6.7.5. Two Correlations
6.7.5.1. Sample Size for Two Correlations
The sample size is estimated using the following formula:
where:
·
is the (one or) two-tailed
critical value from the standard normal distribution for Type I error probability α.
·
is the one-tailed
critical value from the standard normal distribution for Type II error probability β.
The user is expected to enter:
· Correlation Coefficient 1
· Correlation Coefficient 2
· Power of the test
· Confidence Level
· 1 or 2 tailed test
and the program will output the estimated sample size.
This procedure assumes that the two sample sizes are equal. However, when there is a constraint on one of the sample sizes, the other can be found as follows:
If such a constraint exists, then enter the given sample size in the N1 Given (optional) field. Otherwise this field should have a zero entry.
Example
Example 19.8 on p. 393 from Zar, J. H. (2010). The difference between two Fisher’s z transforms is given as 0.5. We substitute this input with the correlation coefficients as 0.75 and 0.9, which give a difference of 0.5 between their respective Fisher’s z transforms. Select Statistics 1 → Sample Size and Power Estimation → Two Correlations and the Sample Size option. Enter the following data at the next dialogue:
Sample Size and Power Estimation: Two Correlations
Sample Size
|
Correlation Coefficient 1 = |
0.7500 |
|
Correlation Coefficient 2 = |
0.9000 |
|
Power of the Test = |
0.9000 |
|
Confidence Level = |
0.9500 |
|
1 or 2 Tailed Test = |
2.0000 |
|
N1 Given (optional) = |
0.0000 |
|
Estimated Sample Size = |
87.3389 |
6.7.5.2. Power of the Test for Two Correlations
Power of the test is one minus the p-value of the following Z-statistic:
where:
The user is expected to enter:
· Sample Size 1
· Sample Size 2
· Correlation Coefficient 1
· Correlation Coefficient 2
· Confidence Level
· 1 or 2 tailed test
and the program will output the estimated Z-statistic and its p-value.
Example
Example 19.7 on p. 392 from Zar, J. H. (2010). Select Statistics 1 → Sample Size and Power Estimation → Two Correlations and the Power of the Test option. Enter the following data at the next dialogue:
Sample Size and Power Estimation: Two Correlations
Power of the Test
|
Sample Size 1 = |
95.0000 |
|
Sample Size 2 = |
98.0000 |
|
Correlation Coefficient 1 = |
0.8400 |
|
Correlation Coefficient 2 = |
0.7800 |
|
Confidence Level = |
0.9500 |
|
1 or 2 Tailed Test = |
2.0000 |
|
Power of the Test: |
|
|
Z-Statistic = |
-0.7581 |
|
2-Tail Probability = |
0.2242 |