Exact Sample Sizes for Groups in the Ratio 2:1

by Rod Bennett West Wales General Hospital Carmarthenshire NHS Trust SA31 2AF

Abstract: Exact Sample Size Tables are given for groups in the ratio 2:1 and 1:1 for alpha = 0.05 and beta = 0.20. The Tables were calculated from Excel Spreadsheets using Fisher's Exact Test and a modified version of Mainland's Method.

Keywords: Ratio of Treatment Groups Fisher’s Exact Test p-values Type I Type II Errors Power Approximations Mesh Size.

For prospective studies there can be difficulty in recruiting sufficient patients to take part within a given time span.

There are instances where randomly distributing the patients to the two treatment groups in the ratio 2:1 may be more time efficient and cost effective than the usual ratio 1:1.

The least conservative a priori values for alpha and beta¹, the type I and type II errors, that are generally acceptable are a value of alpha = 0.05 ( the level set for statistical significance) and Power = 80% ( the value of 1- beta expressed as a percentage ).

One of the simplest arrangements for comparing data collected from two groups is the 2x2 Contingency Table but there is still no agreement amongst Statisticians as to the use of Conditional² or Unconditional³ Tests.

Similarly tables of Samples Sizes may be determined by Exact⁴ or Approximate⁵ Methods and in one case a mixture of both⁶.

The simplicity of the Approximation Formulae of Lehr⁷ contrast with the large number of calculations required for Exact Sample Sizes which increases exponentially⁸.

As with the type of test and the method of calculation several methods have been suggested for the calculation of one-sided probability, amongst which are a Clopper-Pearson type value⁹ which maintains a value of at least alpha in the tail, the mid-p value¹⁰, and an approximation¹¹ based on straddling the tail values.

The cells in the table were calculated as an Excel Spreadsheet and only those less than 150 are given here. The method being adapted from that of Mainland¹².

The tables presented here are for Conditional Exact Sample Sizes using Fisher’s Exact Test to determine Statistical Significance and assuming the proportions in the Contingency Table to be Independent Binomial Proportions to calculate the corresponding Power.

The choice of mesh size 0.05 x 0.05 is simply one of convenience for publication. Both tables are extracts from larger unpublished tables of mesh size 0.01 x 0.01 and Sample Sizes to 1500. [No details have been given of the arc sin approximation¹³.]

To detect a difference of 0.3 between two proportions where p₁ is 0.55 ( (and hence p₂ is 0.85) for the groups in the ratio 2:1 72 patients in groups of 24 and 48 would be required as against two groups of 34.

The total number of patients required is always more for the 2:1 ratio.

Table 1 Treatment Groups in the Ratio 2:1

Difference in Proportions p₂ – p₁ where p₁ is the smaller

p1	0.15	0.2	0.25	0.3	0.35	0.4	0.45	0.5	0.55	0.6	0.65	0.7	0.75	0.8	0.85	0.9	0.95	1	p1
0	102	63	54	45	39	33	30	27	24	18	18	15	15	15	12	9	9	9	0
0.05		102	78	60	45	39	33	30	27	21	18	15	15	15	15	9	9		0.05
0.1		129	90	66	54	42	36	30	27	24	21	18	15	15	15	15			0.1
0.15		147	105	75	60	48	39	36	27	24	21	18	15	15	15				0.15
0.2			114	81	63	51	39	36	30	24	21	18	18	18					0.2
0.25			117	87	63	51	42	36	30	24	21	21	18						0.25
0.3			126	90	63	51	42	36	30	24	21	21							0.3
0.35			126	90	63	51	39	33	27	24	21								0.35
0.4			126	90	63	51	39	30	27	24									0.4
0.45			126	84	63	48	36	30	27										0.45
0.5			120	81	57	45	33	30											0.5
0.55			111	72	54	39	33												0.55
0.6			102	63	45	39													0.6
0.65		147	90	57	45														0.65
0.7		123	72	54															0.7
0.75		99	63																0.75
0.8	147	81																	0.8
0.85	108																		0.85
p1	0.15	0.2	0.25	0.3	0.35	0.4	0.45	0.5	0.55	0.6	0.65	0.7	0.75	0.8	0.85	0.9	0.95	1	p1

Table 2 Treatment Groups in the Ratio 1:1

Difference in Proportions p₂ – p₁ where p₁ is the smaller

p1	0.15	0.2	0.25	0.3	0.35	0.4	0.45	0.5	0.55	0.6	0.65	0.7	0.75	0.8	0.85	0.9	0.95	1	p1
0	88	66	52	42	36	26	24	20	18	16	14	14	12	12	10	8	8	8	0
0.05	134	90	68	50	40	34	28	24	22	18	18	16	12	12	10	8	8		0.05
0.1		112	78	60	48	38	32	26	24	20	18	16	12	12	10	8			0.1
0.15		130	92	68	52	44	34	30	24	20	18	18	12	12	10				0.15
0.2		146	98	72	54	46	34	30	24	20	18	16	12	12					0.2
0.25			108	74	60	46	36	30	24	20	18	16	12						0.25
0.3			110	82	62	46	36	30	24	20	18	14							0.3
0.35			112	82	62	46	34	30	24	18	14								0.35
0.4			112	82	60	46	34	26	22	16									0.4
0.45			110	74	54	46	32	24	20										0.45
0.5			108	72	52	46	28	22											0.5
0.55			98	68	48	34	48												0.55
0.6		146	92	60	40	26													0.6
0.65		130	78	50	36														0.65
0.7		112	68	42															0.7
0.75		90	52																0.75
0.8	134	66																	0.8
0.85	88																		0.85
p1	0.15	0.2	0.25	0.3	0.35	0.4	0.45	0.5	0.55	0.6	0.65	0.7	0.75	0.8	0.85	0.9	0.95	1	p1

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