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) twotailed critical value from the standard normal distribution for Type I error probability α.
· is the onetailed 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 pvalue of the following Zstatistic:
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 Zstatistic and its pvalue.
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: 

ZStatistic = 
0.7581 
2Tail Probability = 
0.2242 