Cohen developed a number of effect size indices that can be used with tables to determine the power associated with a given sample size at a specified significance level. The effect size index for two independent samples with the same standard deviation is termed d.
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
