.82	14	20	20	26	26	32	19	26	26	33	33	39	30	39	39	47	47	55
0.83	14	20	20	25	25	31	18	25	25	32	32	38	30	38	38	46	46	54
0.84	13	19	19	25	25	30	18	25	25	31	31	37	29	37	37	45	45	53
0.85	13	19	19	24	24	30	18	24	24	30	30	36	28	37	37	44	44	51
0.86	13	18	18	24	24	29	17	24	24	30	30	36	28	36	36	43	43	50
0.87	12	18	18	23	23	28	17	23	23	29	29	35	27	35	35	42	42	49
0.88	12	17	17	23	23	28	16	23	23	28	28	34	26	34	34	41	41	48
0.89	12	17	17	22	22	27	16	22	22	28	28	33	26	33	33	40	40	47
0.90	12	17	17	22	22	26	16	22	22	27	27	33	25	33	33	39	39	46
0.91	11	16	16	21	21	26	15	21	21	27	27	32	25	32	32	39	39	45
0.92	11	16	16	21	21	25	15	21	21	26	26	31	24	31	31	38	38	44
0.93	11	16	16	20	20	25	15	20	20	26	26	31	24	31	31	37	37	43
0.94	11	15	15	20	20	24	14	20	20	25	25	30	23	30	30	36	36	42
0.95	10	15	15	19	19	24	14	19	19	24	24	29	23	29	29	35	35	41
0.96	10	15	15	19	19	23	14	19	19	24	24	29	22	29	29	35	35	40
0.97	10	14	14	19	19	23	14	19	19	24	24	28	22	28	28	34	34	40
0.98	10	14	14	18	18	22	13	18	18	23	23	28	21	28	28	33	33	39
0.99	10	14	14	18	18	22	13	18	18	23	23	27	21	27	27	33	33	38