The optimum power is 0.8 (Cohen, 1992, page 156). This equals (1 - (beta)), with beta the probability of a Type II error. A value of 0.8 for power is a compromise between too large a beta versus too large a sample size.
Assumptions:
(1) You have 2 samples of equal size N.
(2) Both samples have the same standard deviation.
(3) The means are not equal.
Parameters affecting the power of the analysis:
(1) significance criterion alpha (one tailed vs two tailed; level at 0.01, 0.05 or 0.10)
(2) effect size index
(3) number composing each sample
effect size index d for 2 samples with independent means =
= ((mean for sample A) - (mean for sample B)) / (standard deviation)
where:
• The mean for sample A is greater than the mean for sample B. Alternatively it can be described as the absolute value for the difference without specifying which is larger.
|
|
number in each sample for different alpha values to give a power = 0.8
|
|
d value
|
alpha 1-tail 0.01
|
alpha 1-tail 0.05
|
alpha 1-tail 0.10
|
alpha 2-tail 0.01
|
alpha 2-tail 0.05
|
|
1.4
|
12
|
< 8
|
< 8
|
14
|
9
|
|
1.2
|
15
|
9
|
< 8
|
18
|
12
|
|
1.0
|
21
|
13
|
10
|
25
|
17
|
|
0.8
|
33
|
20
|
14
|
38
|
26
|
|
0.7
|
42
|
26
|
19
|
49
|
33
|
|
0.6
|
57
|
35
|
26
|
68
|
45
|
|
0.5
|
82
|
50
|
36
|
95
|
64
|
|
0.4
|
130
|
78
|
56
|
150
|
100
|
|
0.3
|
225
|
140
|
100
|
260
|
175
|
|
0.2
|
500
|
310
|
225
|
586
|
393
|
|
0.1
|
NA
|
NA
|
900
|
NA
|
NA
|
from Table 2.3.1 to 2.3.6, pages 28 to 39, Cohen (1988)
The power tables are equivalent when the level for alpha 1 tail equals half the level for alpha 2 tail, as shown in the following table:
|
alpha 1-tail
|
alpha 2-tail
|
|
0.005
|
0.01
|
|
0.01
|
0.02
|
|
0.025
|
0.05
|
|
0.05
|
0.10
|
|
0.10
|
0.20
|
The data was analyzed in JMP, with the following equations:
number for alpha 1-tail 0.01 =
= (19.9665 / ((d value)^2)) + 1.995
number for alpha 1-tail 0.05 =
= (12.3959 / ((d value)^2)) + 0.62919
number for alpha 1-tail 0.10 =
= (8.99555 / ((d value)^2)) + 0.336
number for alpha 2-tail 0.01 =
= (23.354 / ((d value)^2)) + 1.9822
number for alpha 2-tail 0.05 =
= (15.66825 / ((d value)^2)) + 1.3117
