# Exact Sample Sizes for Samples in the Ratio 2:1

```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.
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