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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 ClopperPearson 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. 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 0.05 0.1 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 0.85 102 63 102 129 147 54 78 90 105 114 117 126 126 126 126 120 111 102 90 72 63 45 60 66 75 81 87 90 90 90 84 81 72 63 57 54 39 45 54 60 63 63 63 63 63 63 57 54 45 45 33 39 42 48 51 51 51 51 51 48 45 39 39 30 33 36 39 39 42 42 39 39 36 33 33 27 30 30 36 36 36 36 33 30 30 30 24 27 27 27 30 30 30 27 27 27 18 21 24 24 24 24 24 24 24 18 18 21 21 21 21 21 21 15 15 18 18 18 21 21 15 15 15 15 18 18 15 15 15 15 18 12 15 15 15 9 9 15 9 9 9 0 0.05 0.1 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 p1 0.15 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 p1 147 108 147 123 99 81 0.2 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 0 0.05 0.1 88 134 66 90 112 52 68 78 42 50 60 36 40 48 26 34 38 24 28 32 20 24 26 18 22 24 16 18 20 14 18 18 14 16 16 12 12 12 12 12 12 10 10 10 10 10 10 10 10 10 0 0.05 0.1 130 146 92 98 108 110 112 112 110 108 98 92 78 68 52 68 72 74 82 82 82 74 72 68 60 50 42 52 54 60 62 62 60 54 52 48 40 36 44 46 46 46 46 46 46 46 34 26 34 34 36 36 34 34 32 28 48 30 30 30 30 30 26 24 22 24 24 24 24 24 22 20 20 20 20 20 18 16 18 18 18 18 14 18 16 16 14 12 12 12 12 12 10 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 0.15 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 0.85 134 88 p1 0.15 146 130 112 90 66 0.2 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 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. 1. Hulley SB, Cummings SR, Browner WS, Grady D et al. Designing Clinical Research An Epidemiologic Approach. Lippinott Williams & Wilkins, PA. 2nd Edition 2001. 2. Yates F. Contingency Tables involving small numbers and the chi square test. J Royal Statistical Society 1934;2(1) Suppl 1:217-235. 3. Barnard GA. Significance tests for 2x2 tables. J Royal Statistical Society Series B. 1947;34:123-138. 4. Haseman |JK. Exact Sample Sizes for use with the Fisher-Irwin Test for 2x2 Tables. Biometrics 1978:34:106-109. 5. Machin D, Campbell MJ, Fayers PM, Pinol APY. Sample Size Tables for Clinical Studies. Blackwell Science 2nd Edition 1997. 6. Gail M, Gart JJ. The determination of Sample Sizes for use with the Exact Conditional Test in 2x2 Comparative Trials. Biometrics 1973;29:441-448. 7. Lehr R. Sixteen s-squared over d-squared: a relation for crude sample size estimates. Statistics in Medicine 1992;11:1099-1102. 8. Mehta CR. The Exact Analysis of Contingency Tables in Medical Research. Statistical Methods in Medical Research 1994;3:135-156. 9. Clopper CJ, Pearson ES. The use of Confidence or Fiducial Limits illustrated in the case of the Binomial. Biometrika 1934;26(4):404-413. 10. Berry G, Armitage P. Mid-p confidence intervals: a brief review. The Statistician 1995;44(4):417-423. 11. Phipps MC. Liebermeister’s quasi-exact test for two binomials. Proceedings of the 2003 Hawii International Conference on Statistics and Related Fields. Editors: Gregson T, Yang D. University of Hawaii 2003. 12. Mainland D, Sutcliffe MI. Statistical Methods in Medical Research. II. Sample sizes in Experiments involving All-or-None Responses. Canadian J of Medical Science. 1953;31:406-416. 13. Cohen J. Statistical Power Analysis for the Behavioral Sciences. Academic Press N.Y. Revised Edition 1977